# **Integral Transformation, Operational Calculus and Their Applications**

Edited by Hari Mohan Srivastava Printed Edition of the Special Issue Published in *Symmetry*

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## **Integral Transformation, Operational Calculus and Their Applications**

## **Integral Transformation, Operational Calculus and Their Applications**

Editor

**Hari Mohan Srivastava**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editor* Hari Mohan Srivastava University of Victoria Canada

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This is a reprint of articles from the Special Issue published online in the open access journal *Symmetry* (ISSN 2073-8994) (available at: https://www.mdpi.com/journal/symmetry/special issues/Integral Transformation Operational Calculus Their Applications).

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## **Contents**

#### **Muhammad Shafiq, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Maslina Darus and Samiha Kiran**

An Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions Associated with k-Fibonacci Numbers Reprinted from: *Symmetry* **2020**, *12*, 1043, doi:10.3390/sym12061043 ................. **151**

#### **Vijay Gupta, Ana Maria Acu and Hari M. Srivastava**

## **About the Editor**

#### **Hari Mohan Srivastava**

Hari Mohan Srivastava is Professor Emeritus in the Department of Mathematics and Statistics at the University of Victoria in Canada. He currently holds several advisory, honorary, visiting and chair professorships at universities and research institutes around the world. Prof. Hari Mohan Srivastava's current research interests include (for example) real and complex analysis, fractional calculus and its applications, integral equations and integral transformations, higher transcendental functions and their applications, q-series and q-polynomials, analytic and geometric inequalities, probability and statistics, and inventory modeling and optimization. Further biographical and professional details about Professor Hari Mohan Srivastava are available at the following link: https://www.math.uvic.ca/∼harimsri/

## **Preface to "Integral Transformation, Operational Calculus and Their Applications"**

This volume consists of a collection of a total of 12 peer-reviewed and accepted submissions (including several invited feature articles) from all over the world to the Special Issue of the MDPI journal Symmetry on the general subject area of integral transformation, operational calculus and their applications.

For this Special Issue, we cordially invited and welcomed reviews and expository and original research articles dealing with the recent state-of-the-art advances on the topics of integral transformations and operational calculus as well as their multidisciplinary applications with some relevance to the aspect of symmetry. The theory and applications of integral transformations and associated operational calculus are remarkably widespread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. The topics of interest covered in this Special Issue include the following: integral transformations and integral equations as well as other related operators, including their symmetry properties and characteristics; applications involving mathematical (or higher transcendental) functions, including their symmetry properties and characteristics; applications involving fractional-order differential and integral equations and their associated symmetry; applications involving symmetrical aspects of the geometric function theory of complex analysis; applications involving q-series and q-polynomials and their associated symmetry; applications involving special functions of mathematical physics and applied mathematics and their symmetrical aspect; applications involving analytic number theory and symmetry.

Finally, it gives me great pleasure to thank all of the participating authors, referees and reviewers for their invaluable contributions toward the remarkable success of this Special Issue. I do also express my appreciation for the editorial and managerial help and assistance provided efficiently and generously by Mr. Philip Li and other colleagues and associates in the Editorial Office of Symmetry. The dedicated and wholehearted support and help of one and all are indeed greatly appreciated.

> **Hari Mohan Srivastava** *Editor*

## *Article* **Unified Integrals of Generalized Mittag–Leffler Functions and Their Graphical Numerical Investigation**

**Nabiullah Khan 1, Mohammad Iqbal Khan 1, Talha Usman 2, Kamsing Nonlaopon 3,\* and Shrideh Al-Omari 4**

**Abstract:** In this article, we obtain certain finite integrals concerning generalized Mittag–Leffler functions, which are evaluated in terms of the generalized Fox–Wright function. The integrals of concern are unified in nature and thereby yield some new integral formulas as special cases. Moreover, we numerically compute some integrals using the Gaussian quadrature formula and draw a comparison with the main integrals by using graphical numerical investigation.

**Keywords:** Fox–Wright function; generalized hypergeometric function; Mittag–Leffler function

#### **1. Introduction**

In mathematics, functions and symmetric functions are very common in theory and applications. They have been applied to various fields including group theory, Lie algebras, and algebraic geometry, to mention but a few. In applied mathematics, many functions are defined via integrals or series (or infinite products), which are usually referred to as special functions [1–6]. One of them is the Mittag–Leffler function, which was introduced in connection with a method of summation of some divergent series. The Mittag–Leffler function has recently received the interest of scientists due to its wide applications in pure as well as applied mathematics. It is noted that the importance of the Mittag–Leffler function has been envisaged during the last two decades due to its entanglement in physics, chemistry, biology, engineering and applied sciences. The Mittag–Leffler function naturally occurs as a solution of fractional order differential equations or fractional order integral equations. Problems of physics and applied mathematics involve a notable numerical implementation of the Mittag-Leffler function in general and modified forms; therefore, it remains an engaging object of applied research. The implementation of Mittag-Leffler functions is required in a wide variety of problems of physics and mathematics. Because of their crucial requirement, many research works have been dedicated to them, and various representations and generalizations of Mittag-Leffler functions can be found in the literature. Among the most popular special function of fractional calculus is the simplest *p*Ψ*q* function and p = 0, q = 1, called the Wright function or the Bessel–Maitland function or the Wright–Bessel function. From this point of view, the Mittag–Leffler function, expressible in terms of the Fox–Right function, is a special function of fractional calculus. Therefore the Mittag–Leffler function is called the queen function of fractional calculus. The results obtained in the manuscript, connected with a generalized Mittag=-Leffler function that will be used to solve a variety of problems of fractional calculus, for example, Riemann-–Liouville fractional integrals and derivatives, Laplace and Sumudu fractional and integral derivatives and Marichev–Saigo– Maeda fractional integrals and derivatives, etc. Recently, fractional calculus associated with some special functions has proved itself to be a useful tool for applications in many fields of

**Citation:** Khan, N.; Khan, M.I.; Usman, T.; Nonlaopon, K.; Al-Omari, S. Unified Integrals of Generalized Mittag–Leffler Functions and Their Graphical Numerical Investigation. *Symmetry* **2022**, *14*, 869. https://doi.org/10.3390/sym14050869

Academic Editor: Boris Malomed

Received: 15 March 2022 Accepted: 14 April 2022 Published: 23 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

research such as physical systems, biomedicine, nonlinear electronic circuits, chaos-based cryptography, and image encryption. Examples of systems that can be precisely described by fractional-order differential equations (FODEs) involve viscoelastic material models, electrical components, electronic circuits, diffusion waves, the propagation of waves in non-local elastic continua, hydro-logic systems, earthquakes' nonlinear oscillations, models of world economies, fractional viscoelastic models and continuous random walk and equations of muscular blood vessels (see [7–12]). In the past few years, several integral formulas having a variety of special functions have been achieved by many authors (see [13–30]). The present paper provides the study of finite integrals of the generalized Mittag-Leffler function and investigates some useful formulas. We have computed many new results involving integral transforms of the Mittag–Leffler function and plotted three graphs as the major novelty of our work. The results derived in this paper are of general character and likely to find certain applications in the theory of special functions. Additionally, the results provide unification and extension of known results given earlier by various researchers. We compare the results of analytically evaluated integrals with integrals evaluated numerically using the Gaussian quadrature formula. We conclude that the results obtained will provide a significant step in the theory of integral formulas and can yield some potential applications in the field of classical and applied mathematics. Motivated by the aforementioned research and success of the application of integral formulas, we evaluate a new type of integral formulas involving the generalized Mittag–Leffler function (GMLF) expressed in terms of the Fox–Wright function. The Mittag–Leffler function [31,32] is defined as

$$E\_{\sigma}(w) = \sum\_{n=0}^{\infty} \frac{w^n}{\Gamma(\sigma n + 1)}, \quad \sigma \in \mathbb{C}\_{\prime} \text{ } \text{Re}(\sigma) > 0\tag{1}$$

where *ω* ia a complex variable and Γ(.) is the gamma function [25].

In 1905, A. Wiman [33] established a generalization of *Eσ*(*w*), as follows:

$$E\_{\sigma,\mu}(w) = \sum\_{n=0}^{\infty} \frac{w^n}{\Gamma(\sigma n + \mu)}, \quad (\sigma, \mu \in \mathbb{C}, \text{Re}(\mu) > 0, \text{Re}(\sigma) > 0). \tag{2}$$

In 1971, Prabhakar [23] came up with a further generalization of *Eσ*,*μ*(*w*) in the form

$$E\_{\sigma,\mu}^{\gamma}(w) = \sum\_{n=0}^{\infty} \frac{(\gamma)\_n}{\Gamma(\sigma n + \mu)} \frac{w^n}{n!} \quad (\gamma, \mu, \sigma \in \mathbb{C}, \text{Re}(\sigma) > 0, \text{Re}(\gamma) > 0, \text{Re}(\mu) > 0), \tag{3}$$

where (*γ*)*n* is known as the Pochhammer symbol [25]. The underlying generalization of the Mittag–Leffler function is given by Shukla and Prajapati (2007) [29] as

$$E\_{\sigma,\mu}^{\gamma,b}(w) = \sum\_{n=0}^{\infty} \frac{(\gamma)\_{bn}}{\Gamma(\sigma n + \mu)} \frac{w^n}{n!} \tag{4}$$

and expressed by Salim (2009) [26] in the form

$$E\_{\sigma,\mu}^{\gamma,\delta}(w) = \sum\_{n=0}^{\infty} \frac{(\gamma)\_n |w^n}{\Gamma(\sigma n + \mu)(\delta)\_n}.\tag{5}$$

A certain further generalization of the Mittag–Leffler function was given by Salim and Faraj (2012) [27] as

$$E^{\gamma,\delta,b}\_{\sigma,\mu,a}(w) = \sum\_{n=0}^{\infty} \frac{(\gamma)\_{bn}}{\Gamma(\sigma n + \mu)(\delta)\_{an}}.\tag{6}$$

On the other hand, Khan and Ahmad introduced a new generalization of the Mittag– Leffler function (2013) [34] as

$$E^{\gamma,b}\_{\sigma,\mu,\delta}(w) = \sum\_{n=0}^{\infty} \frac{(\gamma)\_{bn}}{\Gamma(\sigma n + \mu)} \frac{w^n}{(\delta)\_n},\tag{7}$$

where *σ*, *μ*, *γ*, *δ* ∈ C; *Re*(*σ*) > 0, *Re*(*μ*) > 0, *Re*(*γ*) > 0, *Re*(*δ*) > 0; *b* ∈ (0, 1) ∪ N. Consequently, they have introduced a generalization of (7) in the following form [34]

$$E^{\mathbb{E},\lambda,\gamma,b}\_{\sigma,\mu,\nu,\oint\_{\mathbb{P}}\delta,\mu}(w) = \sum\_{n=0}^{\infty} \frac{(\mathfrak{f})\_{\lambda n}(\gamma)\_{bn} \, w^n}{\Gamma(\sigma n + \mu)(\nu)\_{\phi n}(\delta)\_{an}},\tag{8}$$

where *σ*, *μ*, *ν*, *φ*, *δ*, *ξ*, *λ*, *γ* ∈ C; min{*Re*(*σ*), *Re*(*μ*), *Re*(*ν*), *Re*(*φ*), *Re*(*δ*), *Re*(*ξ*), *Re*(*λ*), *Re*(*γ*)} > 0; *a*, *b* > 0, *b* ≤ *Re*(*σ*) + *a*.

Above all, (8) is the most generalized definition of all the above formalizations introduced in (1)–(7). Upon substituting *ξ* = *ν*, *λ* = *φ* and *a* = 1 in (8), it becomes (7), which has been established by Khan and Ahmad (2013) [34]. Upon substituting *ξ* = *ν* and *λ* = *φ*, in (8), it becomes a special case (6), which has been established by Salim and Faraj (2012) [27]. Upon substituting *ξ* = *ν*, *λ* = *φ* and *b* = *a* = 1 in (8), it becomes (5), which has been discussed by Salim (2009) [26]. Upon substituting *ξ* = *ν*, *λ* = *φ* and *δ* = *a* = 1 in equation (8), it is a special case (4); see Shukla and Prajapati (2007) [29]. If *b* = 1, it becomes a special case (3) of Prabhakar (1971) [23]. On substituting *ξ* = *ν*, *λ* = *φ* and *γ* = *δ* = *a* = *b* = 1 in (8), it becomes a special case (2) established by A. Wiman (1905) [33]. Furthermore, if *μ* = 1, we get the Mittag–Leffler function *Eσ*(*w*) defined in (1). Finally, on setting *δ* = *a* = *b* = 1 in (8), we establish a new generalization of the Mittag–Leffler function in the form

$$E^{\vec{\xi}, \lambda, \gamma}\_{\sigma, \mu, \nu, \phi}(w) = \sum\_{n=0}^{\infty} \frac{(\vec{\xi})\_{\lambda n} (\gamma)\_n}{\Gamma(\sigma n + \mu)(\nu)\_{\Phi^n}} \frac{w^n}{n!} \, , \tag{9}$$

where *σ*, *μ*, *ν*, *φ*, *ξ*, *λ*, *γ* ∈ C; *Re*(*μ*) > 0, *Re*(*σ*) > 0, *Re*(*ν*) > 0, *Re*(*φ*) > 0, *Re*(*ξ*) > 0, *Re*(*λ*)> 0 *and Re*(*γ*) > 0.

The Fox–Wright function *r*Ψ*s*[*w*] (see [35–42]), is defined by

$$\,\_r\Psi\_s[w] = \,\_r\Psi\_s \left[ \begin{array}{c} (\lambda\_1, \dot{\lambda}\_1), \dots, (\lambda\_r, \dot{\lambda}\_r); \\\\ (l\_1, \dot{l}\_1), \dots, (l\_s, \dot{l}\_s); \end{array} w \right] \tag{10}$$

$$\hat{\lambda} = \sum\_{k=0}^{\infty} \frac{\Gamma(\lambda\_1 + \hat{\lambda}\_1 k), \dots, \Gamma(\lambda\_r + \hat{\lambda}\_r k)}{\Gamma(l\_1 + \hat{l}\_1 k), \dots, \Gamma(l\_s + \hat{l}\_s k)} \frac{w^k}{k!} \tag{11}$$

$$\mathbf{u} = H\_{r,s+1}^{1,r} \begin{bmatrix} (1 - \lambda\_1 \boldsymbol{\lambda\_1'}) , \dots , (1 - \lambda\_{r'} \boldsymbol{\lambda\_{r}'}) \\\\ (0, 1) , (1 - l\_1 \boldsymbol{l\_1'}) , \dots , (1 - l\_{s'} \boldsymbol{l\_s'}) \end{bmatrix} \tag{12}$$

where *H*1,*r r*,*s*+1[*w*] represents the Fox-*H* function [38]. When *λ*´ 1, ... , *λ*´ *r* = 1, ´ *l*1, ... , ´ *ls* = 1 in (10), the Fox–Wright function reduces to the generalized hypergeometric function *rFs*[w] (see [41])

$${}\_{r}\Psi\_{s}\left[\begin{array}{c}(\lambda\_{1},1),\ldots,(\lambda\_{r},1);\\(l\_{1},1),\ldots,(l\_{s},1);\end{array}w\right] = \frac{\Gamma(\lambda)\_{1},\ldots,\Gamma(\lambda)\_{r}}{\Gamma(l)\_{1},\ldots,\Gamma(l)\_{s}}\ {}\_{r}F\_{s}(\lambda\_{1},\ldots,\lambda\_{r};l\_{1},\ldots,l\_{s};w).\tag{13}$$

Here, we recall the result due to Prudnikov et al. [24] (see also [39], p. 250 (2.8)), by means of which we have established our main result in the present article

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} \, dx$$

$$=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} \, (q-p)^{-1}(1+r\_{1})^{-\alpha}(1+r\_{2})^{-\beta} \, . \tag{14}$$

provided that *Re*(*α*) > 0, *Re*(*β*) > 0, *q* = *p* and the constants *r*1 and *r*2 are such that none of the expression 1 + *r*1, 1 + *r*2, [(*q* − *p*) + *r*1(*x* − *p*) + *r*2(*q* − *x*)], where *p* ≤ *x* ≤ *q* is zero.

#### **2. Main Results**

**Theorem 1.** *Let α and β exist such that Re*(*α*) > 0, *Re*(*β*) > 0, *q* = *p and the constants r*1 *and r*2 *are such that none of the expressions* 1 + *r*1*,* 1 + *r*2*,* [(*q* − *p*) + *r*1(*x* − *p*) + *r*2(*q* − *x*)]*, where p* ≤ *x* ≤ *q is zero. Let σ*, *μ*, *ν*, *φ*, *δ*, *ξ*, *λ*, *γ* ∈ C*; if min*{*Re*(*σ*), *Re*(*μ*), *Re*(*ν*), *Re*(*φ*), *Re*(*δ*), *Re*(*ξ*), *Re*(*λ*), *Re*(*γ*)} > 0*; a*, *b* > 0*, b* ≤ *Re*(*σ*) + *a, then the following identity holds:*

$$\int\_{p}^{q} \frac{(x-p)^{\alpha-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{\alpha+\beta}} E\_{\sigma,\mu,\nu,\delta,\delta}^{\xi,\lambda,\gamma,b} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] dx$$

$$= \frac{1}{(q-p)\ (1+r\_{1})^{\alpha}\ (1+r\_{2})^{\beta}} \frac{\Gamma(\nu)\Gamma(\delta)}{\Gamma(\xi)\Gamma(\gamma)}$$

$$\times \,\_5\Psi\_4 \left[ \begin{array}{c} (\xi,\lambda), \ (\gamma,b), \ (a,m), \ (\beta,m), \ (1,1); \\\\ (\mu,\sigma), \ (\nu,\phi), \ (\delta,a), \ (a+\beta,2m); \end{array} \frac{w}{(1+r\_{1})^{\alpha}(1+r\_{2})^{\overline{w}}} \right],\tag{15}$$

$$\text{where } E\_{\sigma,\mu,\nu,\delta,\rho,\delta,\delta}^{\xi,\lambda,\gamma,b}(w) \text{ is a GMLF given by (8).}$$

**Proof.** Denoting the left hand side of (15) by I, writing *Eξ*,*λ*,*γ*,*b σ*,*μ*,*ν*,*φ*,*δ*,*a*(*w*) in its summation formula in the integrand with the help of (8), we obtain

$$I = \int\_{p}^{q} \frac{(\mathbf{x} - \mathbf{p})^{a-1} (q - \mathbf{x})^{\beta - 1}}{[(q - p) + r\_1(\mathbf{x} - \mathbf{p}) + r\_2(q - \mathbf{x})]^{a + \beta}}$$

$$\times \sum\_{n=0}^{\infty} \frac{(\frac{\mathbf{x}}{\delta})\_{\mathrm{an}} (\gamma)\_{\mathrm{bn}} \left(1\right)\_{n} n^{\mathrm{yn}} (\mathbf{x} - \mathbf{p})^{\mathrm{mn}} (q - \mathbf{x})^{\mathrm{mn}}}{\Gamma(\sigma n + \mu) \left(\nu \right)\_{\mathrm{\rho n}} (\delta)\_{\mathrm{an}} \left[(q - p) + r\_1(\mathbf{x} - \mathbf{p}) + r\_2(q - \mathbf{x})\right]^{2 \mathrm{nm}}} \, d\mathbf{x}, \tag{16}$$

which, by further simplification, yields

$$I = \sum\_{n=0}^{\infty} \frac{(\frac{\pi}{\delta})\_{\lambda n} (\gamma)\_{bn}}{\Gamma(\sigma n + \mu)} \frac{w^n}{(\nu)\_{\theta n}} \frac{\int\_p^q}{n!} \frac{(\mathbf{x} - p)^{mn + a - 1} (q - \mathbf{x})^{mn + \beta - 1}}{[(q - p) + r\_1(\mathbf{x} - p) + r\_2(q - \mathbf{x})]^{2mn + a + \beta}} \, d\mathbf{x}. \tag{17}$$

We apply the result of (14), and, through simplifying, this yields

$$I = \frac{1}{\left(q - p\right)\left(1 + r\_1\right)^n \left(1 + r\_2\right)^\beta} \frac{\Gamma(\nu)\,\Gamma(\delta)}{\Gamma(\frac{\nu}{\delta})\,\Gamma(\gamma)}$$

$$\times \sum\_{n=0}^\infty \frac{\Gamma(\frac{\nu}{\delta} + \lambda n)\,\Gamma(\gamma + bn)\,\Gamma(a + mn)\,\Gamma(\beta + mn)\,\Gamma(1 + n)\left(\frac{w}{(1 + r\_1)^n \left(1 + r\_2\right)^m}\right)^n}{\Gamma(\mu + \sigma n)\,\Gamma(\nu + \lambda n)\,\Gamma(\delta + an)\,\Gamma(a + \beta + 2mn)\,\ n!} . \tag{18}$$

Finally, after summing up, with the help of (11), we arrive at (15). This completes the proof of Theorem 1.

**Corollary 1.** *For b* = *δ* = *a* = 1 *and all the conditions already stated in (15), the following identity holds:*

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} E\_{\nu,\mu,\nu,\theta}^{L,\lambda,\gamma} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] dx$$

$$= \frac{1}{(q-p)} \frac{\Gamma(\nu)}{(1+r\_{1})^{a} \ (1+r\_{2})^{\beta} \ \Gamma(\xi) \Gamma(\gamma)}$$

$$\times \,\_4\Psi\_3 \left[ \begin{array}{c} (\not\kern-5. \color[rgb]{1.0pt}{ $\psi$ },\lambda), \ (\gamma,1), \ (a,m), \ & (\beta,m);\\ (\mu,\sigma), \ (\nu,\phi), & (a+\beta,2m); \end{array} \right] . \tag{19}$$

**Theorem 2.** *Let α and β be such that Re*(*α*) > 0, *Re*(*β*) > 0, *q* = *p and the constants r*1 *and r*2 *are such that none of the expressions* 1 + *r*1*,* 1 + *r*2*,* [(*q* − *p*) + *r*1(*x* − *p*) + *r*2(*q* − *x*)]*, where p* ≤ *x* ≤ *q is zero. Let σ*, *μ*, *ν*, *φ*, *δ*, *ξ*, *λ*, *γ* ∈ C*; if min*{*Re*(*σ*), *Re*(*μ*), *Re*(*ν*), *Re*(*φ*), *Re*(*δ*), *Re*(*ξ*), *Re*(*λ*), *Re*(*γ*)} > 0*; a*, *b* > 0*, b* ≤ *Re*(*σ*) + *a, the following identity holds:*

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} E\_{\sigma,\mu,\nu,\rho,\delta,a}^{\mathbb{E},\lambda,\gamma\delta} \left[ \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} w \right] dx$$

$$=\frac{1}{(q-p)} \frac{\Gamma(a)\Gamma(\beta)}{(1+r\_{1})^{a}\left(1+r\_{1}\right)^{\beta}} \frac{\Gamma(a)\Gamma(\beta)}{\Gamma(a+\beta)\Gamma(\mu)} {}\_{2m+\lambda+b+1} \Gamma\_{2m+\sigma+\rho+a} \left[ \begin{matrix} \Delta(m;a), & \Delta(m;\beta), & \Delta(n;\xi), \\ & \Delta(\sigma;\mu), & \Delta(\phi;\nu), & \Delta(a;\delta), \end{matrix} \right]$$

$$\begin{matrix} \Delta(b;\gamma), & 1; & w \lambda^{1}b^{b} \\ \Delta(2m;a+\beta); & \sigma^{\sigma}\sigma^{\theta}\sigma^{a}\mathfrak{a}^{\mu}(1+r\_{1})^{w}(1+r\_{2})^{w} \end{matrix} \right],\tag{20}$$

*where* Δ(*m*; *λ*) *abbreviates the arrangement of m parameters λ m λ*+1 *m ... λ*+*m*−1 *m and m* ≥ 1*.*

**Proof.** By using the formulas

$$
\Gamma(\lambda + n) = \Gamma(\lambda)(\lambda)\_n \tag{21}
$$

and

$$(\lambda)\_{mn} = m^{mn} \left(\frac{\lambda}{m}\right)\_n \left(\frac{\lambda+1}{m}\right)\_n \dots \left(\frac{\lambda+m-1}{m}\right)\_{n'} \tag{22}$$

and after a little simplification, the required result (20) can be obtained. Therefore, we omit the proof.

**Corollary 2.** *On putting a* = *b* = *δ* = 1 *under the condition already set out in (20), the following identity holds:*

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} E\_{\sigma,\mu,\nu,\theta}^{\mathcal{E},\lambda,\gamma} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] dx$$

$$= \frac{1}{(q-p)} \frac{\Gamma(a)\Gamma(\beta)}{(1+r\_{1})^{a}\left(1+r\_{2}\right)^{\beta}} \frac{\Gamma(a)\Gamma(\beta)}{\Gamma(a+\beta)\Gamma(\mu)}$$

$$\times\_{2m+\lambda+1} \Gamma\_{2m+\sigma+\phi} \left[ \begin{array}{c} \Delta(m;a), \quad \Delta(m;\beta), \quad \Delta(\lambda;\xi), \\ \Delta(\sigma;\mu), \quad \Delta(q;\upsilon), \end{array} \right. \qquad \qquad \qquad \qquad \qquad \frac{w\ \lambda^{\lambda}}{\sigma^{\sigma}\,\theta^{\theta}\,4^{m}(1+r\_{1})^{m}(1+r\_{2})^{m}} \right]. \tag{23}$$

#### **3. Special Cases**

Here, we compute certain integral formulas as special cases of our key results.

(i) On setting *ξ* = *ν*, *λ* = *φ* in (15), the following identity holds:

$$\begin{aligned} \,\_3F\_p \left[ \begin{array}{cc} \frac{(\mathbf{x} - \mathbf{p})^{\alpha - 1} (q - \mathbf{x})^{\beta - 1} \\ \hline [(q - p) + r\_1 (\mathbf{x} - \mathbf{p}) + r\_2 (q - \mathbf{x})]^{\alpha + \beta} \end{array} \,\_4E\_{\mathbf{w}, \mathbf{p}, \mathbf{c}}^{\gamma, b, b} \left[ w \left\{ \frac{(\mathbf{x} - \mathbf{p}) (q - \mathbf{x})}{[(q - p) + r\_1 (\mathbf{x} - \mathbf{p}) + r\_2 (q - \mathbf{x})]^2} \right\}^m \right] \,\_4E\_{\mathbf{w}, \mathbf{c}} \\ &= \frac{1}{(q - p)} \frac{\Gamma(\delta)}{(1 + r\_1)^{\alpha} (1 + r\_2)^{\beta}} \frac{\Gamma(\delta)}{\Gamma(\gamma)} \\ &\times \,\_4\Psi\_3 \left[ \begin{array}{cc} (\gamma, b), & (a, m), & (\beta, m) & (1, 1); & w \\ & (\mu, \sigma), & (\delta, a), & (a + \beta, \, 2m); & \frac{w}{(1 + r\_1)^{m} (1 + r\_2)^{m}} \end{array} \right]; \end{aligned} \tag{24}$$

(ii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = 1 in (15), the following identity holds:

$$\int\_{p}^{q} \frac{(\mathbf{x} - \mathbf{p})^{\mathbf{a} - 1} (q - \mathbf{x})^{\mathbf{f} - 1}}{[(q - p) + r\_1(\mathbf{x} - \mathbf{p}) + r\_2(q - \mathbf{x})]^{\mathbf{a} + \mathbf{f}}} E\_{\sigma, \mu, \delta}^{\gamma, \delta} \left[ w \left\{ \frac{(\mathbf{x} - \mathbf{p})(q - \mathbf{x})}{[(q - p) + r\_1(\mathbf{x} - \mathbf{p}) + r\_2(q - \mathbf{x}) + ]^2} \right\}^{m} \right] d\mathbf{x} = 0$$

$$=\frac{1}{\left(q-p\right)\left(1+r\_1\right)^{u}\left(1+r\_2\right)^{\beta}}\frac{\Gamma(\delta)}{\Gamma(\gamma)}$$

$$\times \,\_4\Psi\_3\left[ \begin{array}{c} (\gamma,b), \ (a,m), \ (\beta,m) \end{array} \begin{array}{c} (\beta,m) \\ (\beta,m) \end{array} \begin{array}{c} (1,1); \ \end{array} \begin{array}{c} w \\ \hline (1+r\_1)^{m}(1+r\_2)^{m} \end{array} \right];\tag{25}$$

(iii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = 1 in (15), the following identity holds:

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} \,\_{4}^{\gamma\beta} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] \,\_{4}^{m} \,\_{2}$$

$$= \frac{1}{(q-p)} \frac{\Gamma(\delta)}{(1+r\_{1})^{a} \left(1+r\_{2}\right)^{\beta}} \frac{\Gamma(\delta)}{\Gamma(\gamma)}$$

$$\times \,\_4\Psi\_3 \left[ \begin{array}{c} (\gamma,1), \ (a,m), \ (\beta,m) & (1,1); & w\\ & (\mu,\sigma), \ (\delta,1), & (a+\beta,\ 2m); \end{array} \frac{w}{(1+r\_{1})^{w}(1+r\_{2})^{w}} \right];\tag{26}$$

(iv) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *δ* = 1 in (15), the following identity holds:

$$\int\_{\mathcal{V}} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_1(x-p)+r\_2(q-x)]^{a+\beta}} E\_{\sigma,\mu}^{\gamma,\delta} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_1(x-p)+r\_2(q-x)]^{\beta}} \right\}^m \right] dx$$

$$= \frac{1}{(q-p)} \frac{1}{(1+r\_1)^a \left(1+r\_2\right)^{\beta}} \frac{1}{\Gamma(\gamma)}$$

$$\times \,\_3\Psi\_2 \left[ \begin{array}{c} (\gamma,b), \ (a,m), & (\beta,m); \\\\ (\mu,\sigma), & (a+\beta, \ 2m); \end{array} \frac{w}{(1+r\_1)^m (1+r\_2)^m} \right];\tag{27}$$

(v) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *δ* = 1 in (15), the following identity holds:

$$\int\_{\mathcal{V}} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_1(x-p)+r\_2(q-x)]^{a+\beta}} E\_{\sigma,\mu}^{\gamma} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_1(x-p)+r\_2(q-x)]^2} \right\}^m \right] dx$$

$$= \frac{1}{(q-p)} \frac{1}{(1+r\_1)^a \left(1+r\_2\right)^\beta} \frac{1}{\Gamma(\gamma)}$$

$$\times \,\_3\Psi\_2 \left[ \begin{array}{c} (\gamma,1), \ (a,m), \\\\ (\mu,\sigma), \qquad (a+\beta,\ 2m); \end{array} \frac{w}{(1+r\_1)^m (1+r\_2)^m} \right];\tag{28}$$

(vi) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = 1 in (15), the following identity holds:

$$\int\_{\mathcal{P}} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_1(x-p)+r\_2(q-x)]^{a+\beta}} \,\_2E\_{\mathcal{P},\mathcal{U}} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_1(x-p)+r\_2(q-x)]^2} \right\}^m \right] \,d\mathbf{x}$$

$$=\frac{1}{(q-p)\left(1+r\_1\right)^a\left(1+r\_2\right)^b}$$

$$\times \,\_3\Psi\_2 \left[ \begin{array}{c} (x,m), & (\beta,m), & (1,1); \\ & (\mu,\sigma), & (a+\beta, \ 2m); \end{array} \frac{w}{(1+r\_1)^m(1+r\_2)^m} \right];\tag{29}$$

(vii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = *μ* = 1 in (15), the following identity holds:

$$\int\_{p}^{q} \frac{(\mathbf{x} - p)^{a-1} (q - \mathbf{x})^{\beta - 1}}{[(q - p) + r\_1(\mathbf{x} - p) + r\_2(q - \mathbf{x})]^{a + \beta}} \,\_2E\_{\mathbf{f}} \left[ w \left\{ \frac{(\mathbf{x} - p)(q - \mathbf{x})}{[(q - p) + r\_1(\mathbf{x} - p) + r\_2(q - \mathbf{x})]^2} \right\}^m \right] d\mathbf{x}$$

$$= \frac{1}{(q - p) \left(1 + r\_1\right)^a \left(1 + r\_2\right)^b}$$

$$\times \, \_3\Psi\_2 \left[ \begin{array}{ll} (\mathfrak{a}, m), & (\mathfrak{f}, m), & (1, 1); \\\\ & (1, \sigma), & (\mathfrak{a} + \mathfrak{f}, 2m); \end{array} \frac{w}{(1 + r\_1)^m (1 + r\_2)^m} \right]; \tag{30}$$

(viii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = *μ* = *σ* = 1 in (15), the following identity holds:

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} e^{\left[w\left\{\frac{(x-p)(q-z)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-z)]^{2}}\right\}^{m}\right]} dx$$

$$=\frac{1}{(q-p)\left(1+r\_{1}\right)^{a}(1+r\_{2})^{\beta}}$$

$$\times \,\_{2}\Psi\_{1}\left[\begin{array}{c} (a,m), \ (\beta,m), \\\\ (a+\beta,2m); \end{array}\frac{w}{(1+r\_{1})^{m}(1+r\_{2})^{m}}\right];\tag{31}$$

(ix) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = *μ* = 1, *σ* = 0 in (15), the following identity holds:

$$\int\_{p}^{q} \frac{(x-p)^{\alpha-1}(q-x)^{\beta-1}}{\left[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)\right]^{\alpha+\beta}} \frac{1}{\left[1-w\left\{\frac{(x-p)(q-x)}{\left[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)\right]^{2}}\right\}^{m}\right]} dx$$

$$=\frac{1}{(q-p)\left(1+r\_{1}\right)^{\alpha}(1+r\_{2})^{\beta}}$$

$$\times \,\_3\Psi\_2\left[ \begin{array}{c} (a,m), \ (\beta,m), \ (1,1); \ & \begin{array}{c} w \\ (1+r\_{1})^{m}(1+r\_{2})^{m} \end{array} \end{array} \right];\tag{32}$$

(x) Setting *ξ* = *ν*, *λ* = *φ* in (20), the following identity holds:

$$\int\_{p}^{q} \frac{(\mathbf{x} - p)^{a-1} (q - \mathbf{x})^{\beta - 1}}{[(q - p) + r\_1 (\mathbf{x} - p) + r\_2 (q - \mathbf{x})]^{a + \beta}} E\_{\sigma, p, a}^{\gamma, \delta, \beta} \left[ w \left\{ \frac{(\mathbf{x} - p)(q - \mathbf{x})}{[(q - p) + r\_1 (\mathbf{x} - p) + r\_2 (q - \mathbf{x})]^2} \right\}^m \right] d\mathbf{x}$$

$$= \frac{1}{(q - p)} \frac{\Gamma(a) \Gamma(\beta)}{(1 + r\_1)^a \left(1 + r\_2\right)^{\beta}} \frac{\Gamma(a) \Gamma(\beta)}{\Gamma(a + \beta) \Gamma(\mu)}$$

$$\times \quad\_{2m + b + 1} F\_{2m + \sigma + a} \begin{bmatrix} \Delta(m; a), & \Delta(m; \beta), \Delta(b; \gamma), & & & & w \, b^b\\ \Delta(\sigma; \mu), & & & & \overline{u} \, \mathbf{e}^{\sigma} a^{\mu} (1 + r\_1)^m (1 + r\_2)^n} \\ \Delta(\sigma; \mu), & & & & \overline{u}^{\text{in } \sigma} a^{\sigma} (1 + r\_1)^m (1 + r\_2)^n \end{bmatrix}; \tag{33}$$

$$\text{(xi)} \quad \text{Setting } \tilde{\xi} = \nu, \lambda = \phi \text{ and } a = 1 \text{ in (2)), the following identity holds:}$$

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} E\_{\sigma,\mu,\delta}^{\gamma,b} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] dx$$

$$= \frac{1}{(q-p)} \frac{\Gamma(a)\Gamma(\beta)}{(1+r\_{1})^{a}\left(1+r\_{2}\right)^{\beta}} \frac{\Gamma(a)\Gamma(\beta)}{\Gamma(a+\beta)\Gamma(\mu)}$$

$$\times \quad\_{2m+b+1} E\_{2m+\sigma+1} \left[ \begin{matrix} \Delta(m;a), & \Delta(m;\beta), \ \Delta(b;\gamma), & 1; & w \, b^{b} \\ & \Delta(r;\mu), & \Delta(2m;a+\beta), & \delta; & \frac{4^{m}\sigma^{\sigma}(1+r\_{1})^{m}(1+r\_{2})^{m}}{4^{m}\sigma^{\sigma}(1+r\_{1})^{m}(1+r\_{2})^{m}} \end{matrix} \right];\tag{34}$$

(xii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = 1 in (20), the following identity holds:

$$\int\_{p}^{q} \frac{(\mathbf{x} - p)^{a-1} (q - \mathbf{x})^{\beta - 1}}{[(q - p) + r\_1 (\mathbf{x} - p) + r\_2 (q - \mathbf{x})]^{a + \beta}} E\_{\sigma, \mu}^{\gamma, \delta} \left[ w \left\{ \frac{(\mathbf{x} - p)(q - \mathbf{x})}{[(q - p) + r\_1 (\mathbf{x} - p) + r\_2 (q - \mathbf{x})]^2} \right\}^m \right] d\mathbf{x}$$

$$= \frac{1}{(q - p) \left(1 + r\_1\right)^a (1 + r\_2)^{\beta}} \frac{\Gamma(a) \Gamma(\beta)}{\Gamma(a + \beta) \Gamma(\mu)}$$

× 2*m*+2*F*2*m*+*σ*+1 ⎡ ⎣ Δ(*m*; *α*), Δ(*m*; *β*), *γ*, Δ(*σ*; *μ*), Δ(2*m*; *α* + *β*), 1; *δ*; *w* 4*mσσ*(1 + *r*1)*m*(1 + *r*2)*m* ⎤ ⎦; (35)

(xiii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *δ* = 1 in (20), the following identity holds:

$$\int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} E\_{\sigma,p}^{\gamma,\beta} \left[ w \left\{ \frac{(x-p)(q-x)}{[+(q-p)r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] dx$$

$$= \frac{1}{(q-p)} \frac{\Gamma(a)\Gamma(\beta)}{(1+r\_{1})^{a}(1+r\_{2})^{\beta}} \frac{\Gamma(a)\Gamma(\beta)}{\Gamma(a+\beta)\Gamma(\mu)}$$

$$\times \,\_{2m+b} \mathrm{F}\_{2m+\sigma} \left[ \begin{matrix} \Delta(m;a), & \Delta(m;\beta), \ \Delta(b;\gamma); \\\\ \Delta(\sigma;\mu), & \Delta(2m;a+\beta); \end{matrix} \right. \qquad \frac{w \, b^{b}}{4^{m}\sigma^{\sigma}(1+r\_{1})^{m}(1+r\_{2})^{m}} \right];\tag{36}$$

(xiv) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *δ* = 1 in (20), the following identity holds:

$$\begin{split} \int\_{p}^{q} \frac{(\mathbf{x} - p)^{a-1} (q - \mathbf{x})^{\beta - 1}}{[(q - p) + r\_{1}(\mathbf{x} - p) + r\_{2}(q - \mathbf{x})]^{a + \beta}} E\_{\sigma, \mu}^{\gamma} \left[ w \left\{ \frac{(\mathbf{x} - p)(q - \mathbf{x})}{[(q - p) + r\_{1}(\mathbf{x} - p) + r\_{2}(q - \mathbf{x})]^{2}} \right\}^{m} \right] d\mathbf{x} \\ &= \frac{1}{(q - p)} \frac{\Gamma(a) \Gamma(\beta)}{(1 + r\_{1})^{a} \left(1 + r\_{2}\right)^{\beta}} \frac{\Gamma(a) \Gamma(\beta)}{\Gamma(a + \beta) \Gamma(\mu)} \\ &\times \,\_ \begin{bmatrix} \Delta(m; a), & \Delta(m; \beta), & \gamma; & & w \\ \Delta(\sigma; \mu), & \Delta(2m; a + \beta); & & \overline{4^{n} \sigma^{\prime} (1 + r\_{1})^{m} (1 + r\_{2})^{m}} \end{bmatrix}; \end{split} \tag{37}$$

(xv) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = 1 in (20), the following identity holds:

$$\begin{split} \int\_{p}^{q} \frac{(x-p)^{a-1}(q-x)^{\beta-1}}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{a+\beta}} E\_{\sigma,\mu} \left[ w \left\{ \frac{(x-p)(q-x)}{[(q-p)+r\_{1}(x-p)+r\_{2}(q-x)]^{2}} \right\}^{m} \right] dx \\ &= \frac{1}{(q-p)} \frac{\Gamma(a)\Gamma(\beta)}{(1+r\_{1})^{a}\left(1+r\_{2}\right)^{\beta}} \frac{\Gamma(a)\Gamma(\beta)}{\Gamma(a+\beta)\Gamma(\mu)} \\ &\times \,\_{2m+1}F\_{2m+r} \begin{bmatrix} \Delta(m;a), & \Delta(m;\beta), \,1; \\\\ \Delta(\sigma;\mu), & \Delta(2m;a+\beta); \end{bmatrix} ; \end{split} \tag{38}$$

(xvi) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = *μ* = 1 in (20), the following identity holds:

$$\int\_{\mathbb{T}}^{q} \frac{(\mathbf{x} - p)^{\mathbf{x} - 1} (q - \mathbf{x})^{\beta - 1}}{[(q - p) + r\_1(\mathbf{x} - p) + r\_2(q - \mathbf{x})]^{\mathbf{a} + \beta}} \,\_2E\_{\mathbf{r}} \left[ w \left\{ \frac{(\mathbf{x} - p)(q - \mathbf{x})}{[(q - p) + r\_1(\mathbf{x} - p) + r\_2(q - \mathbf{x})]^2} \right\}^m \right] d\mathbf{x}$$

$$= \frac{\Gamma(\mathbf{a})\Gamma(\beta)}{\Gamma(\mathbf{a} + \beta)} \,\_ { (q - p)} \frac{1}{(1 + r\_1)^{\mathbf{a}} \, (1 + r\_2)^{\beta}}$$

$$\times \,\_ { 2m + 1} \Gamma\_{2m + \sigma} \begin{bmatrix} \Delta(m; \mathbf{a}), & \Delta(m; \beta), \, 1; & & w \\ & \Delta(\sigma; 1), & \Delta(2m; \mathbf{a} + \beta); & & \frac{4}{4^m \sigma^{\sigma} (1 + r\_1)^m (1 + r\_2)^m} \end{bmatrix}; \tag{39}$$

(xvii) Setting *ξ* = *ν*, *λ* = *φ* and *a* = *b* = *γ* = *δ* = *μ* = *σ* = 1 in (20), the following identity holds:

$$\int\_{p}^{q} \frac{(\mathbf{x} - \mathbf{p})^{\mathbf{a} - 1} (q - \mathbf{x})^{\mathbf{f} - 1}}{\left[ (q - p) + r\_1 (\mathbf{x} - \mathbf{p}) + r\_2 (q - \mathbf{x}) \right]^{\mathbf{a} + \boldsymbol{\beta}}} e^{\left[ w \left\{ \frac{(\mathbf{x} - \mathbf{p})(q - \mathbf{x})}{\left[ (q - p) + r\_1 (\mathbf{x} - \mathbf{p}) + r\_2 (q - \mathbf{x}) \right]^2} \right\} \right]}} \, d\mathbf{x}$$

$$= \frac{\Gamma(\boldsymbol{\alpha}) \Gamma(\boldsymbol{\beta})}{\Gamma(\boldsymbol{\alpha} + \boldsymbol{\beta})} \, \frac{1}{(q - p) \left( 1 + r\_1 \right)^{\mathbf{a}} (1 + r\_2)^{\boldsymbol{\beta}}}$$

$$\times \, \_{2m}F\_{2m}\left[ \begin{array}{c} \Delta(m;a), & \Delta(m;\beta); \\\\ \Delta(2m;a+\beta); \end{array} \bigg| \begin{array}{c} w \\\\ \overline{4^m v^\sigma (1+r\_1)^m (1+r\_2)^m} \end{array} \right].\tag{40}$$

#### **4. Graphical Representation**

Here, in terms of the parameter *β*, we illustrate Equations (14) and (15) using graphical simulations. For this, we evaluate the integrals numerically using the Gaussian quadrature Method (see [37]) and compare this with the main results. We choose *k* = 5 and *n* = 8 to get more precise results.

#### **5. Conclusions**

It is worth stressing that the generalized Mittag–Leffler function obtained and the integral formulas computed are amenable to further generalizations and future investigation. We have attempted to exploit the close connection of the generalized Mittag–Leffler functions with several important special functions and compute the integrals of the functions mentioned above in the form of the generalized Mittag–Leffler, linking different families of special functions. Our main results (15) and (20) and some special cases (24)–(32) can yield several new integrals in terms of Fox-*H* functions obtained from Equations (11) and (13). For instance, we write

 *q p* (*x* − *p*)*α*−1(*q* − *x*)*β*−1 [(*q* − *p*) + *r*1(*x* − *p*) + *r*2(*q* − *x*)] *α*+*β Eξ*,*λ*,*γ*,*b σ*,*μ*,*ν*,*φ*,*δ*,*a w* (*x* − *p*)(*q* − *x*) [(*q* − *p*) + *r*1(*x* − *p*) + *r*2(*q* − *x*)]2 *m dx* = 1 (*q* − *p*) (1 + *r*1)*α* (1 + *r*2)*β* Γ(*ν*) Γ(*δ*) Γ(*ξ*) Γ(*γ*) × 5Ψ4 ⎡ ⎣ (*ξ*, *λ*), (*γ*, *b*), (*α*, *m*), (*β*, *m*), (1, 1); (*μ*, *σ*), (*ν*, *φ*), (*δ*, *a*), (*α* + *β*, 2*m*); *w* (1 + *r*1)*m*(1 + *r*2)*m* ⎤ ⎦ (41) = 1 (*q* − *p*) (1 + *r*1)*α* (1 + *r*2)*β* Γ(*ν*) Γ(*δ*) Γ(*ξ*) Γ(*γ*) × *H*1, 5 5, 5 ⎡ ⎣ −*w* (1 + *r*1)*m*(1 + *r*2)*m* (1 − *ξ*, *λ*), (1 − *γ*, *b*), (1 − *α*, *m*), (1 − *β*, *m*), (0, 1) (0, 1), (1 − *μ*, *σ*), (1 − *ν*, *φ*), (1 − *δ*, *a*), {1 − (*α* + *β*, 2*m*)]} ⎤ ⎦, (42)

with all the conditions prescribed in Theorem 1. We have also proved that Figures 1–3 show a good compatibility of the numerical solution obtained by the Gaussian quadrature method and the analytic expression. We conclude that the results obtained will provide a significant step in the theory of integral transforms and can yield some potential applications in the field of the classical and applied mathematics.

**Figure 2.** Solution of (15) (for *q* = 1) for *α* = 6, *r*1 = 2, *r*2 = 4, *p* = 0, *ξ* = 1, *λ* = 2, *γ* = 2 *b* = 3, *σ* = 2, *μ* = 5, *v* = 2, *φ* = 3, *δ* = 2, *a* = 4, *w* = 3 and *m* = 2.

**Figure 3.** Solution of (15) (for all *q*) for *α* = 6, *r*1 = 2, *r*2 = 4, *p* = 0, *ξ* = 1, *λ* = 2, *γ* = 2 *b* = 3, *σ* = 2, *μ* = 5, *v* = 2, *φ* = 3, *δ* = 2, *a* = 4, *w* = 3 and *m* = 2.

**Author Contributions:** Conceptualization, N.K. and S.A.-O.; methodology, T.U.; software, M.I.K.; validation, K.N. and M.I.K.; formal analysis, T.U.; investigation, N.K.; resources, S.A.-O.; data curation, S.A.-O.; writing—original draft preparation, T.U.; writing—review and editing, M.I.K.; visualization, N.K.; supervision, M.I.K.; project administration, K.N.; funding acquisition, K.N.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has received funding support from the National Science, 43 Research and Innovation Fund (NSRF), Thailand.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are thankful to the editors and reviewers for their valuable suggestions and comments.

**Conflicts of Interest:** The authors declare that they have no competing interest.

#### **References**

### *Article* **Non-Separable Linear Canonical Wavelet Transform**

#### **Hari M. Srivastava 1,2,3,4,\*, Firdous A. Shah 5, Tarun K. Garg 6, Waseem Z. Lone 5 and Huzaifa L. Qadri 5**

**Abstract:** This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in *L*2(R*n*). The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, we formulate several uncertainty inequalities, including the Heisenberg's, logarithmic, and Nazorav's inequalities for the proposed transform in the linear canonical domain. The obtained results are reinforced with illustrative examples.

**Keywords:** non-separable linear canonical wavelet; symplectic matrix; non-separable linear canonical transform; uncertainty principle

**MSC:** 42C40; 42B10; 53D22; 65R10

#### **1. Introduction**

The origin of the multi-dimensional linear canonical transform (LCT) dates back to the early 1970s with the foundational work of Moshinsky and Quesne [1] in quantum mechanics to study the linear maps of phase space. Soon after its inception in quantum mechanics, the linear canonical transform has been exclusively studied both in theory and applications [2,3]. The theory of multi-dimensional non-separable LCT involving a general 2*n* × 2*n* real, symplectic matrix **M** = (*A*, *B* : *C*, *D*) with *n*(2*n* + 1) independent parameters offers a canonical formalism for the representation of several physical systems in a lucid and insightful way. For any *f* ∈ *L*2(R*n*), the non-separable LCT with respect to a real, symplectic matrix **M** is given by [4,5]

$$\mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) = \frac{1}{|\det B|^{1/2}} \int\_{\mathbb{R}^n} f(\mathbf{t}) e^{i\pi \left(\mathbf{w}^T D B^{-1} \mathbf{w} - 2\mathbf{w}^T B^{T-1} \mathbf{t} + \mathbf{t}^T B^{-1} A \mathbf{t}\right)} \, d\mathbf{t}, \quad |\det B| \neq 0. \tag{1}$$

The importance of the arbitrary real symplectic matrices involved in Equation (1) lies in the fact that an appropriate choice of the matrix can be taken to inculcate a sense of rotation and shift into both the time and frequency axes, resulting in an efficient representation of the chirp-like signals, which are ubiquitous both in nature and man-made systems. Due to the extra degrees of freedom, the non-separable LCT has been successfully employed in diverse problems arising in various branches of science and engineering, such as harmonic analysis, reproducing kernel Hilbert spaces, optical systems, quantum mechanics, sampling, image processing, and so on [6,7].

**Citation:** Srivastava, H.M.; Shah, F.A.; Garg, T.K.; Lone, W.Z.; Qadri, H.L. Non-Separable Linear Canonical Wavelet Transform. *Symmetry* **2021**, *13*, 2182. https://doi.org/10.3390/ sym13112182

Academic Editor: Marin Marin

Received: 23 September 2021 Accepted: 29 October 2021 Published: 9 Novemeber 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Undoubtedly, wavelet transforms have fascinated the scientific, engineering, and research communities both with their versatile applicability and lucid mathematical framework [8,9]. In recent years, the classical wavelet transform has been extended and employed in different domains. The most prompt ones are the fractional wavelet transform [10], linear canonical wavelet transform [11,12], special affine wavelet transform [13,14], quaternion linear canonical wavelet transform [15], and quadratic-phase wavelet transform [16]. Unfortunately, all these transforms only perform well at representing point singularities and are incompetent at handling the distributed singularities, such as curves or edges in higher-dimensional signals [17–20]. The intuitive reason for this inadequacy is that wavelets are isotropic entities generated by isotropically dilating the mother wavelet, and as such, they ignore the geometric properties of the structures to be analyzed. Therefore, the conventional wavelet approach is inadequate while dealing with multi-dimensional signals, wherein the primary interest is to efficiently capture the geometric features, such as edges and corners, appearing due to the spatial occlusion between different objects. As such, the key problem in multi-dimensional signal analysis is to extract and characterize the relevant geometric information regarding the occurrence of curves and boundaries in signals. Subsequently, a higher-dimensional variant of the standard wavelet transform has been proposed, which serves as a potent tool for representing non-transient multidimensional signals in the time-frequency domain. Mathematically, the multi-dimensional wavelet transform of any *f* ∈ *L*2(R*n*) is defined by [21]

$$\mathcal{W}\_{\Psi}[f](a,\mathbf{b}) = \frac{1}{\sqrt{a}} \int\_{\mathbb{R}^{n}} f(\mathbf{t}) \,\overline{\psi\left(\frac{\mathbf{t}-\mathbf{b}}{a}\right)} e^{-i\mathbf{w}\cdot\mathbf{t}} \,d\mathbf{t}, \quad a \in \mathbb{R}^{+}, \mathbf{b} \in \mathbb{R}^{n},\tag{2}$$

where *a* is called the scaling parameter, which controls the degree of compression or scale, and **b** is the translation parameter that determines the time location of the wavelet. The multi-dimensional wavelet transform in Equation (2) has found numerous applications across diverse fields of science and engineering, particularly in video image processing, medical imaging, singular detection problems, fluid dynamics, shape recognition, and so on [21,22]. In the context of higher-dimensional wavelet theory, the symmetry property of wavelets is often desirable in practical applications, and as such, wavelets can reveal different patterns and singularities of highly nonstationary signals, such as Brownian motions, patterns on the water surfaces, fractal properties of the velocity field, computations of Renyi dimensions, Hurst and H¨older exponents. Some prominent examples of the symmetric wavelets include biorthogonal wavelets, quincunx wavelets, and carinal Bsplines.

Keeping in view the profound characteristics of the multi-dimensional wavelet transform and more degrees of freedom of non-separable linear canonical transforms, we are deeply motivated to intertwine these integral transforms into a novel integral transform coined as a non-separable linear canonical wavelet transform. The novel integral transform can efficiently localize any non-transient signal in the time-frequency plane with more degrees of freedom. With major modifications to the existing multi-dimensional wavelet transform in Equation (2), we propose the non-separable linear canonical wavelet transform of any *f* ∈ *L*2(R*n*) concerning the free symplectic matrix **M** = (*A*, *B* : *C*, *D*) as

$$\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f]\left(\boldsymbol{a},\mathbf{b}\right) = \frac{1}{|\det B|^{1/2}} \int\_{\mathbb{R}^{n}} f(\mathbf{t}) \overline{\psi\left(\frac{\mathbf{t}-\mathbf{b}}{a}\right)} e^{-\pi i \left(\boldsymbol{\Lambda}\_{\boldsymbol{a}}^{\mathrm{T}} D\boldsymbol{\mathcal{B}}^{-1} \boldsymbol{\Lambda}\_{\boldsymbol{a}} - 2\boldsymbol{\Lambda}\_{\boldsymbol{a}}^{\mathrm{T}} \boldsymbol{\mathcal{B}}^{\mathrm{T}^{-1}} \mathbf{t} + \mathbf{t}^{\mathrm{T}} \boldsymbol{\mathcal{B}}^{-1} A \mathbf{t}\right)} d\mathbf{t},\tag{3}$$

where Λ*a* = (*a a* ... *a*)*T*. Besides studying all the fundamental properties of the novel wavelet transform, we derive some well-known theorems, including the Rayleigh's theorem, inversion formula, and range theorem. In the sequel, we also formulate several uncertainty inequalities such as the Heisenberg's, logarithmic, and Nazorav-type inequalities for the non-separable linear canonical wavelet transform in Equation (3).

The rest of the article is structured as follows: Section 2 is concerned with the preliminary aspects of the study and the formulation of the non-separable linear canonical wavelet

transform. Section 3 is devoted to formulating several variants of the uncertainty principles, such as Heisenberg's, logarithmic, and Nazorav-type inequalities, for the proposed transform. Finally, a conclusion is extracted in Section 4.

#### **2. Non-Separable Linear Canonical Wavelet Transform in** *L*2(R*n*)

In this section, we first provide a healthy overview of the non-separable linear canonical transform. Then, we introduce the notion of the non-separable linear canonical wavelet transform in *L*2(R*n*), followed by some fundamental properties of the proposed transform, including the orthogonality relation, energy preserving relation, range theorem, and the inversion formula.

#### *2.1. Non-Separable Linear Canonical Transform*

For typographical convenience, we shall denote a real 2*n* × 2*n* matrix

$$\mathbf{M} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} a\_{11} & a\_{12} & \dots & a\_{1n} & b\_{11} & b\_{12} & \dots & b\_{1n} \\ a\_{21} & a\_{22} & \dots & a\_{2n} & b\_{21} & b\_{22} & \dots & b\_{2n} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ a\_{n1} & a\_{n2} & \dots & a\_{nn} & b\_{n1} & b\_{n2} & \dots & b\_{nn} \\ c\_{11} & c\_{12} & \dots & c\_{1n} & d\_{11} & d\_{12} & \dots & d\_{1n} \\ c\_{21} & c\_{22} & \dots & c\_{2n} & d\_{21} & d\_{22} & \dots & d\_{2n} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ c\_{n1} & c\_{n2} & \dots & c\_{nn} & d\_{n1} & d\_{n2} & \dots & d\_{nn} \end{pmatrix} \tag{4}$$

as **M** = (*A*, *B* : *C*, *D*), where *A*, *B*, *C*, and *D* are *n* × *n* sub-matrices with real entries. Moreover, the matrix **M** = (*A*, *B* : *C*, *D*) is said to be free symplectic if **M***T***JM** = **J** and |det *B*| = 0, where **J** = (**0**, *In* : −*In*, **0**), and *In* denotes the *n*-dimensional identity matrix. Furthermore, the sub-matrices corresponding to the free symplectic matrix **M** = (*A*, *B* : *C*, *D*) satisfy

$$AB^T = BA^T, \mathbf{CD}^T = DC^T,\\ AD^T - BC^T = I\_{n\prime} \tag{5}$$

or equivalently

$$A^T \mathbb{C} = \mathbb{C}^T A,\\ B^T D = D^T B,\\ A^T D - \mathbb{C}^T B = I\_{\mathbb{N}}.\tag{6}$$

The transpose and inverse corresponding to the free symplectic matrix *M* = (*A*, *B* : *C*, *D*) are given by **M***T* = *AT*, *CT* : *BT*, *DT* and **M**−1 = *DT*, −*BT* : −*CT*, *AT* , respectively. Moreover, we have

$$\begin{aligned} \mathbf{M} \mathbf{M}^{-1} &= \begin{pmatrix} A & B \\ \mathbb{C} & D \end{pmatrix} \begin{pmatrix} D^T & -B^T \\ -\mathbb{C}^T & A^T \end{pmatrix} \\ &= \begin{pmatrix} A D^T - B \mathbb{C}^T & -A B^T + B A^T \\ \mathbb{C} D^T - D \mathbb{C}^T & -C B^T + D A^T \end{pmatrix} \\ &= \begin{pmatrix} I\_{\mathbb{II}} & 0 \\ 0 & I\_{\mathbb{II}} \end{pmatrix} .\end{aligned}$$

A typical example of a 4 × 4 free symplectic matrix is given below

$$\mathbf{M} = \begin{pmatrix} 1 & 2 & -1/2 & -1/2 \\ -2 & 1 & -1/2 & 1/2 \\ -1 & -3 & 1 & 1 \\ -1 & 0 & -1/2 & 1/2 \end{pmatrix}.$$

**Definition 1.** *Given a free symplectic matrix* **M** = (*A*, *B* : *C*, *D*)*, the non-separable linear canonical transform of any f* ∈ *L*2(R*n*) *is denoted by* F**M** *f and is defined as*

$$\mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) = \int\_{\mathbb{R}^{n}} f(\mathbf{t}) \, \mathbb{K}^{\mathbf{M}}(\mathbf{t}, \mathbf{w}) \, d\mathbf{t},\tag{7}$$

*where the kernel* K**M**(**t**, **w**) *is given by*

$$\mathcal{K}^{\mathbf{M}}(\mathbf{x},\mathbf{w}) = \frac{1}{|\det B|^{1/2}} \exp\left\{ \pi i \left( \mathbf{w}^T D B^{-1} \mathbf{w} - 2 \mathbf{w}^T B^{T^{-1}} \mathbf{t} + \mathbf{t}^T B^{-1} A \mathbf{t} \right) \right\}, \quad |\det B| \neq 0. \tag{8}$$

The additive property of the non-separable LCT (Equation (7)) is very crucial for its understanding and application and is given by

$$
\mathcal{F}^{\mathbf{M}\_1} \left[ \mathcal{F}^{\mathbf{M}\_2} \left[ f(\mathbf{t}) \right] \right](\mathbf{w}) = \mathcal{F}^{\mathbf{M}\_1 \mathbf{M}\_2} \left[ f(\mathbf{t}) \right](\mathbf{w}).
$$

The Plancheral and inversion formulae corresponding to Equation (7) are given by

$$\left< f, \mathbf{g} \right>\_2 = \left< \mathcal{F}^{\mathbf{M}}[f], \mathcal{F}^{\mathbf{M}}[\mathbf{g}] \right>\_{2'} \quad \forall f, \mathbf{g} \in L^2(\mathbb{R}^n) \qquad \text{and} \tag{9}$$

$$f(\mathbf{t}) = \mathcal{F}^{\mathbf{M}^{-1}} \left[ \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \right](\mathbf{x}) = \int\_{\mathbb{R}^n} \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \, \mathbb{K}^{\mathbf{M}^{-1}}(\mathbf{w}, \mathbf{t}) \, d\mathbf{w},\tag{10}$$

respectively, where **M**−1 = *DT*, −*BT* : −*CT*, *AT* . Furthermore, the kernel in Equation (8) satisfies the following properties:

(i) K**M**−1 (**w**, **t**) = K**M**(**t**, **w**), (ii) R*n* K**M**(**t**, **w**) K**M**−1 (**t**, **z**) *d***t** = *δ*(**z** − **w**), (iii) R*n* K**M**(**t**, **w**) K**M**−1 (**z**, **w**) *d***w** = *δ*(**z** − **t**), (iv) R*n* K**M**(**t**, **w**) K**N**(**t**, **z**) *d***t** = K**MN**(**w**, **z**).

The non-separable linear canonical transform (Equation (7)) encompasses several well-known integral transforms, including the Fourier transform (FT), fractional Fourier transform (FrFT), linear canonical transform (LCT), and the Fresnel transforms (FrT) [4]. Table 1 shows some special cases of the non-separable linear canonical transform.

**Table 1.** Some special cases of the non-separable linear canonical transform.

#### *2.2. Non-Separable Linear Canonical Wavelet Transform*

Wavelets act as window functions whose radius increases in time (reduces in frequency) while resolving the low-frequency contents and decreases in time (increases in frequency) while resolving high-frequency contents of a non-transient signal. Mathematically, a doubly indexed family of wavelets *ψa*,**b** is generated by restricting the scaling parameter *a* belonging to R+ and the translation parameter **b** belonging to R*n* as [8]:

$$
\psi\_{a, \mathbf{b}}(\mathbf{t}) = \frac{1}{\sqrt{a}} \psi \left(\frac{\mathbf{t} - \mathbf{b}}{a}\right), \quad a \in \mathbb{R}^+, b \in \mathbb{R}^n. \tag{11}
$$

The scaling parameter *a* measures the degree of compression or scale, whereas the translation parameter **b** determines the location of the wavelet. With major modifications of the family (Equation (4)), we define a new family of functions *ψ***M** *a*,**b**(**t**) with respect to a free symplectic matrix **M** = (*A*, *B* : *C*, *D*) as:

$$
\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) = \frac{1}{\sqrt{a}} \psi\left(\frac{\mathbf{t} - \mathbf{b}}{a}\right) \mathcal{K}^{\mathbf{M}}(\mathbf{t}, a), \tag{12}
$$

where

$$\mathcal{K}^{\mathbf{M}}(\mathbf{t},a) = \frac{1}{|\det B|^{1/2}} \exp\left\{ \pi i \left( \boldsymbol{\Lambda}\_a^T D \boldsymbol{B}^{-1} \boldsymbol{\Lambda}\_d - 2 \boldsymbol{\Lambda}\_a^T \boldsymbol{B}^{T^{-1}} \mathbf{t} + \mathbf{t}^T \boldsymbol{B}^{-1} A \mathbf{t} \right) \right\},\tag{13}$$

where Λ*a* = (*a a* ... *a*)*T*. Having formulated a family of analyzing functions, we are now ready to introduce the definition of the non-separable linear canonical wavelet transform in *L*2(R*n*).

**Definition 2.** *For any f* ∈ *L*2(R*n*)*, the non-separable linear canonical wavelet transform of f with respect to an analyzing wavelet ψ and the free symplectic matrix* **M** = (*A*, *B* : *C*, *D*) *is defined by*

$$\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) = \frac{1}{\sqrt{a|\det B|}} \int\_{\mathbb{R}^{\pi}} f(\mathbf{t}) \overline{\psi\left(\frac{\mathbf{t}-\mathbf{b}}{a}\right)} e^{-\pi i \left(\boldsymbol{\Lambda}\_{\mathrm{z}}^{\mathrm{T}} D\mathbb{B}^{-1}\boldsymbol{\Lambda}\_{\mathrm{z}} - 2\boldsymbol{\Lambda}\_{\mathrm{z}}^{\mathrm{T}} \boldsymbol{\mathcal{B}}^{\mathrm{T}^{-1}} \mathbf{t} + \mathbf{t}^{\mathrm{T}} \boldsymbol{\mathcal{B}}^{-1} A \mathbf{t}\right)} d\mathbf{t}. \tag{14}$$

Definition 2 allows us to make the following comments:

(i) The non-separable linear canonical wavelet transform can be written in the innerproduct form as

$$\mathcal{W}^{\mathbf{M}}\_{\boldsymbol{\Psi}}[f](a,\mathbf{b}) = \left\langle f, \,\psi^{\mathbf{M}}\_{a,\mathbf{b}} \right\rangle\_{\boldsymbol{\Psi}}$$

where *ψ***M** *a*,**b**(**t**) is given by Equation (12).

(ii) It is worth noticing that the proposed transform in Equation (7) encompasses several existing integral transforms, such as the classical wavelet transform, fractional wavelet transform, linear canonical wavelet transform, and so on [8,9]. The corresponding wavelet transforms can be obtained by choosing an appropriate symplectic matrix **M** = (*A*, *B* : *C*, *D*).

We now present an example for the lucid illustration of the proposed non-separable linear canonical wavelet transform in Equation (14).

**Example 1.** *(a) Consider the function f*(**t**) = *e*( 12 11 *t*1− 30 11 *t*2) 2 *and the* 2*D-Morlet function ψ*(**t**) = *ei*Λ·**t**−|**t**| 2/2, Λ = (*λ*1, *λ*2) > 0*. Then, the translated and scaled versions of ψ*(**t**) *are given by*

$$\begin{split} \psi \left( \frac{\mathbf{t} - \mathbf{b}}{a} \right) = \exp \left\{ -\frac{i \left( \lambda\_1 b\_1 + \lambda\_2 b\_2 \right)}{a} - \frac{\left( b\_1^2 + b\_2^2 \right)}{2a^2} \right\} \exp \left\{ -\frac{t\_1^2}{2a^2} + \left( \frac{i \lambda\_1}{a} + \frac{b\_1}{a^2} \right) t\_1 \right\} \\ & \quad \times \exp \left\{ -\frac{t\_2^2}{2a^2} + \left( \frac{i \lambda\_2}{a} + \frac{b\_2}{a^2} \right) t\_2 \right\}. \end{split} \tag{15}$$

*Consequently, the family of non-separable linear canonical wavelets ψ***M** *a*,**b**(**t**) *is obtained as:*

$$\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) = \frac{1}{\sqrt{a \mid \det B \mid}} \psi\left(\frac{\mathbf{t} - \mathbf{b}}{a}\right) \exp\left\{i\pi \left(\boldsymbol{\Lambda}\_{a}^{T} D B^{-1} \boldsymbol{\Lambda}\_{a} - 2\boldsymbol{\Lambda}\_{a}^{T} B^{T-1} \mathbf{t} + \mathbf{t}^{T} B^{-1} A \mathbf{t}\right)\right\}. \tag{16}$$

*To compute the non-separable linear canonical wavelet transform of f*(**t**) *with respect to the window function ψ*(**t**)*,* Λ = (1, 1)*, and a real symplectic matrix*

$$\mathbf{M} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1/6 & 1 & -2 & 1/6 \\ -5/6 & -1/6 & 1/6 & 5/3 \\ 1 & 0 & 12/29 & -31/29 \\ -6/29 & -36/29 & 36/29 & 0 \end{pmatrix}.$$

,

*we proceed as:*

$$\begin{split} \mathcal{W}\_{\boldsymbol{\uprho}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) &= \frac{1}{\sqrt{a \mid \det B \mid}} \exp\left\{ \frac{i \left(b\_{1} + b\_{2}\right)}{a} - \frac{\left(b\_{1}^{2} + b\_{2}^{2}\right)}{2a^{2}} \right\} \\ &\times \int\_{\mathbb{R}^{n}} e^{\left(\frac{i \left\Vert \frac{\boldsymbol{\uprho}}{4}t\_{1} - \left\Vert \boldsymbol{\uprho} \right\Vert\_{2}}{2a^{2}}\right)^{2}} \exp\left\{ -\frac{t\_{1}^{2}}{2a^{2}} + \left(\frac{b\_{1}}{a^{2}} - \frac{i}{a}\right)t\_{1} \right\} \exp\left\{ -\frac{t\_{2}^{2}}{2a^{2}} + \left(\frac{b\_{2}}{a^{2}} - \frac{i}{a}\right)t\_{2} \right\} \\ &\times \exp\left\{ -\pi i \left(\boldsymbol{\Lambda}\_{a}^{\mathrm{T}} \boldsymbol{D} \boldsymbol{B}^{-1} \boldsymbol{\Lambda}\_{a} - 2\boldsymbol{\Lambda}\_{a}^{\mathrm{T}} \boldsymbol{B}^{\mathrm{T}^{-1}} \boldsymbol{t} + \boldsymbol{\upgamma}^{\mathrm{T}} \boldsymbol{B}^{-1} \boldsymbol{A} \boldsymbol{t}\right) \right\} dt\_{1} \, dt\_{2} . \end{split} \tag{17}$$

*Moreover, we have*

$$\begin{aligned} \Lambda\_a^T D B^{-1} \Lambda\_a &= -\frac{36}{121} \begin{pmatrix} a & a \end{pmatrix} \begin{pmatrix} 12/29 & -31/29 \\ 36/29 & 0 \end{pmatrix} \begin{pmatrix} 5/3 & -1/6 \\ -1/6 & -2 \end{pmatrix} \begin{pmatrix} a \\ a \end{pmatrix} \\ &= -\frac{36}{121} \begin{pmatrix} a & a \end{pmatrix} \begin{pmatrix} 5579/6786 & 1720/1131 \\ 60/29 & -6/29 \end{pmatrix} \begin{pmatrix} a \\ a \end{pmatrix} \\ &= -\frac{756a^2}{605} \\ \Lambda\_a^T B^{T-1} \mathbf{t} &= -\frac{36}{121} \begin{pmatrix} a & a \end{pmatrix} \begin{pmatrix} 5/3 & -1/6 \\ -1/6 & -2 \end{pmatrix} \begin{pmatrix} t\_1 \\ t\_2 \end{pmatrix} \end{aligned} \tag{18}$$

$$\mathbf{A}\_{a}^{\;I\;}B^{\;I\;}\;\mathbf{t} = -\frac{\mathbf{G}a}{121} \begin{pmatrix} a & a \end{pmatrix} \begin{pmatrix} \mathbf{t} \\ -1/6 & -2 \end{pmatrix} \begin{pmatrix} \mathbf{t}\_{2} \\ \mathbf{t}\_{2} \end{pmatrix}$$

$$= -\frac{6a}{121} \begin{pmatrix} 9t\_{1} - 13t\_{2} \end{pmatrix} \tag{19}$$

$$\begin{split} \mathbf{t}^T B^{-1} A \mathbf{t} &= -\frac{36}{121} \begin{pmatrix} t\_1 & t\_2 \end{pmatrix} \begin{pmatrix} 5/3 & -1/6 \\ -1/6 & -2 \end{pmatrix} \begin{pmatrix} 1/6 & 1 \\ -5/6 & -1/6 \end{pmatrix} \begin{pmatrix} t\_1 \\ t\_2 \end{pmatrix} \\ &= -\frac{36}{121} \begin{pmatrix} t\_1 & t\_2 \end{pmatrix} \begin{pmatrix} 5/2 & 61/6 \\ 59/6 & 1 \end{pmatrix} \begin{pmatrix} t\_1 \\ t\_2 \end{pmatrix} \\ &= -\frac{36}{121} \left( \frac{5t\_1^2}{2} + 20t\_1t\_2 + t\_2^2 \right) . \end{split} \tag{20}$$

*Implementing Equations* (18)*–*(20) *in Equation* (17)*, we obtain*

W**M** *ψ f* (*a*, **b**) = 11 6 √*a* exp% *i b*1 + *b*2 *a* − *b*2 1 + *b*2 2 2*a*2 + 756*πia*2 605 & × R exp − 1 2*a*2 − 90*πi* + 144 121 *t* 2 1 + *b*1 *a*2 − *i a* − 12*aπi* 121 *t*1 *dt*1 × R exp − 1 2*a*2 − 36*πi* + 900 121 *t* 2 2 + *b*2 *a*2 − *i a* − 156*aπi* 121 *t*2 *dt*2 = 11*π* 6 ' *a* 1 2*a*2 − 90*πi* + 144 121 1 2*a*2 − 36*πi* + 900 121 × exp% *i b*1 + *b*2 *a* − *b*2 1 + *b*2 2 2*a*2 + 756*πia*2 605 & × exp ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ *b*1 *a*2 − *i a* − 12*aπi* 121 2 4 1 2*a*2 − 90*πi* + 144 121 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ exp ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ *b*2 *a*2 − *i a* − 156*aπi* 121 2 4 1 2*a*2 − 36*πi* + 900 121 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ . (21)

*For different values of a and* **b***, the corresponding non-separable linear canonical wavelet transforms are plotted in Figures 1–3.*

*(b) Consider the constant function f*(**t**) = *K and the two-dimensional Morlet wavelet ψ*(**t**) = *ei*Λ·**t**−|**t**| 2/2, Λ = (*λ*1, *λ*2) > 0*. Then, the non-separable linear canonical wavelet transform of f*(**t**) *with respect to the real symplectic matrix*

$$\mathbf{M} = \begin{pmatrix} A & B \\ \mathbb{C} & D \end{pmatrix} = \begin{pmatrix} 2 & -1/8 & 1/4 & -1 \\ 1/8 & -2 & 1 & 1/4 \\ -2/15 & -31/30 & 1 & 0 \\ 1 & 2/3 & -4/5 & -14/15 \end{pmatrix}$$

*is given by*

$$\begin{split} \mathcal{W}\_{\boldsymbol{\uprho}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) &= \frac{1}{\sqrt{a \mid \det B \mid}} \exp\left\{ \frac{i \left(\lambda\_{1} b\_{1} + \lambda\_{2} b\_{2}\right)}{a} - \frac{\left(b\_{1}^{2} + b\_{2}^{2}\right)}{2a^{2}} \right\} \\ &\times \int\_{\mathbb{R}^{d}} \boldsymbol{K} \exp\left\{ -\frac{t\_{1}^{2}}{2a^{2}} + \left(\frac{b\_{1}}{a^{2}} - \frac{i\lambda\_{1}}{a}\right) t\_{1} \right\} \exp\left\{ -\frac{t\_{2}^{2}}{2a^{2}} + \left(\frac{b\_{2}}{a^{2}} - \frac{i\lambda\_{2}}{a}\right) t\_{2} \right\} \\ &\times \exp\left\{ -i\pi \left(\Lambda\_{a}^{T} D B^{-1} \Lambda\_{d} - 2\Lambda\_{a}^{T} B^{T-1} \mathbf{t} + \mathbf{t}^{T} B^{-1} A \mathbf{t}\right) \right\} d\mathbf{t}. \end{split} \tag{22}$$

*Moreover, we have*

$$
\Lambda\_a^T D B^{-1} \Lambda\_a = \frac{16}{15} \begin{pmatrix} a & a \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -4/15 & -14/15 \end{pmatrix} \begin{pmatrix} -1/4 & 1 \\ -1 & 1/4 \end{pmatrix} \begin{pmatrix} a \\ a \end{pmatrix} = \frac{12}{25}, \tag{23}
$$

$$
\Lambda\_a^T B^{T^{-1}} \mathbf{t} = \frac{16}{15} (a \quad a) \begin{pmatrix} -1/4 & 1 \\ -1 & 1/4 \end{pmatrix} \begin{pmatrix} t\_1 \\ t\_2 \end{pmatrix} = -\frac{4a}{3} \begin{pmatrix} t\_1 - t\_2 \end{pmatrix} \tag{24}
$$

$$\mathbf{t}^T \mathbf{t}^{-1} A \mathbf{t} = \frac{16}{15} \begin{pmatrix} t\_1 & t\_2 \end{pmatrix} \begin{pmatrix} -1/4 & 1\\ -1 & 1/4 \end{pmatrix} \begin{pmatrix} 2 & -1/8\\ 1/8 & -2 \end{pmatrix} \begin{pmatrix} t\_1\\ t\_2 \end{pmatrix} = -\frac{2}{5} \begin{pmatrix} t\_1^2 + t\_2^2 \end{pmatrix}.\tag{25}$$

*Implementing Equations* (23)*–*(25) *in Equation* (22)*, we obtain*

W**M** *ψ f* (*a*, **b**) = *K* / 15 16*a* exp% *i λ*1*b*1 + *λ*2*b*2 *a* − *b*2 1 + *b*2 2 2*a*2 − 12*πia*2 25 & × R exp − 1 2*a*2 − 2*πi* 5 *t* 2 1 + *b*1 *a*2 − *iλ*1 *a* + 8*aπi* 3 *t*1 *dt*1 × R exp − 1 2*a*2 − 2*πi* 5 *t* 2 2 + *b*2 *a*2 − *iλ*2 *a* − 8*aπi* 3 *t*2 *dt*2 = *Kπ* 1 2*a*2 − 2 5 / 15 16*a* exp% *i λ*1*b*1 + *λ*2*b*2 *a* − *b*2 1 + *b*2 2 2*a*2 − 12*πia*2 25 & × exp ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ *b*1 *a*2 − *iλ*1 *a* + 8*aπi* 3 2 4 1 2*a*2 − 2*πi* 5 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ exp ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ *b*2 *a*2 − *iλ*2 *a* − 8*aπi* 3 2 4 1 2*a*2 − 2*πi* 5 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ . (26)

**Figure 1.** Real and imaginary parts of the non-separable linear canonical wavelet transform of *f* corresponding to a fixed scale *a* = 1/4.

**Figure 2.** Real and imaginary parts of the non-separable linear canonical wavelet transform of *f* corresponding to a fixed scale *a* = 1.

**Figure 3.** (**a**) Frequency representation of *f* corresponding to a position **b** = (0, 0). (**b**) Frequency representation of *f* corresponding to a position **b** = (1, 1).

The non-separable linear canonical wavelet transforms shown in Equation (26) of *f* corresponding to Λ = (1, 1) are plotted in Figures 4–6.

**Figure 4.** Real and imaginary parts of the non-separable linear canonical wavelet transform of *f* corresponding to a fixed scale *a* = 1/4.

**Figure 5.** Real and imaginary parts of the non-separable linear canonical wavelet transform of *f* corresponding to a fixed scale *a* = 1.

**Figure 6.** (**a**) Frequency representation of *f* corresponding to a position **b** = (0, 0). (**b**) Frequency representation of *f* corresponding to a position **b** = (1, 1).

Next, we shall derive a fundamental relationship between the non-separable linear canonical wavelet transform (Equation (7)) and the non-separable linear canonical transform (Equation (1)). With the aid of this formula, we shall study the fundamental properties of the proposed transform.

**Proposition 1.** *Let* W**M** *ψ f* (*a*, **b**) *and* F**M** *f* (*a*) *be the non-separable linear canonical wavelet transform and the non-separable linear canonical transform of any f* ∈ *L*2(R*n*)*, respectively. Then, we have*

$$\mathcal{F}^{\mathbf{M}}\Big[\mathcal{W}^{\mathbf{M}}\_{\boldsymbol{\Psi}}[f]\Big](\mathbf{w}) = \sqrt{a \,|\det B|} \,\mathcal{K}^{\mathbf{M}}(\mathbf{b}, \Lambda\_{\mathbf{d}}) \, e^{\pi i \boldsymbol{a}^{2} \mathbf{w}^{T} D B^{-1} \mathbf{w}} \,\mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \,\overline{\mathcal{F}^{\mathbf{M}}[\mathbf{w}](a\mathbf{w})},\tag{27}$$

*where*

$$\Psi(\mathbf{t},a) = \mathbf{e}^{\pi i \left(2(a\Lambda\_d)^T B^{T^{-1}} \mathbf{t} - \mathbf{t}^T B^{-1} A \mathbf{t}\right)} \,\psi(\mathbf{t}).\tag{28}$$

**Proof.** Applying the definition of the non-separable linear canonical transform, we have

F**M** *ψ***M** *a*,**b**(**t**) (**w**) = R*n* 1 √*a ψ* **t** − **b** *a* 1 \$| det *B*| exp −*πi* Λ*T a DB*−1Λ*a* − 2Λ*T a BT*−1 **t** + **t** *TB*−1*A***t** × 1 \$| det *B*| exp *πi* **w***TDB*−1**w** − 2**w***TBT*−1 **t** + **t** *TB*−1*A***t** *d***t** = √*a* | det *B*| R*n ψ*(**z**) exp −*πi* Λ*T a DB*−1Λ*a* − 2Λ*T a BT*−1 (**b** + *a***z**) × exp *πi* **w***TDB*−1**w** − 2**w***TBT*−1 (**b** + *a***z**) *d***z** = √*a* | det *B*| R*n ψ*(**z**) exp *πi* **w***TDB*−1**w** − 2(*a***w***T*)*BT*−1 **z** + **z***TB*−1*A***z** × exp −*πi* Λ*T a DB*−1Λ*a* − 2(*a*Λ*a*)*TBT*−1 **z** + **z***TB*−1*A***z** × exp *πi* 2Λ*T a BT*−1 **b** − 2**w***TDB*−1**w** *d***z** = √*a* | det *B*| R*n ψ*(**z**) exp *πi* (*a***w**)*TDB*−1(*a***w**) − 2(*a***w**)*TBT*−1 **z** + **z***TB*−1*A***z** × exp *πi* **w***TDB*−1**w** − 2**w***TBT*−1 **b** + **b***TB*−1*A***b** × exp −*πi* Λ*T a DB*−1Λ*a* − 2Λ*T a BT*−1 **b** + **b***TB*−1*A***b** × exp −*πi* (*a***w**)*TDB*−1(*a***w**) − 2(*a*Λ*a*)*TBT*−1 **z** + **z***TB*−1*A***z** *d***z** = 0 *a* | det *B*|*e* −*πi a*2**w***TDB*−1**w** K**M**(**b**, **w**) K**M**(**b**, Λ*a*) × R*n e πi* 2(*a*Λ*a* )*T BT*−1 **z**−**z***T B*−1*A***z** *ψ*(**z**) K**M**(**z**, *a***w**) *d***z** = 0 *a* | det *B*|*e* −*πi a*2**w***TDB*−1**w** K**M**(**b**, **w**) K**M**(**b**, Λ*a*) F**M** Ψ (*a***w**), (29)

where Ψ(**t**, *a*) = *e πi* 2(*a*Λ*a* )*T BT*−1 **t**−**z***T B*−1*A***t** *ψ*(**t**).

Invoking the Plancheral theorem for the non-separable linear canonical transform and using Equation (29), we have

$$\begin{split} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) &= \sqrt{\boldsymbol{a} \,|\det{\boldsymbol{B}}|} \, \mathsf{K}^{\mathbf{M}}(\mathbf{b},\Lambda\_{\mathrm{d}}) \int\_{\mathbb{R}^{d}} e^{\boldsymbol{\eta} \boldsymbol{l} \, \boldsymbol{a}^{\mathsf{T}} \boldsymbol{D} \boldsymbol{B}^{-1} \mathbf{w}} \, \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \, \overline{\mathcal{F}^{\mathbf{M}}[\!\!\!/{} \!\!/]} \, \overline{\mathcal{K}^{\mathbf{M}}(\mathbf{b},\mathbf{w})} \, \mathsf{K}^{\mathbf{M}}(\mathbf{b},\mathbf{w}) \\ &= \mathcal{F}^{\mathbf{M}^{-1}} \Big[ \sqrt{\boldsymbol{a} \,|\det{\boldsymbol{B}}|} \, \mathsf{K}^{\mathbf{M}}(\mathbf{b},\Lambda\_{\mathrm{d}}) \, \epsilon^{\operatorname{\operatorname{\bf M}} \boldsymbol{a}^{\mathsf{T}} \boldsymbol{D} \boldsymbol{B}^{-1} \mathbf{w}} \, \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \, \overline{\mathcal{F}^{\mathbf{M}}[\!\!\!/{} \!\!/]}(\mathbf{b}). \end{split}$$

Consequently,

$$\mathcal{F}^{\mathbf{M}}\Big[\mathcal{W}^{\mathbf{M}}\_{\boldsymbol{\Psi}}[f](a,\mathbf{b})\Big](\mathbf{w}) = \sqrt{a \,|\det B|}\,\mathcal{K}^{\mathbf{M}}(\mathbf{b},\Lambda\_{a})\,e^{\pi i a^{2} \mathbf{w}^{T} D\mathbf{b}^{-1} \mathbf{w}}\mathcal{F}^{\mathbf{M}}[f](\mathbf{w})\,\overline{\mathcal{F}^{\mathbf{M}}[\mathbf{w}](a\mathbf{w})}.$$

This completes the proof of Proposition 1.

#### *2.3. Basic Properties of the Non-Separable Linear Canonical Wavelet Transform*

In this subsection, we shall study some mathematical properties of the proposed nonseparable linear canonical wavelet transform (Equation (7)), including Rayleigh's theorem, inversion formula, and the range theorem. In this direction, we have the following theorem, which assembles some of the basic properties of the proposed transform.

**Theorem 1.** *For any f* , *g* ∈ *L*2(R*n*) *and α*, *β* ∈ R*,* **k** ∈ R*n, and μ* ∈ R+*, the non-separable linear canonical wavelet transform as defined by Equation* (7) *satisfies the following properties:*

$$\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f(\mathbf{t}-\mathbf{k})](\boldsymbol{a},\mathbf{b}) = e^{2\pi i \Lambda\_{4} \boldsymbol{B}^{T} \boldsymbol{-}^{1} \mathbf{k}} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ f(\mathbf{t}) e^{\pi i \left( \mathbf{k}^{T} \boldsymbol{B}^{-1} A \mathbf{x} + \mathbf{x}^{T} \boldsymbol{B}^{-1} A \mathbf{k} \right)} \right](\boldsymbol{a},\mathbf{b}-\mathbf{k})$$

*(iv) Scaling:* W**M** *ψ f*(*μ***t**) (*a*, **b**) = |*μ*| 1− *n* 2 W**M** *ψ f*(*μa*, *μ***b**), **M** = (*A*/*μ*, *B*/*μ* : *μC*, *μD*)

$$\text{2(v)}\quad \text{Conjugation: } \mathcal{W}\_{\Psi}^{\mathbf{M}} \left[ \bar{f} \right](a, \mathbf{b}) = \frac{1}{\sqrt{a}} \overline{\mathcal{W}\_{\Psi}^{\mathbf{M}'} \left[ f \right](a, \mathbf{b})}, \ \mathbf{M}' = (A, -B:-\mathbb{C}, D).$$

**Proof.** For the sake of brevity, we omit the proof of the theorem.

Next, we shall define the admissibility condition for a function *ψ* ∈ *L*2(R*n*).

**Definition 3.** *A function ψ* ∈ *L*2(R*n*) *is said to be admissible with respect to a real free symplectic matrix* **M** = (*A*, *B* : *C*, *D*) *if*

$$\mathbb{C}\_{\Psi} = \int\_{\mathbb{R}^+} \frac{\left| \mathcal{F}^{\mathbf{M}}[\Psi](a\mathbf{w}) \right|^2}{a} da < \infty, \quad a.e. \quad \mathbf{w} \in \mathbb{R}^n,\tag{30}$$

*where* Ψ(**t**, *a*) *is given by Equation* (28)*.*

We are now in a position to derive the orthogonality relation for the proposed transform defined in Equation (7). As a consequence of the orthogonality relation, we will demonstrate that the non-separable wavelet transform is an isometry from the space of square-integrable functions *L*2(R*n*) to the space of transforms *L*2(R*n* × R+).

**Theorem 2.** *Let* W**M** *ψ f* (*a*, **b**) *and* W**M** *ψ g* (*a*, **b**) *be the non-separable linear canonical wavelet transforms of f and g belonging to L*2(R*n*)*, respectively. Then, we have*

$$\int\_{\mathbb{R}^n \times \mathbb{R}^+} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a}, \mathbf{b}) \, \overline{\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[\boldsymbol{g}]}(\boldsymbol{a}, \mathbf{b}) \, \frac{d\mathbf{b} \, da}{a^2} = \mathbb{C}\_{\boldsymbol{\Psi}} \, \langle f, \mathbf{g} \rangle,\tag{31}$$

*where Cψ is given by Equation* (30)*.*

**Proof.** For any pair of square integrable functions *f* and *g*, Proposition 1 implies that

$$\mathcal{W}\_{\boldsymbol{\Phi}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) = \sqrt{\boldsymbol{a} \,|\,\mathsf{det}\,\mathsf{B}|} \int\_{\mathbb{R}^{d}} \boldsymbol{\varepsilon}^{\operatorname{mid}\boldsymbol{a}^{2}\mathbf{w}^{\top}\mathsf{D}\boldsymbol{B}^{-1}\mathbf{w}} \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \,\overline{\mathcal{K}^{\mathbf{M}}(\mathbf{b},\mathbf{w})} \,\bar{\mathcal{K}}^{\mathbf{M}}(\mathbf{b},\Lambda\_{\mathfrak{a}}) \,\overline{\mathcal{F}^{\mathbf{M}}[\mathbf{y}]}(\boldsymbol{a}\mathbf{w}) \,d\mathbf{w}$$

and

$$\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[\boldsymbol{\varrho}](\boldsymbol{a},\mathbf{b}) = \sqrt{\boldsymbol{a} \,|\det{\boldsymbol{B}}|} \int\_{\mathbb{R}^{d}} e^{\boldsymbol{\pi} \boldsymbol{1} \boldsymbol{d}^{2} \mathbf{x}^{\top} \boldsymbol{D} \boldsymbol{B}^{-1} \mathbf{x}} \, \mathcal{F}^{\mathbf{M}}[\boldsymbol{\varrho}](\mathbf{x}) \, \overline{\mathcal{K}^{\mathbf{M}}(\mathbf{b},\mathbf{x})} \, \mathcal{K}^{\mathbf{M}}(\mathbf{b},\Lambda\_{\boldsymbol{\varrho}}) \, \overline{\mathcal{F}^{\mathbf{M}}[\boldsymbol{\Psi}](\boldsymbol{a}\mathbf{x})} \, d\mathbf{x}, \quad$$

where Ψ are given by Equation (28). Consequently, we have

 R*n*×R+ W**M** *ψ f* (*a*, **b**) W**M** *ψ g* (*a*, **b**) *d***b** *da a*2 = | det *B*| R*n*×R*n*×R*n*×R+ *e πia*2 (**w***TDB*−1**w**−**x***TDB*−1**x**) F**M** *f* (**w**) F**M** *g* (**x**) × F**M** Ψ (*a***w**) F**M** Ψ (*a***x**) K**M**(**b**, Λ*a*) K**M**(**b**, Λ*a*) K**M**(**b**, **x**) K**M**(**b**, **w**) *d***b** *d***w** *d***x** *da a* = R*n*×R*n*×R+ *e πia*2 (**w***TDB*−1**w**−**x***TDB*−1**x**) F**M** *f* (**w**) F**M** *g* (**x**) × F**M** Ψ (*a***w**) F**M** Ψ (*a***x**) R*n* K**M**(**b**, **x**) K**M**(**b**, **w**) *d***b** *d***w** *d***x** *da a* = R*n*×R*n*×R+ *e πia*2 (**w***TDB*−1**w**−**x***TDB*−1**x**) F**M** *f* (**w**) F**M** *g* (**v**) × F**M** Ψ (*a***w**) F**M** Ψ (*a***x**) *δ*(**w** − **x**) *d***w** *d***x** *da a* = R*n*×R+ F**M** *f* (**w**) F**M** *g* (**w**) F**M** Ψ (*a***w**) 2 *d***w** *da a* = R*n* F**M** *f* (**w**) F**M** *g* (**w**) % R+ F**M** Ψ (*a***w**) 2 *a da*& *d***w** = *Cψ* " F**M** *f* (**w**), F**M** *g* (**w**) # 2 = *Cψ f* , *g* ! 2.

This completes the proof of Theorem 2.

**Remark 1.** *(i). For f* = *g, Theorem 2 yields the energy preserving relation associated with the non-separable linear canonical wavelet transform (Equation* (10)*):*

$$\int\_{\mathbb{R}^n \times \mathbb{R}^+} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a}, \mathbf{b}) \right|^2 \frac{d\mathbf{b} \, da}{a^2} = \mathbb{C}\_{\boldsymbol{\Psi}} \left\| \boldsymbol{f} \right\|\_{2}^2. \tag{32}$$

*(ii). The operator* W**M** *ψ is a bounded-linear operator. Moreover, for Cψ* = 1 *and* | det *B*| = 1*, the operator* W**M** *ψ becomes an isometry from L*2(R*n*) *to L*2(R*n* × R+)*.*

In our next theorem, we demonstrate that the non-separable linear canonical wavelet transform W**M** *ψ f* (*a*, **b**) of any function *f* ∈ *L*2(R*n*) is reversible in the sense that *f* can be easily recovered from the transformed domain *L*2(R*n* × R+).

**Theorem 3.** *Let* W**M** *ψ f* (*a*, **b**) *be the non-separable linear canonical wavelet transform of an arbitrary function f* ∈ *L*2(R*n*)*. Then, f can be reconstructed via*

$$f(\mathbf{t}) = \frac{1}{\mathbb{C}\_{\Psi}} \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} \mathcal{W}\_{\Psi}^{\mathbf{M}}[f](a, \mathbf{b}) \,\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \, \frac{d\mathbf{b} \, da}{a^{2}}, \quad a.e. \tag{33}$$

**Proof.** According to Theorem 2, we can write

$$
\begin{split}
\left<\langle f,\mathbf{g}\right> &= \frac{1}{\mathsf{C}\_{\boldsymbol{\Psi}}} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}\left[f\right](a,\mathbf{b}) \,\overline{\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}\left[\boldsymbol{g}\right]}(a,\mathbf{b}) \,\frac{d\mathbf{b}\,da}{a^{2}} \\&= \frac{1}{\mathsf{C}\_{\boldsymbol{\Psi}}} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}\left[f\right](a,\mathbf{b}) \left<\int\_{\mathbb{R}^{n}} \overline{\overline{\mathbf{g}}(\mathbf{t})} \,\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \,d\mathbf{t}\right> \frac{d\mathbf{b}\,da}{a^{2}} \\&= \frac{1}{\mathsf{C}\_{\boldsymbol{\Psi}}} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{+}} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}\left[f\right](a,\mathbf{b}) \,\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \,\overline{\overline{\mathbf{g}}(\mathbf{t})} \,\frac{d\mathbf{t}\,d\mathbf{b}\,da}{a^{2}} \\&= \frac{1}{\mathsf{C}\_{\boldsymbol{\Psi}}} \left<\int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}\left[f\right](a,\mathbf{b}) \,\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \,\frac{d\mathbf{b}\,da}{a^{2}},\mathbf{g}(\mathbf{t})\right>.
\end{split}
$$

Since *g* is chosen arbitrarily from *L*2(R*n*), using the elementary properties of inner products, one can obtain

$$f(\mathbf{t}) = \frac{1}{\mathbb{C}\_{\Psi}} \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} \mathcal{W}\_{\Psi}^{\mathbf{M}}[f](a, \mathbf{b}) \,\psi\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \, \frac{d\mathbf{b} \, da}{a^{2}} \quad a.e.$$

This completes the proof of Theorem 3.

Finally, we investigate the characterization of the range for the proposed transform (Equation (7)). As a consequence of the range theorem, we shall demonstrate that the range of the non-separable linear canonical wavelet transforms; that is, W**M** *ψ* (*L*2(R*n*)) is a reproducing kernel Hilbert space.

**Theorem 4.** *If f* ∈ *L*2(R*n* × R+)*, then f belongs to the range* W**M** *ψ L*2(R*n*) *if and only if*

$$f\left(a',\mathbf{b'}\right) = \frac{1}{\mathbb{C}\_{\Psi}} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} f\left(a,\mathbf{b}\right) \left\langle \psi\_{a,\mathbf{b'}}^{\mathbf{M}}, \psi\_{a',\mathbf{b'}}^{\mathbf{M}} \right\rangle\_{2} \frac{d\mathbf{b}\,da}{a^{2}},\tag{34}$$

*where Cψ satisfies Equation* (27)*.*

**Proof.** Assume that *f* ∈ W**M** *ψ* (*L*2(R*n*)). Then, there exists a square integrable function *g* such that W**M** *ψ g* = *f* . In order to show that *f* satisfies Equation (34), we proceed as

*f a* , **b** = W**M** *ψ g a* , **b** = R*n g*(**t**) *ψ***M** *a* ,**b** (**t**) *d***t** = 1 *Cψ* R*n* R*n*×R+ W**M** *ψ g* (*a*, **b**) *ψ***M** *a*,**b**(**t**) *d***b** *da a*2 *ψ***M** *a* ,**b** (**t**) *d***t** = 1 *Cψ* R*n*×R+ W**M** *ψ g* (*a*, **b**) R *ψ***M** *a*,**b**(**t**) *ψ***M** *a* ,**b** (**t**) *d***t** *d***b** *da a*2 = 1 *Cψ* R*n*×R+ *f*(*a*, **b**) " *ψ***M** *a*,**b**, *ψ***M** *a* ,**b** # 2 *d***b** *da a*2 ,

which evidently verifies our claim. Conversely, suppose that the function *f* satisfies Equation (34). To verify that *f* ∈ W**M** *ψ* (*L*2(R*n*)), it is sufficient to find out a function *g* ∈ *L*2(R*n*) such that W**M** *ψ g* = *f* . Therefore, the desired function *g* will be constructed as follows:

Let

$$g(\mathbf{t}) = \frac{1}{\mathcal{C}\_{\psi}} \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} f(a, \mathbf{b}) \,\psi\_{a, \mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \, \frac{d\mathbf{b} \, da}{a^{2}}.\tag{35}$$

Then, it is straightforward to obtain 1 1*g* 1 1 2 ≤ 1 1 *f* 1 1 2 < ∞; that is *g* ∈ *L*2(R*n*). Furthermore, by virtue of the Fubini theorem, we have

$$\begin{split} \mathcal{W}\_{\boldsymbol{\upPsi}}^{\mathbf{M}}[\mathcal{J}] \left(a',\mathbf{b}'\right) &= \int\_{\mathbb{R}^{n}} \boldsymbol{g}(\mathbf{x}) \, \overline{\boldsymbol{\upPsi}\_{a',\mathbf{b'}}^{\mathbf{M}}}(\mathbf{t}) \, d\mathbf{t} \\ &= \frac{1}{\mathsf{C}\_{\boldsymbol{\upPsi}}} \int\_{\mathbb{R}^{n}} \left\{ \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} f(\boldsymbol{a},\mathbf{b}) \, \boldsymbol{\uppsi}\_{a,\mathbf{b}}^{\mathbf{M}}(\mathbf{t}) \, \frac{d\mathbf{b} \, da}{a^{2}} \right\} \overline{\boldsymbol{\uppsi}\_{a',\mathbf{b'}}^{\mathbf{M}}}(\mathbf{t}) \, d\mathbf{t} \\ &= \frac{1}{\mathsf{C}\_{\boldsymbol{\upPsi}}} \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} f(\boldsymbol{a},\mathbf{b}) \left\langle \boldsymbol{\uppsi}\_{a,\mathbf{b'}}^{\mathbf{M}}, \boldsymbol{\uppsi}\_{a',\mathbf{b'}}^{\mathbf{M}} \right\rangle\_{2} \frac{d\mathbf{b} \, da}{a^{2}} \\ &= f\left(a',\mathbf{b'}\right). \end{split}$$

This completes the proof of Theorem 4.

**Corollary 1.** *For any admissible wavelet ψ* ∈ *L*2(R*n*)*, the range of the proposed non-separable linear canonical wavelet transform; that is,* W**M** *ψ* (*L*2(R*n*)) *is a reproducing kernel Hilbert space embedded as a subspace in L*2 R*n* × R+ *with the kernel given by*

$$\mathcal{K}\_{\boldsymbol{\Psi}}^{\Lambda}(\boldsymbol{a},\mathbf{b};\boldsymbol{a}',\mathbf{b}') = \left\langle \psi\_{\boldsymbol{a},\mathbf{b}'}^{\mathbf{M}} \psi\_{\boldsymbol{a}',\mathbf{b}'}^{\mathbf{M}} \right\rangle\_2. \tag{36}$$

#### **3. Uncertainty Principles for the Non-Separable Linear Canonical Wavelet Transform**

The uncertainty principle lies at the heart of harmonic analysis, which asserts that "the position and the velocity of a particle cannot be both determined precisely at the same time" [23]. The harmonic analysis version of this principle states that "a non-trivial function cannot be properly localized in both the time and frequency domains at the same time" [24]. This standard inequality has been extensively studied in numerous domains and vistas [25–27]. Keeping in view the fact that the theory of uncertainty principles for the nonseparable linear canonical wavelet transform is yet to be explored exclusively; therefore, it is both theoretically and practically fascinating to develop some new uncertainty principles, including the Heisenberg's, logarithmic, and Nazaros uncertainty principles for the nonseparable linear canonical wavelet transform 7.

**Theorem 5.** *Let* W**M** *ψ f* (*a*, **b**) *be the non-separable linear canonical wavelet transform of any non-trivial function f* ∈ *L*2(R*n*) *with respect to a real free symplectic matrix* **M** = (*A*, *B* : *C*, *D*)*, then the following uncertainty inequality holds:*

$$\frac{1}{2} \left\{ \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} \left| \mathbf{b} \right|^{2} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ f \right] \left( a, \mathbf{b} \right) \right|^{2} \frac{d \mathbf{a} \, d\mathbf{b}}{a^{2}} \right\}^{1/2} \left\{ \int\_{\mathbb{R}^{n}} \left| \mathbf{w} \right|^{2} \left| \mathcal{F}^{\mathbf{M}} \left[ f \right] \left( \mathbf{w} \right) \right| d \mathbf{w} \right\}^{1/2} \geq \frac{n \, \sigma\_{\min}(\boldsymbol{\mathcal{B}}) \sqrt{\mathcal{C}\_{\boldsymbol{\Psi}}}}{4 \pi} \left\| f \right\|\_{2}^{2}. \tag{37}$$

*where σ*min(*B*) *denotes the minimum singular value of matrix B.*

**Proof.**

**Remark 2.** *The uncertainty inequality in Equation* (37) *embodies a wide class of uncertainty relations including the ones corresponding to the separable linear canonical wavelet transform, fractional wavelet transform, and classical wavelet transforms. The corresponding uncertainty principles can be obtained by choosing an appropriate matrix parameter* **M** = (*A*, *B* : *C*, *D*)*.*

**Example 2.** *For the sake of computational convenience, we restrict ourselves to the two-dimensional space. From the inequality in Equation* (37)*, we observe that the lower bound can be adjusted suitably by choosing a real, free symplectic matrix* **M** = (*A*, *B* : *C*, *D*) *and the analyzing function ψ.*

**(i).** *Consider the real, free symplectic matrix*

$$\mathbf{M}\_1 = \begin{pmatrix} A\_1 & B\_1 \\ C\_1 & D\_1 \end{pmatrix} = \begin{pmatrix} 1/2 & -3/2 & 1 & -1 \\ 3/2 & 1/2 & -1 & -1 \\ 0 & -1 & 1 & -1 \\ -1 & 0 & 1 & 1 \end{pmatrix}$$

*and the two-dimensional Morlet wavelet ψ*1(**t**) *given by*

$$\psi\_1(\mathbf{t}) = e^{i\boldsymbol{\Lambda}\cdot\mathbf{t} - |\mathbf{t}|^2/2}, \quad \boldsymbol{\Lambda} = (\lambda\_1, \lambda\_2) > 0.$$

*Then, by virtue of Equation* (28)*, we obtain*

$$\begin{split} \Psi(\mathbf{t},a) &= \exp\left\{\pi i \left(2a\Lambda\_a^T B^{-1} \mathbf{t} - \mathbf{t}^T B^{-1} A \mathbf{t}\right)\right\} \Psi(\mathbf{t}) \\ &= \exp\left\{\pi i \left(-2a^2 t\_2 + \frac{t\_1^2 + t\_2^2}{2}\right)\right\} \exp\left\{i(\lambda\_1 t\_1 + \lambda\_2 t\_2) - \frac{t\_1^2 + t\_2^2}{2}\right\} \\ &= \exp\left\{-\left(\frac{1-\pi i i}{2}\right)t\_1^2 + \lambda\_1 t\_1\right\} \exp\left\{-\left(\frac{1-\pi i i}{2}\right)t\_2^2 + (\lambda\_2 - 2a^2)t\_2\right\}. \end{split}$$

*Subsequently, we have*

F**M** Ψ (*a***w**) = 1 | det *B*|1/2 R2 Ψ(**t**, *a*) exp *πi* (*a***w**)*TDB*−1(*a***w**) − 2(*a***w**)*TBT*−1 **t** + **t** *TB*−1*A***t** *d***t** = 1 | det *B*|1/2 R2 exp − 1 − *πi* 2 *t* 2 1 + *λ*1*t*1 exp − 1 − *πi* 2 *t* 2 2 + (*λ*2 − 2*a*2)*t*2 × exp% *πi a*2 *ω*2 1 − *ω*2 2 + *aω*1(*t*2 − *t*1) + *aω*2(*t*1 + *t*2) − *t* 2 1 + *t* 2 2 2 &*dt*1 *dt*2 = √ 2 *e πia*2 (*ω*2 1−*ω*2 2 ) R exp% −*t* 2 1 2 + *λ*1 − *aπi*(*ω*1 − *ω*2) *t*1 & *dt*1 × R exp% −*t* 2 2 2 + *λ*2 − 2*a*2 + *aπi*(*ω*1 + *ω*2) *t*1 & *dt*2 = 2*π* √ 2*a eπia*2 (*ω*2 1−*ω*2 2 ) exp% *λ*1 − *aπi*(*ω*1 − *ω*2) 2 2 & exp% *λ*2 − 2*a*2 + *aπi*(*ω*1 + *ω*2) 2 2 & .

*Taking λ*1 = *aπi and λ*2 = 2*a*2*, we obtain*

$$\left| \mathcal{F}^{\mathbf{M}}[\Psi](a\mathbf{w}) \right|^2 = 8\pi^2 a \exp\left\{-a^2 \pi^2 (1 + 2\omega\_1^2 + 2\omega\_2^2 + \omega\_2)\right\}.\tag{41}$$

*Implementing Equation* (41) *in Equation* (30) *yields*

$$\mathcal{C}\_{\Psi} = 8\pi^2 \int\_{\mathbb{R}^+} \exp\left\{-\pi^2 \left(1 + 2\omega\_1^2 + 2\omega\_2^2 + \omega\_2\right) a^2\right\} da = \frac{4\pi^{3/2}}{\sqrt{1 + 2\omega\_1^2 + 2\omega\_2^2 + \omega\_2}} \cdot \frac{1}{2}$$

*In particular, for* (*ω*1, *ω*2)=(1, 1)*, we obtain*

$$\mathcal{C}\_{\Psi} = \frac{4\pi^{3/2}}{\sqrt{6}}.\tag{42}$$

*Therefore, for any normalized function f* ∈ *L*2(R2)*, an application of Equation* (42) *in Equation* (37) *yields the lower bound for the Heisenberg's inequality in Equation* (37) *of the form*

$$\left\{ \int\_{\mathbb{R}^{\mathbf{z}} \times \mathbb{R}^{+}} |\mathbf{b}|^{2} \Big| \mathcal{W}\_{\Psi}^{\mathbf{M}}[f](a,\mathbf{b}) \Big| \, \frac{da \, d\mathbf{b}}{a^{2}} \right\}^{1/2} \left\{ \int\_{\mathbb{R}^{\mathbf{z}}} |\mathbf{w}|^{2} \Big| \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \Big| d\mathbf{w} \right\}^{1/2} \geq \left(\frac{2}{3\pi}\right)^{1/4}.\tag{43}$$

**(ii).** *Consider the real, free symplectic matrix*

$$\mathbf{M}\_2 = \begin{pmatrix} A\_2 & B\_2 \\ C\_2 & D\_2 \end{pmatrix} = \begin{pmatrix} 1/6 & 1 & -2 & 1/6 \\ -5/6/2 & -1/6 & 1/6 & 5/3 \\ 1 & 0 & 12/29 & -31/29 \\ -6/29 & -36/29 & 36/29 & 6 \end{pmatrix}'$$

*and the two-dimensional DOG wavelet ψ*2 *given by*

$$
\psi\_2(\mathbf{t}) = \frac{1}{2a^2} e^{-|\mathbf{t}|^2/(2a^2)} - e^{-|\mathbf{t}|^2/2}, \quad 0 < a < 1.
$$

*Similar to computations carried out in (i), we can show that*

$$\mathbb{C}\_{\psi\_2} = \frac{6\pi}{11} \sqrt{\frac{1 + 3a^2}{1 + a^2}}, \quad \text{and,}$$

$$\left\{ \int\_{\mathbb{R}^3 \times \mathbb{R}^+} |\mathbf{b}|^2 \Big| \mathcal{W}\_{\psi}^{\mathbf{M}}[f](a, \mathbf{b}) \Big|^2 \frac{da \, d\mathbf{b}}{a^2} \right\}^{1/2} \left\{ \int\_{\mathbb{R}^n} |\mathbf{w}|^2 \Big| \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \Big| d\mathbf{w} \right\}^{1/2} \geq \sqrt{\frac{101(1 + 3a^2)}{66\pi(1 + a^2)}}. \tag{45}$$

**(iii).** *Finally, for the real free symplectic matrix*

$$\mathbf{M}\_3 = \begin{pmatrix} A\_3 & B\_3 \\ C\_3 & D\_3 \end{pmatrix} = \begin{pmatrix} 2 & -1/8 & 1/4 & -1 \\ 1/8 & -2 & 1 & 1/4 \\ -2/15 & -31/30 & 1 & 0 \\ 1 & 2/3 & -4/5 & -14/15 \end{pmatrix}$$

*and the two-dimensional Maxican-hat wavelet ψ*3

$$\psi\_3(\mathbf{t}) = \left(2 - |\mathbf{t}|^2\right) e^{-|\mathbf{t}|^2/2}.$$

*The admissibility constant Cψ*3 *and inequality in Equation* (37) *turn out to be*

$$C\_{\Psi} = \frac{\pi^{5/2}}{2} \sqrt{\frac{16}{17}}, \quad and,\tag{46}$$

$$\left\{ \int\_{\mathbb{R}^{3} \times \mathbb{R}^{+}} |\mathbf{b}|^{2} \Big| \mathcal{W}\_{\Psi}^{\mathbf{M}}[f](a, \mathbf{b}) \Big| \frac{2a \, d\mathbf{b}}{a^{2}} \right\}^{1/2} \left\{ \int\_{\mathbb{R}^{3}} |\mathbf{w}|^{2} \Big| \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \Big| d\mathbf{w} \right\}^{1/2} \geq \left( \frac{16\pi}{17} \right)^{1/4} \sqrt{\frac{17}{32}}.\tag{47}$$

The lower bounds of the Heisenberg's uncertainty inequality in Equation (37) corresponding to the aforementioned parametric symplectic matricies and analyzing functions are summarized in Table 2.

**Table 2.** Lower bounds associated with the Heisenberg's inequality in Equation (37).

In our next theorem, we shall establish the logarithmic uncertainty principle for the non-separable linear canonical wavelet transform in Equation (14).

**Theorem 6.** *Let ψ be an admissible function and suppose that* W**M** *ψ f* (·, **b**) ∈ S(R*n*)*, then the non-separable linear canonical wavelet transform (Equation* (14)*) of any f* ∈ S(R*n*) *satisfies the following logarithmic estimate of the uncertainty inequality:*

$$\begin{split} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \ln\left|\mathbf{b}\right| \left|\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}\left[f\right](\boldsymbol{a},\mathbf{b})\right|^{2} \frac{da\,d\mathbf{b}}{a^{2}} + \mathbb{C}\_{\boldsymbol{\Psi}} \int\_{\mathbb{R}^{n}} \ln\left|\mathbf{w}\mathcal{B}^{T^{-1}}\right| \left|\mathcal{F}^{\mathbf{M}}\left[f\right](\mathbf{w})\right|^{2} \,d\mathbf{w} \\ \geq \left[\frac{\Gamma'(n/2)}{\Gamma(n/2)} - \ln\pi\right] \mathbb{C}\_{\boldsymbol{\Psi}}\left[\left|f\right|\_{2}^{2}.\end{split} \tag{48}$$

*whenever the L.H.S of Equation* (48) *is defined.*

**Proof.** For any *f* ∈ S(R) ⊆ *L*2(R*n*), the logarithmic uncertainty principle for the nonseparable linear canonical transform (Equation (7)) is given by

$$\int\_{\mathbb{R}^n} \ln |\mathbf{t}| \left| f(\mathbf{t}) \right|^2 d\mathbf{t} + \int\_{\mathbb{R}^n} \ln \left| \mathbf{w} B^{T^{-1}} \right| \left| \mathcal{F}^{\mathbf{M}}[f](\mathbf{w}) \right|^2 d\mathbf{w} \ge \left( \frac{\Gamma'(n/2)}{\Gamma(n/2)} - \ln \pi \right) \int\_{\mathbb{R}^n} \left| f(\mathbf{t}) \right|^2 d\mathbf{t}.\tag{49}$$

Identifying W**M** *ψ f* (*a*, **b**) as a function of the translation parameter **b** and then replace *f* ∈ S(R*n*) with W**M** *ψ f* (*a*, **b**), we have

$$\begin{split} \int\_{\mathbb{R}^{n}} \ln |\mathbf{b}| \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f] \left( a, \mathbf{b} \right) \right|^{2} d\mathbf{b} + \int\_{\mathbb{R}^{n}} \ln \left| \mathbf{w} B^{T^{-1}} \right| \left| \mathcal{F}^{\mathbf{M}}[\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f] \left( a, \mathbf{b} \right)] \left( \mathbf{w} \right) \right|^{2} d\mathbf{w} \\ \geq \left( \frac{\Gamma'(n/2)}{\Gamma(n/2)} - \ln \pi \right) \int\_{\mathbb{R}^{n}} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f] \left( a, \mathbf{b} \right) \right|^{2} d\mathbf{b}. \end{split} \tag{50}$$

Integrating Equation (50) with respect to the measure *da*/*a*2, we obtain

$$\begin{split} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \ln|\mathsf{b}| \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) \right|^{2} \frac{da \, d\mathbf{b}}{a^{2}} + \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \ln\Big|\mathsf{w}\mathcal{B}^{T^{-1}} \left| \left[\mathcal{F}^{\mathbf{M}}[\mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b})](\mathbf{w})\right]^{2} \frac{da \, d\mathbf{w}}{a^{2}} \\ \geq \left(\frac{\Gamma'(n/2)}{\Gamma(n/2)} - \ln\pi\right) \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f](\boldsymbol{a},\mathbf{b}) \right|^{2} \frac{da \, d\mathbf{b}}{a^{2}}.\end{split} \tag{51}$$

As a consequence of Proposition 1, we can simplify Equation (51) as:

$$\begin{split} \frac{1}{2} \int\_{\mathbb{R}^{\mathfrak{s}} \times \mathbb{R}^{+}} \ln |\mathsf{b}| \left| \mathcal{W}\_{\Psi}^{\mathbf{M}}[f] \left( a, \mathbf{b} \right) \right|^{2} \frac{da \, d\mathbf{b}}{a^{2}} + \int\_{\mathbb{R}^{\mathfrak{s}}} \ln \left| \mathsf{w} B^{T^{-1}} \right| \left| \mathcal{F}^{\mathbf{M}}[f] (\mathbf{w}) \right|^{2} \left\{ \int\_{\mathbb{R}^{+}} \frac{\left| \mathcal{F}^{\mathbf{M}}[\Psi](a\mathbf{w}) \right|^{2}}{a} da \right\} d\mathbf{w} \\ \geq \left( \frac{\Gamma'(n/2)}{\Gamma(n/2)} - \ln \pi \right) \mathbb{C} \boldsymbol{\uprho} \left\| f \right\|\_{2}^{2}. \end{split} \tag{52}$$

Equivalently,

$$\begin{split} \int\_{\mathbb{R}^{n}\times\mathbb{R}^{+}} \ln|\mathbf{b}| \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}}[f] \left( \boldsymbol{a}, \mathbf{b} \right) \right|^{2} \frac{d\boldsymbol{a} \, d\mathbf{b}}{a^{2}} + \mathbb{C}\_{\boldsymbol{\Psi}} \int\_{\mathbb{R}^{n}} \ln \left| \mathbf{w} \mathbf{B}^{T^{-1}} \right| \left| \boldsymbol{\mathcal{F}}^{\mathbf{M}}[f] \left( \mathbf{w} \right) \right|^{2} d\mathbf{w} \\ \geq \left( \frac{\Gamma'(n/2)}{\Gamma(n/2)} - \ln \pi \right) \mathbb{C}\_{\boldsymbol{\Psi}} \left\| \left| f \right| \right\|\_{2}^{2} . \end{split} \tag{53}$$

This completes the proof of Theorem 6.

Nazarov's uncertainty principle measures the localization of a non-trivial function *f* by taking into consideration the notion of support of the function instead of the dispersion as used in the Heisenberg–Pauli–Weyl inequality (38). In this direction, we have the following theorem.

**Theorem 7.** *Let* W**M** *ψ f* (*a*, **b**) *be the non-separable linear canonical wavelet transform of any function f* ∈ *L*2(R*n*)*.  For an arbitrary function *f* ∈ *L*2(R*n*) and a pair of finite measurable subsets *T*1 and *T*2 of R*n*, Nazarov's uncertainty principle in the linear canonical domain reads [5]

$$\mathbb{C}e^{\mathbb{C}(T\_1, T\_2)} \left( \int\_{\mathbb{R}^n \backslash T\_1} \left| f(\mathbf{t}) \right|^2 d\mathbf{t} + \int\_{\mathbb{R}^n \backslash \left( \mathbb{T}\_2 \mathbf{B}^T \right)} \left| \mathcal{F}^M \left[ f \right](\mathbf{w}) \right|^2 d\mathbf{w} \right) \geq \int\_{\mathbb{R}^n} \left| f(\mathbf{t}) \right|^2 d\mathbf{t},\tag{55}$$

where *C*(*T*1, *T*2) = *C* min |*T*1||*T*2|, |*T*1| 1/*n*W(*T*2),W(*T*1)*T*1/*n* 2 , W(·) is the mean width of the measurable subset, and |·| denotes the Lebesgue measure.

By identifying W**M** *ψ f* (*a*, **b**) as a function of **b** followed by invoking Equation (55), we obtain

$$\begin{split} \left( \mathbf{C} e^{\mathbf{C} \left( T\_{1}, T\_{2} \right)} \right) \left( \int\_{\mathbb{R}^{n} \left| \boldsymbol{T}\_{1} \right|} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ \boldsymbol{f} \right] \right| (\boldsymbol{a}, \mathbf{b}) \right)^{2} d\mathbf{b} + \int\_{\mathbb{R}^{n} \left\langle \boldsymbol{T}\_{2} \mathbf{B}^{\mathrm{T}} \right\rangle} \left| \mathcal{F}^{\mathbf{M}} \left[ \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ \boldsymbol{f} \right] \left( \boldsymbol{a}, \mathbf{b} \right) \right] (\mathbf{w}) \right|^{2} d\mathbf{w} \right) \\ &\geq \int\_{\mathbb{R}^{n}} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ \boldsymbol{f} \right] (\boldsymbol{a}, \mathbf{b}) \right|^{2} d\mathbf{b}, \end{split} \tag{56}$$

Upon integrating Equation (56) with respect to the measure *da*/*a*2, we have

$$\begin{split} \left| \mathbb{C} e^{\mathbb{C}(T\_{1},T\_{2})} \left( \int\_{\mathbb{R}^{n} \backslash T\_{1} \times \mathbb{R}^{+}} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ f \right] (\boldsymbol{a}, \mathbf{b}) \right|^{2} \frac{da \, d\mathbf{b}}{a^{2}} + \int\_{\mathbb{R}^{n} \backslash \left( T\_{2} \mathbb{R}^{T} \right) \times \mathbb{R}^{+}} \left| \mathcal{F}^{\mathbf{M}} \left[ \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ f \right] (\boldsymbol{a}, \mathbf{b}) \right] (\mathbf{w}) \right|^{2} \frac{da \, d\mathbf{w}}{a^{2}} \right) \right| \\ & \geq \int\_{\mathbb{R}^{n} \times \mathbb{R}^{+}} \left| \mathcal{W}\_{\boldsymbol{\Psi}}^{\mathbf{M}} \left[ f \right] (\boldsymbol{a}, \mathbf{b}) \right|^{2} \frac{da \, d\mathbf{b}}{a^{2}}. \end{split}$$

Finally, as a consequence of orthogonality relation in Equation (31) and Proposition 1, we obtain the desired result

*CeC*(*T*1,*T*2) R*n*\*T*1×R+ W**M** *ψ f* (*a*, **b**) 2 *da d***b** *a*2 + *Cψ* R*n*\(*T*2*BT*) F *M f* (**w**) 2 *d***w** ≥ *Cψ* R*n f*(**t**) 2 *d***t**.

This completes the proof of Theorem 7.

#### **4. Conclusions**

In the present article, we introduced the notion of a kernel-based non-separable linear canonical wavelet transform in *L*2(R*n*) for obtaining an efficient time-frequency representation of higher-dimensional non-transient signals that has more degrees of freedom. Besides studying all the fundamental properties, such as Rayleigh's theorem, inversion formula, and range theorem, we have also formulated several uncertainty inequalities for the proposed transform containing Heisenberg's, logarithmic, and Nazarov's inequalities in the non-separable linear canonical domain.

**Author Contributions:** Writing original draft preparation, H.M.S.; Conceptualization, methodology, F.A.S.; Software and editing, T.K.G.; Methodology and software, W.Z.L. and H.L.Q.; Funding acquisition and research support, T.K.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The authors are deeply indebted to the anonymous referees for meticulously reading the manuscript, pointing out many inaccuracies, and giving several valuable suggestions to improve the initial version of the manuscript to the present stage. The second named author is supported by SERB (DST), Government of India under Grant No. EMR/2016/007951.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **A Double Logarithmic Transform Involving the Exponential and Polynomial Functions Expressed in Terms of the Hurwitz–Lerch Zeta Function**

**Robert Reynolds \* and Allan Stauffer**

Department of Mathematics and Statistics, York University, Toronto, ON M3J1P3, Canada<EMAIL_ADDRESS>**\*** Correspondence<EMAIL_ADDRESS>
**Abstract:** The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

**Keywords:** Lerch function; double integral; Catalan's constant; Aprey's constant

#### **1. Significance Statement**

P.S. Laplace (1749–1827) introduced the Laplace transform as part of their famous study of probability theory and celestial mechanics [1]. R.H Mellin (1854–1933) first gave a systematic formulation of the Mellin transformation and its inverse [2]. He used their transform to develop applications of the theory of special functions to the solution of hypergeometric differential equations.

The double Laplace transform is studied in the work of Debnath [3] to solve initial and boundary value problems in applied mathematics, and mathematical physics. The double Laplace transform has been used to study European vulnerable options under constant as well as stochastic (the Hull–White) interest rates [4].

In this work, we derive a double integral whose kernel involves generalized exponential, logarithmic and polynomial functions. The form of this kernel can be viewed as a double Laplace transform of logarithmic and polynomial functions when the powers of the variable *x* in the exponential function is equal to one. On the other hand, since a polynomial is involved in this kernel, we can also consider this double integral as a double Mellin transform of the logarithm and exponential functions when *c* = *e* = 1.

These integral types are also used to derive geometric probability constants by calculating the expected Euclidean distance *δ*(*n*) to the center of an *n*-dimensional cube [5], and have some similarities with the generalized hyperterminants in globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents [6].

#### **2. Introduction**

In this paper, we derive the double integral given by

$$\int\_0^\infty \int\_0^\infty t^{c\mu - 1} x^{-c\mu + c - 1} e^{-bt^c - dx^c} \log^k \left(at^c x^{-c}\right) dx dt \tag{1}$$

where the parameters *k*, *a*, *b*, *d*, *m* ∈ C, *Re*(*c*) > 0, *Re*(*e*) > 0. The derivations follow the method used by us in [7]. This method involves using a form of the generalized Cauchy's integral formula given by

$$\frac{y^k}{\Gamma(k+1)} = \frac{1}{2\pi i} \int\_{\mathcal{C}} \frac{e^{wy}}{w^{k+1}} dw. \tag{2}$$

**Citation:** Reynolds, R.; Stauffer, A. A Double Logarithmic Transform Involving the Exponential and Polynomial Functions Expressed in Terms of the Hurwitz–Lerch Zeta Function. *Symmetry* **2021**, *13*, 1983. https://doi.org/10.3390/sym 13111983

Academic Editor: Hari Mohan Srivastava

Received: 26 September 2021 Accepted: 18 October 2021 Published: 20 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

where *C* is in general an open contour in the complex plane, which has the same value at the end points of the contour. We then multiply both sides by a function of *x* and *t*, then take a definite double integral of both sides. This yields a definite integral in terms of a contour integral. Then, we multiply both sides of Equation (2) by another function of *y* and take the infinite sums of both sides such that the contour integral of both equations are the same.

#### **3. Definite Integral of the Contour Integral**

We use the method shown in [7]. The variable of integration in the contour integral is *α* = *w* + *m*. The cut and contour are in the first quadrant of the complex *α*-plane. The cut approaches the origin from the interior of the first quadrant and the contour goes round the origin with zero radius and is on opposite sides of the cut. Using a generalization of Cauchy's integral formula, we replace equations by replacing *y* by log*k* (*atcx*−*e* ) and multiplying by *t cm*−1*xe*(−*m*)+*e*−1*e*−*btc*−*dxe* then taking the definite integral with respect *x* ∈ [0, ∞) and *t* ∈ [0, ∞) to obtain

$$\begin{split} &\frac{1}{\Gamma(k+1)}\int\_{0}^{\infty}\int\_{0}^{\infty}t^{cm-1}x^{-cm+\epsilon-1}e^{-bt^{\epsilon}-dx^{\epsilon}}\log^{k}(at^{\epsilon}x^{-\epsilon})dxdt \\ &=\frac{1}{2\pi i}\int\_{0}^{\infty}\int\_{0}^{\infty}\int\_{\mathcal{C}}a^{w}w^{-k-1}t^{c(m+w)-1}x^{\epsilon(-(m+w))+\epsilon-1}e^{-bt^{\epsilon}-dx^{\epsilon}}dwdxdt \\ &=\frac{1}{2\pi i}\int\_{\mathcal{C}}\int\_{0}^{\infty}\int\_{0}^{\infty}a^{w}w^{-k-1}t^{c(m+w)-1}x^{\epsilon(-(m+w))+\epsilon-1}e^{-bt^{\epsilon}-dx^{\epsilon}}dxdtdw \\ &=\frac{1}{2\pi i}\int\_{\mathcal{C}}\frac{\pi a^{w}w^{-k-1}b^{-m-w}d^{m+w-1}\csc(\pi(m+w))}{\epsilon\epsilon}dw \end{split} \tag{3}$$

from Equation (3.381.10) in [8], where *Re*(*w* + *m*) > 0, *Re*(*c*) > 0, *Re*(*e*) > 0, *Re*(*b*) > 0, *Re*(*d*) > 0, *Re*(*m*) > 0 and using the reflection formula for the Gamma function (25.4.1) in [9]. We are able to switch the order of integration over *α*, *x* and *t* using Fubini's theorem since the integrand is of bounded measure over the space C × [0, ∞) × [0, ∞).

#### **4. The Hurwitz–Lerch Zeta Function and Infinite Sum of the Contour Integral**

#### *4.1. The Hurwitz–Lerch Zeta Function*

The Hurwitz–Lerch zeta function, shown in Section (25.14) in [9–11] has a series representation given by

$$\Phi(z,s,v) = \sum\_{n=0}^{\infty} (v+n)^{-s} z^n \tag{4}$$

where |*z*| < 1, *vs*. = 0, −1, .. and is continued analytically by its integral representation given by

$$\Phi(z,s,\upsilon) = \frac{1}{\Gamma(s)} \int\_0^\infty \frac{t^{s-1} e^{-\upsilon t}}{1 - z e^{-t}} dt = \frac{1}{\Gamma(s)} \int\_0^\infty \frac{t^{s-1} e^{-(\upsilon-1)t}}{e^t - z} dt \tag{5}$$

where *Re*(*v*) > 0, and either |*z*| ≤ 1, *z* = 1, *Re*(*s*) > 0, or *z* = 1, *Re*(*s*) > 1.

#### *4.2. Infinite Sum of the Contour Integral*

Using Equation (2) and replacing *y* by log(*a*) − log(*b*) + log(*d*) + *iπ*(2*y* + 1) then multiplying both sides by

$$-\frac{2i\pi b^{-m}d^{m-1}e^{i\pi m(2y+1)}}{c\varepsilon}\tag{6}$$

then taking the infinite sum over *y* ∈ [0, ∞) and simplifying in terms of the Hurwitz–Lerch zeta function to obtain

$$\begin{split} -\frac{(2i\pi)^{k+1}\epsilon^{imb-m}d^{m-1}\Phi\left(\frac{2i\pi w}{c\pi}, -k, \frac{-ik\log(a) + i\log(b) - i\log(d) + n}{2\pi}\right)}{cc\Gamma(k+1)} \\ = -\frac{1}{2\pi i}\sum\_{g=0}^{\infty}\int\_{C}\frac{1}{c\epsilon}2i\pi b^{-m}d^{m-1}w^{-k-1}\exp(w(\log(a) - \log(b) + \log(d))) \\ + i\pi(2y+1)(m+w))dw \\ = -\frac{1}{2\pi i}\int\_{C}\sum\_{g=0}^{\infty}\frac{1}{c\epsilon}2i\pi b^{-m}d^{m-1}w^{-k-1}\exp(w(\log(a) - \log(b) + \log(d))) \\ + i\pi(2y+1)(m+w))dw \\ = \frac{1}{2\pi i}\int\_{C}\frac{\pi a^{w}w^{-k-1}b^{-m-w}d^{m+v-1}\csc(\pi(m+w))}{cc}dw \end{split} \tag{7}$$

from Equation (1.232.3) in [8], where *Im*(*w* + *m*) > 0 in order for the sum to converge.

#### **5. Definite Integral in Terms of the Hurwitz–Lerch Zeta Function**

**Theorem 1.** *For all k*, *a*, *b*, *d*, *m* ∈ C, *Re*(*c*) > 0, *Re*(*e*) > 0*,*

$$\begin{split} & \int\_{0}^{\infty} \int\_{0}^{\infty} t^{\varepsilon m - 1} x^{-\varepsilon m + \varepsilon - 1} e^{-bt^{\varepsilon} - dx^{\varepsilon}} \log^{k} (at^{\varepsilon} x^{-\varepsilon}) dx dt \\ & = -\frac{1}{c\varepsilon} (2i\pi)^{k + 1} e^{i\pi m} b^{-m} d^{m - 1} \Phi \left( e^{2im\pi}, -k, \frac{-i\log(a) + i\log(b) - i\log(d) + \pi}{2\pi} \right) \end{split} \tag{8}$$

**Proof.** Observe the right-hand side of Equations (3) and (7) are equal so we may equate the left-hand sides and simplify the factorial to yield the stated result.

**Lemma 1.**

$$\begin{split} \int\_{0}^{\infty} \int\_{0}^{\infty} t^{\frac{\xi}{2} - 1} x^{\frac{\xi}{2} - 1} e^{-bt^{\varepsilon} - dx^{\varepsilon}} \log^{k} \left( -t^{\varepsilon} x^{-\varepsilon} \right) dx dt \\ = & - \frac{1}{\sqrt{\delta} c \sqrt{d} \varepsilon} i^{k + 2} (2\pi)^{k + 1} \left( 2^{k} \zeta \left( -k, \frac{i \log(b) - i \log(d) + 2\pi}{4\pi} \right) \\ & \qquad \qquad \qquad - 2^{k} \zeta \left( -k, \frac{1}{2} \left( \frac{i \log(b) - i \log(d) + 2\pi}{2\pi} + 1 \right) \right) \right) \end{split} \tag{9}$$

**Proof.** Use Equation (8) and set *m* = 1/2, *a* = −1 and simplify using entry (4) in the table below (64:12:7) in [12].

#### **6. Special Cases**

In this section, we evaluate Equation (8) for various values of the parameters in terms of special functions and fundamental constants. In this section, we use the following functions and fundamental constants; Euler's constant *γ*, Catalan's constant *C*, Glaisher's constant *A*, Aprey's constant *ζ*(3), Hurwitz zeta function *ζ*(*s*, *a*), hypergeometric function 2*F*1(*a*, *b*; *c*; *z*), Polylogarithm function *Lin*(*z*) and Riemann zeta function *ζ*(*s*).

**Example 1.**

$$\int\_0^\infty \int\_0^\infty t^{cm-1} x^{-cm+c-1} e^{-bt^c - dx^c} dx dt = \frac{\pi b^{-m} d^{m-1} \csc(\pi m)}{c\epsilon} \tag{10}$$

**Proof.** Use Equation (8) and set *k* = 0 and simplify using entry (2) in the table below (64:12:7) in [12].

**Example 2.**

$$\begin{aligned} \int\_0^\infty \int\_0^\infty \frac{e^{-2\left(t^2+x^3\right)} \left(t^{2p}x^{2-3p} - t^{2m}x^{2-3m}\right)}{t\log\left(\frac{t^2}{x^3}\right)} dx dt\\ = \frac{1}{5} \left(\tanh^{-1}\left(e^{i\pi m}\right) - \tanh^{-1}\left(e^{i\pi p}\right)\right) \end{aligned} \tag{11}$$

**Proof.** Use Equation (8) and form a second equation by replacing *m* → *p* and taking their difference and setting *k* = −1, *a* = 1, *b* = *c* = *d* = 2,*e* = 3 and simplifying using entry (3) in the table below (64:12:7) in [12].

#### **Example 3.**

$$\begin{split} \int\_{0}^{\infty} \int\_{0}^{\infty} \sqrt{x} e^{-2t^{2} - 3x^{3}} \log\left(\log\left(-\frac{t^{2}}{x^{3}}\right)\right) dx dt \\ = \frac{\pi}{12\sqrt{6}} \left( 4\log\Gamma\left(-\frac{i\log\left(\frac{3}{2}\right)}{4\pi}\right) - 4\log\Gamma\left(-\frac{1}{2} - \frac{i\log\left(\frac{3}{2}\right)}{4\pi}\right) + 3i\pi + \log(16) \right. \\ \left. + 2\log(\pi) + 4\log\left(\log\left(\frac{3}{2}\right)\right) + \log\left(\frac{1}{\left(\log\left(\frac{3}{2}\right) - 2i\pi\right)^{4}}\right) \right) \end{split} \tag{12}$$

**Proof.** Use Equation (9) and take the first partial derivative with respect to *k* and set *k* = 0, *b* = *c* = 2, *d* = *e* = 3 and simplify using Equation (25.11.18) in [9].

**Example 4.**

$$\int\_0^\infty \int\_0^\infty \frac{\sqrt{\mathbf{x}} e^{-5\mathbf{x}^3 - it^2}}{\log^2\left(\frac{it^2}{5\mathbf{x}^3}\right)} d\mathbf{x} dt = \frac{(-1)^{3/4} \mathcal{C}}{3\sqrt{5}\pi} \tag{13}$$

**Proof.** Use Equation (9) and set *k* = −2, *a* = *i*/5, *b* = *i*, *c* = 2, *d* = 5,*e* = 3 and simplify using Equation (3) in [13].

**Example 5.**

$$\int\_0^\infty \int\_0^\infty \frac{t^{\frac{1}{6}\left(\sqrt{2}-6\right)} x^{\frac{1}{4}\left(\sqrt{2}-4\right)} e^{-t\frac{1}{\sqrt{2}}^2 - x\sqrt{2}} \left(x^{\frac{1}{6\sqrt{2}}} - t^{\frac{1}{6\sqrt{2}}}\right)}{\log\left(t^{\frac{1}{\sqrt{2}}} x^{-\frac{1}{\sqrt{2}}}\right)} dx dt = -\log(3) \tag{14}$$

**Proof.** Use Equation (8) and form a second equation by replacing *m* → *p* and taking their difference and setting *k* = −1, *a* = 1, *b* = 1, *d* = 1, *m* = 1/2, *p* = 1/3, *c* = 1/ √2,*e* = 1/ √2 and simplify using Equation (7) in [14].

**Example 6.**

$$\int\_0^\infty \int\_0^\infty \frac{\frac{1}{t^{-t-\sum\_{-x}\sqrt{2}}} \frac{1}{x^{-\frac{m+p-1}{\sqrt{2}}-1} \left(x^{\frac{m}{\sqrt{2}}}t^{\frac{p}{\sqrt{2}}} - t^{\frac{m}{\sqrt{2}}}x^{\frac{p}{\sqrt{2}}}\right)}{t\log\left(t^{\frac{1}{\sqrt{2}}}x^{-\frac{1}{\sqrt{2}}}\right)} dx dt$$

$$= 2\log\left(\cot\left(\frac{\pi m}{2}\right)\tan\left(\frac{\pi p}{2}\right)\right)$$

**Proof.** Use Equation (8) and form a second equation by replacing *m* → *p* and taking their difference and setting *k* = −1, *a* = 1, *b* = 1, *d* = 1, *c* = 1/ √2,*e* = 1/ √2 and simplifying using entry (1) in the table below (64:12:7) in [12].

**Example 7.**

$$\int\_0^\infty \int\_0^\infty \frac{t^{\frac{1}{2}\left(\sqrt{2}-8\right)} x^{\frac{1}{2}\left(\sqrt{2}-8\right)} e^{-t^{\frac{1}{\sqrt{2}}} - x^{\frac{1}{\sqrt{2}}}} \left(t^{\frac{1}{2\sqrt{2}}} - x^{\frac{1}{2\sqrt{2}}}\right)}{\log\left(t^{\frac{1}{\sqrt{2}}} x^{-\frac{1}{\sqrt{2}}}\right)} dx dt = 4 \sinh^{-1}(1) \tag{16}$$

**Proof.** Use Equation (15) and set *m* = 2 cot−1(1+ √2) *π* , *p* = 2 tan−1(1+ √2) *π* and simplify. Note this is a double integral representation for the Universal Parabolic constant, *P*, given in [15]. This integral has similarities with the generalized hyperterminants in [6].

**Example 8.**

$$\int\_0^\infty \int\_0^\infty \frac{x^{3/2} e^{-t^2 - x^5}}{4 \log^2\left(\frac{t^2}{x^5}\right) + \pi^2} dx dt = \frac{\pi + \log\left(3 - 2\sqrt{2}\right)}{20\sqrt{2}\pi} \tag{17}$$

and

$$\int\_0^\infty \int\_0^\infty \frac{\mathbf{x}^{3/2} e^{-t^2 - x^5} \log\left(\frac{t^2}{x^5}\right)}{\log^2\left(\frac{t^2}{x^5}\right) + \frac{x^2}{4}} dx dt = 0$$

**Proof.** Use Equation (8) and set *k* = −1, *a* = *i*, *c* = 2,*e* = 5, *b* = 1, *d* = 1, *m* = 1/2 and simplify using entry (1) in table below (64:12:7) in [12]; then, rationalize the denominator and equate real and imaginary parts.

**Example 9.**

$$\int\_0^\infty \int\_0^\infty \frac{\sqrt{t} x^{3/2} e^{-t^3 - x^5} \left(\log^2\left(\frac{\rho^3}{x^5}\right) - \pi^2\right)}{\left(\log^2\left(\frac{\rho^3}{x^5}\right) + \pi^2\right)^2} dx dt = -\frac{\pi}{360} \tag{19}$$

and

$$\int\_0^\infty \int\_0^\infty \frac{\sqrt{t} x^{3/2} e^{-t^3 - x^5} \log\left(\frac{t^3}{x^5}\right)}{\left(\log^2\left(\frac{t^3}{x^5}\right) + \pi^2\right)^2} dx dt = 0$$

**Proof.** Use Equation (8) and set *k* = −1, *a* = −1, *c* = 3,*e* = 5, *b* = 1, *d* = 1, *m* = 1/2 and simplify using entry (1) in table below (64:12:7) in [12]; then, rationalize the denominator and equate real and imaginary parts.

**Example 10.**

$$\begin{split} \int\_{0}^{\infty} \int\_{0}^{\infty} \sqrt{x} e^{-5t^{2} - 5x^{3}} \log\left(-\frac{t^{2}}{x^{3}}\right) \log\left(\log\left(-\frac{t^{2}}{x^{3}}\right)\right) dxdt \\ = \frac{1}{30} i\pi^{2} \log(2i\pi) - \frac{2}{15} i\pi^{2} \log\left(\frac{A^{3}}{\sqrt[3]{2}\sqrt[4]{\varepsilon}}\right) \end{split} \tag{21}$$

**Proof.** Use Equation (8) and take the first partial derivative with respect to *k* and set *m* = 1/2, *k* = 1, *b* = *d* = 5, *a* = −1, *c* = 2,*e* = 3 and simplify using Equation (8) in [14].

**Example 11.**

$$\int\_0^\infty \int\_0^\infty \sqrt{x}e^{-5t^2 - 5x^3} \log^2\left(-\frac{t^2}{x^3}\right) \log\left(\log\left(-\frac{t^2}{x^3}\right)\right) dxdt = \frac{7\pi\zeta(3)}{15} \tag{22}$$

**Proof.** Use Equation (8) and take the first partial derivative with respect to *k* and set *m* = 1/2, *k* = 2, *b* = *d* = 5, *a* = −1, *c* = 2,*e* = 3 and simplify using Equation (9) in [14].

**Example 12.**

$$\int\_0^\infty \int\_0^\infty \frac{\sqrt{t}e^{-t^3 - x^2}}{\sqrt{\log\left(\frac{\bar{x}^3}{x^2}\right)}} dx dt = \frac{1}{6} (-1)^{3/4} \sqrt{\pi} \left( \zeta\left(\frac{1}{2}, \frac{7}{8}\right) - \zeta\left(\frac{1}{2}, \frac{3}{8}\right) \right) \tag{23}$$

**Proof.** Use Equation (8) and set *m* = 1/2, *k* = −1/2, *b* = 1, *a* = *i*, *c* = 3, *d* = 1,*e* = 2 and simplify using entry (4) in the table below (64:12:7) in [12].

**Example 13.**

$$\begin{split} & \int\_{0}^{\infty} \int\_{0}^{\infty} \sqrt{t} e^{-\mathbf{x}^{2} + (-1 - i)t^{3}} \log^{\frac{i}{2}} \left( \frac{it^{3}}{\mathbf{x}^{7}} \right) dx dt \\ & = \frac{1}{3} (-1)^{\frac{7}{6} + \frac{i}{4}} 2^{-\frac{1}{4} + i} \pi^{1 + \frac{i}{2}} \left( \zeta \left( -\frac{i}{2}, \frac{13}{16} + \frac{i \log(2)}{8\pi} \right) - \zeta \left( -\frac{i}{2}, \frac{5}{16} + \frac{i \log(2)}{8\pi} \right) \right) \end{split} \tag{24}$$

**Proof.** Use Equation (8) and set *m* = 1/2, *k* = −*i*/2, *b* = 1 + *i*, *a* = *i*, *c* = 3, *d* = 1,*e* = 2 and simplify using entry (4) in the table below (64:12:7) in [12].

**Example 14.**

$$\int\_0^\infty \int\_0^\infty \frac{t \varepsilon^{-1^3 - x^2}}{\sqrt[3]{x} \log\left(\frac{it^3}{x^2}\right)} dx dt = -\frac{2}{9} \left( -\frac{1}{2} + \frac{i\sqrt{3}}{2} \right) \,\_2F\_1\left(\frac{3}{4}, 1; \frac{7}{4}; -\frac{1}{2} - \frac{i\sqrt{3}}{2}\right) \tag{25}$$

**Proof.** Use Equation (8) and set *m* = 2/3, *k* = −1, *b* = 1, *a* = *i*, *c* = 3, *d* = 1,*e* = 2 and simplify using Equation (9.559) in [8].

**Example 15.**

$$\begin{split} \int\_{0}^{\infty} \int\_{0}^{\infty} t^{3m-1} x^{1-2m} e^{-d\left(t^{3} + x^{2}\right)} \log^{k} \left(-\frac{t^{3}}{x^{2}}\right) dx dt &= \\ -\frac{i(2i)^{k} \pi^{k+1} e^{-i\pi m} L i\_{-k} \left(e^{2im\pi}\right)}{3d} \end{split} \tag{26}$$

**Proof.** Use Equation (8) and set *b* = *d*, *a* = −1, *c* = 3,*e* = 2 and simplify using Equation (25.14.3) in [9].

**Example 16.**

$$\int\_0^\infty \int\_0^\infty \sqrt{t} e^{-d\left(t^3 + x^2\right)} \log^k\left(-\frac{t^3}{x^2}\right) dxdt = -\frac{(2i)^k \left(2^{k+1} - 1\right) \pi^{k+1} \zeta(-k)}{3d} \tag{27}$$

**Proof.** Use Equation (26) and set *m* = 1/2 and simplify using

**Example 17.**

$$\int\_0^\infty \int\_0^\infty \frac{\sqrt{t}e^{-t^3 - x^2} \left(\pi^2 - 3\log^2\left(\frac{t^3}{x^2}\right)\right)}{\left(\log^2\left(\frac{t^3}{x^2}\right) + \pi^2\right)^3} dx dt = \frac{\zeta(3)}{32\pi^3} \tag{28}$$

and