# Differential Calculus on Deformed Generalized Fibonacci Polynomials and the
Functional-Difference Equation $\mathbf{D}_{s,t}f(x)=af(ux)$

Ronald Orozco López

###### Abstract

We give a differential calculus defined on deformed generalized Fibonacci
polynomials. The main goal is to generalize the $q$-calculus and the Golden
calculus or Fibonacci calculus and thus obtain the Pell calculus, Jacobsthal
calculus, Chebysheff calculus, Mersenne calculus, among others. This calculus
will serve as a framework for the solutions of equations in differences with
proportional delay. For this reason, we define the deformed
$(s,t)$-exponential functions and we also construct a family of functions that
are solutions of a linear functional difference equation with proportional
delay of first order.

2020 Mathematics Subject Classification: Primary 11B39. Secondary 11B65,
30B10, 34K06.

_Keywords:_ Fibonacci polynomial, deformed functions, Pell calculus,
Jacobsthal calculus, Chebysheff calculus, Mersenne calculus, functional-
difference equation.

##  1 Introduction

Fontené in [10] published a paper in which he generalized the binomial
coefficients by replacing $\binom{n}{k}=\frac{n(n-1)\cdots(n-k+1)}{1\cdot
2\cdots k}$, consisting of natural numbers, with
$\binom{n}{k}_{\psi}=\frac{\psi_{n}\psi_{n-1}\cdots\psi_{n-k+1}}{\psi_{1}\psi_{2}\cdots\psi_{k}}$,
formed by an arbitrary sequence $\psi=\\{\psi_{n}\\}$ of real or complex
numbers. He gave a fundamental recurrence relation for these coefficients such
that when we make $\psi_{n}=n$ we recover the ordinary binomial coefficients
and when we make $\psi_{n}=\brk[s]{n}_{q}=\frac{q^{n}-1}{q-1}$ we recover the
$q$-binomial coefficients studied by Gauss, Euler, Jackson and others.

Subsequently, Ward [15] developed a symbolic calculus on sequences
$\psi=\\{\psi_{n}\\}$ with $\psi_{0}=0$, $\psi_{1}=1$, and $\psi_{n}\neq 0$
for all $n\geq 1$, and thus generalized the ordinary calculus and the
$q$-calculus of Jackson [2, 13]. Other well-studied calculus emerged from his
work, the $(p,q)$-calculus and the Fibonomial calculus, where $\psi_{n}=F_{n}$
is the Fibonacci sequence defined recursively by $F_{0}=0$, $F_{1}=1$,
$F_{n+1}=F_{n}+F_{n-1}$. For more details on some works on this subject see
[23, 24, 25]. In this paper we investigate a Ward calculus defined on
generalized Fibonacci polynomials

$\displaystyle F_{n+1}$ $\displaystyle=sF_{n}+tF_{n-1},$ (1)

with initial values $F_{0}=0$ and $F_{1}=1$, in the variables $s,t$. As
special cases we obtain the differential calculus of Pell, Jacobsthal,
Chebysheff of second kind, Mersenne, Repunits, among others. When $s=P$ and
$t=-Q$, with $P$ and $Q$ integers, we obtain the $(P,-Q)$-Lucas differential
calculus and if $Q=-1$, we obtain the $P$-Fibonacci differential calculus. We
will now give two justifications for writing this paper.

Euler [12] in the 1740s initiated the $q$-calculus with the study of
partitions or additive number theory. Also Gauss [19, 20] got involved in the
$q$-calculus since he studied the $q$-hypergeometric series and the
$q$-analogue of binomial formula. However, who actually studied the
$q$-calculus systematically was Jackson, [2, 3, 4, 5, 6, 7, 8], introducing
the $q$-difference operator, some $q$-functions and the $q$-analogs of the
integral. Since then there has been an extensive number of papers and books
devoted to the $q$-calculus and its applications in mathematics and physics,
for example, in the theory of special functions, difference and differential
equations, combinatorics, analytic number theory, quantum theory, quantum
group, numerical analysis, operator theory and other related theories.
Recently, Chakrabarti and Jagannathan [26], Brodimas et al. [27], Wachs and
White [28], and Arik et al. [29] developed a new calculus, extension of the
$q$-calculus, called the $(p,q)$-calculus or Post Quantum Calculus. We have
the following definition of $(p,q)$-number

$\brk[s]{n}_{p,q}=\frac{p^{n}-q^{n}}{p-q}.$

When $p=1$, the $(p,q)$-numbers reduce to the $q$-numbers $\brk[s]{n}_{q}$. In
general, the $(p,q)$-calculus reduce to the $q$-calculus when $p=1$. The
$(p,q)$-numbers satisfy the following arithmetic rule

$\displaystyle\brk[s]{n+m}_{p,q}$
$\displaystyle=p^{n}\brk[s]{m}_{p,q}+q^{m}\brk[s]{n}_{p,q}=p^{m}\brk[s]{n}_{p,q}+q^{n}\brk[s]{m}_{p,q},$
$\displaystyle\brk[s]{-n}_{p,q}$ $\displaystyle=-(pq)^{-n}\brk[s]{n}_{p,q}.$
(2)

The $(p,q)$-analogue of $n!$ is

$\brk[s]{n}_{p,q}!=\begin{cases}\brk[s]{1}_{p,q}\brk[s]{2}_{p,q}\cdots\brk[s]{n-1}_{p,q}\brk[s]{n}_{p,q},&\text{
if }n\geq 1;\\\ 1,&\text{ if }n=0.\end{cases}$

From here the $(p,q)$-binomial coefficients are

$\binom{n}{k}_{p,q}=\frac{\brk[s]{n}_{p,q}}{\brk[s]{k}_{p,q}\brk[s]{n-k}_{p,q}}$

with Pascal identity given by

$\displaystyle\binom{n}{k}_{p,q}$
$\displaystyle=p^{k}\binom{n-1}{k}_{p,q}+q^{n-k}\binom{n-1}{k-1}_{p,q},$
$\displaystyle=q^{k}\binom{n-1}{k}_{p,q}+p^{n-k}\binom{n-1}{k-1}_{p,q}.$

The $(p,q)$-derivative of a function $f$ is

$\mathbf{D}_{p,q}f(x)=\begin{cases}\frac{f\left(px\right)-f\left(qx\right)}{(p-q)x},&\text{
if }x\neq 0;\\\ f^{\prime}(0),&\text{ if }x=0,\end{cases}$ (3)

provided that $f$ is differentiable at 0. Some basic $(p,q)$-functions are:
the $(p,q)$-analogue of $(x-a)^{n}$

$(x\ominus a)_{p,q}^{n}=\begin{cases}\prod_{k=0}^{n-1}(p^{k}x-q^{k}a),&\text{
if }n\geq 1;\\\ 1,&\text{ if }n=0,\end{cases}$

and the $(p,q)$-exponential functions

$\displaystyle\mathrm{e}_{p,q}(x)$
$\displaystyle=\sum_{n=0}^{\infty}p^{\binom{n}{2}}\frac{x^{n}}{\brk[s]{n}_{p,q}!},$
(4) $\displaystyle\mathop{\mbox{{E}}}\nolimits_{p,q}(x)$
$\displaystyle=\sum_{n=0}^{\infty}q^{\binom{n}{2}}\frac{x^{n}}{\brk[s]{n}_{p,q}!}.$
(5)

For more on $(p,q)$-calculus see [26, 27, 28, 29].

Pashaev et al. [23] introduced the Fibonomial calculus or Golden $q$-calculus
as a special case of the $q$-calculus. They defined the golden derivative of
the function $f(x)$ as

$\mathbf{D}_{F}f(x)=\begin{cases}\frac{f\left(\varphi
x\right)-f\left(-x/\varphi\right)}{(\varphi+\frac{1}{\varphi})x},&\text{ if
}x\neq 0;\\\ f^{\prime}(0),&\text{ if }x=0.\end{cases}$

Also, the Golden exponential functions are

$\displaystyle e^{x}_{F}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{x^{n}}{F_{n}!},$ (6)

and

$\displaystyle E^{x}_{F}$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{\frac{n(n-1)}{2}}\frac{x^{n}}{F_{n}!},$
(7)

where $F_{n}!=F_{1}F_{2}\cdots F_{n}$ is the $F$-analogue of $n!$. The Golden
Binomial is

$(x+y)_{F}^{n}=(x+\varphi^{n-1}y)(x-\varphi^{n-3}y)\cdots(x+(-1)^{n-1}\varphi_{-n+1}y).$

For more on Fibonomial calculus see also [24, 25].

The Fibonomial calculus also turns out to be a special case of the
$(p,q)$-calculus. In fact, every calculus obtained from Eq. (1) can be seen as
a special case of the $(p,q)$-calculus. It is enough to make $p+q=s$ and
$pq=-t$ to obtain

$\displaystyle p$ $\displaystyle=\frac{s+\sqrt{s^{2}+4t}}{2},$ $\displaystyle
q$ $\displaystyle=\frac{s-\sqrt{s^{2}+4t}}{2}.$

In this paper, we will study the case $s^{2}+4t\neq 0$ and the degenerate case
$s^{2}+4t=0$, which is obtained when $s\rightarrow\pm 2i\sqrt{t}$. With the
latter we obtain a family of calculus analogous to ordinary calculus or
Newton’s calculus. Just as the Fibonacci calculus also has direct application
in quantum mechanics, we hope with this paper to construct new calculus with
possible applications in mathematics and physics.

On the other hand, Morris et al. [22] studied the function

$\mathop{\mbox{{Exp}}}\nolimits(x,y)=\sum_{n=0}^{\infty}y^{\frac{n(n-1)}{2}}\frac{x^{n}}{n!}$

which is the solution of the Pantograph functional differential equation [31,
32]

$f^{\prime}(x)=f(yx),\ \ f(0)=1.$ (8)

The function $\mathop{\mbox{{Exp}}}\nolimits(x,y)$ is a deformed exponential
function since when $y\rightarrow 1$, then
$\mathop{\mbox{{Exp}}}\nolimits(x,y)\rightarrow e^{x}$. It is closely related
to the generating function for the Tutte polynomials of the complete graph
$K_{n}$ in combinatorics, the Whittaker and Goncharov constant in complex
analysis, and the partition function of one-site lattice gas with fugacity $x$
and two-particle Boltzmann weight $q$ in statistical mechanics [21].

As the functions in Eqs. (4),(5),(6), and (7) satisfy the equations

$\displaystyle\mathbf{D}_{p,q}f(x)$ $\displaystyle=f(ux),\ \ u=p,q,$ (9)
$\displaystyle\mathbf{D}_{F}f(x)$ $\displaystyle=f(ux),\ \ u=1,-1$ (10)

with $f(0)=1$, which are analogous to the Pantograph differential equation in
Eq. (8), we can then call such functions deformed exponential functions. What
we want to do in this paper is to introduce a framework for the solutions of
the difference equation $\mathbf{D}_{s,t}f(x)=f(ux)$. For this reason,
deformed exponential functions on generalized Fibonacci polynomials are
defined.

We divide this paper as follows. In Section 2, we introduce deformed
generalized Fibonacci polynomials in the variables $s,t$, with their
respective most important specializations. In addition, we give the definition
of Fibotorials and Fibonomials with their respective properties. In Section 3,
we define the Ward ring of $(s,t)$-exponential generating functions together
with their respective specializations and we give the definitions of positive
and negative deformations of these functions and the relationship between
them. In section 4, we give different specializations of deformed derivatives
and we construct the kernel of the deformed derivatives. In Section 5, we
introduce the deformed $(s,t)$-exponential functions. Finally, we solve the
$(s,t)$-analog of the Eq.(8).

##  2 Deformed generalized Fibonacci polynomials

The generalized Fibonacci polynomials depending on the variables $s,t$ are
defined by

$\displaystyle\brk[c]{0}_{s,t}$ $\displaystyle=0,$
$\displaystyle\brk[c]{1}_{s,t}$ $\displaystyle=1,$
$\displaystyle\brk[c]{n+2}_{s,t}$
$\displaystyle=s\\{n+1\\}_{s,t}+t\\{n\\}_{s,t}.$

Since $n\geq 0$, we will call $\brk[c]{n}_{s,t}$ positive Fibonacci
polynomials. Other polynomials related to Fibonacci polynomials are the
generalized Lucas polynomials. These polynomials are defined by

$\displaystyle\brk[a]{0}_{s,t}$ $\displaystyle=2,$
$\displaystyle\brk[a]{1}_{s,t}$ $\displaystyle=1,$
$\displaystyle\brk[a]{n+2}_{s,t}$
$\displaystyle=s\brk[a]{n+1}_{s,t}+t\brk[a]{n}_{s,t}.$

Below are some important specializations of Fibonacci polynomials.

  1. 1.

When $s=0,t=0$, then $\brk[c]{0}_{0,0}=0$, $\brk[c]{1}_{0,0}=1$ and
$\brk[c]{n}_{0,0}=0$ for all $n\geq 2$.

  2. 2.

When $s=0,t\neq 0$, then $\brk[c]{2n}_{0,t}=0$ and
$\brk[c]{2n+1}_{0,t}=t^{n}$.

  3. 3.

When $s\neq 0,t=0$, then $\brk[c]{n}_{s,0}=s^{n-1}$.

  4. 4.

When $s=2,t=-1$, then $\brk[c]{n}_{2,-1}=n$, the positive integer.

  5. 5.

When $s=1,t=1$, then $\brk[c]{n}_{1,1}=F_{n}$, the Fibonacci numbers.

  6. 6.

When $s=2,t=1$, then $\brk[c]{n}_{2,1}=P_{n}$, where $P_{n}$ are the Pell
numbers

$P_{n}=(0,1,2,5,12,29,\ldots).$

  7. 7.

When $s=1,t=2$, then $\brk[c]{n}_{1,2}=J_{n}$, where $J_{n}$ are the
Jacosbthal numbers

$J_{n}=(0,1,1,2,3,5,11,21,43,85,171,\ldots).$

  8. 8.

When $s=p+q,t=-pq$, then $\brk[c]{n}_{p+q,-pq}=\brk[s]{n}_{p,q}$, where
$\brk[s]{n}_{p,q}$ are the $(p,q)$-numbers

$\brk[s]{n}_{p,q}=(0,1,\brk[s]{2}_{p,q},[3]_{p,q},[4]_{p,q},[5]_{p,q},[6]_{p,q},[7]_{p,q},[8]_{p,q}\ldots).$

  9. 9.

When $s=2t,t=-1$, then $\brk[c]{n}_{2t,-1}=U_{n-1}(t)$, where $U_{n}(t)$ are
the Chebysheff polynomials of the second kind, with $U_{-1}(t)=0$

$U_{n}(t)=(0,1,2t,4t^{2}-1,8t^{3}-4t,16t^{4}-12t^{2}+1,32t^{5}-32t^{3}+6t,\ldots).$

  10. 10.

When $s=3,t=-2$, then $\brk[c]{n}_{3,-2}=M_{n}$, where $M_{n}=2^{n}-1$ are the
Mersenne numbers

$M_{n}=(0,1,3,7,15,31,63,127,255,\ldots).$

  11. 11.

When $s=b+1,t=-b$, then $\brk[c]{n}_{b+1,-b}=R_{n}^{(b)}$, where $R_{n}^{(b)}$
are the Repunit numbers in base $b$

$R_{n}^{(b)}=(0,1,b+1,b^{2}+b+1,b^{3}+b^{2}+b+1,\ldots).$

  12. 12.

When $s=P,t=-Q$, then $\brk[c]{n}_{P,-Q}=U_{n}(P,Q)$, where $U_{n}(P,Q)$ is
the Lucas sequence, with $P,Q$ integer numbers,

$U_{n}(P,Q)=(0,1,P,P^{2}-Q,P^{3}-2PQ,P^{4}-3P^{2}Q+Q^{2},\ldots).$

If $Q=-1$, then the sequence $U_{n}(P,-1)$ reduces to the $P$-Fibonacci
sequence. If $s=x$ and $t=1$, we obtain the Fibonacci polynomials

$F_{n}(x)=(0,1,x,x^{2}+1,x^{3}+2x,x^{4}+3x^{2}+1,\ldots).$

The $(s,t)$-Fibonacci constant is the ratio toward which adjacent
$(s,t)$-Fibonacci polynomials tends. This is the only positive root of
$x^{2}-sx-t=0$. We will let $\varphi_{s,t}$ denote this constant, where

$\varphi_{s,t}=\frac{s+\sqrt{s^{2}+4t}}{2}$

and

$\varphi_{s,t}^{\prime}=s-\varphi_{s,t}=-\frac{t}{\varphi_{s,t}}=\frac{s-\sqrt{s^{2}+4t}}{2}.$

Some specializations of the constants $\varphi_{s,t}$ and
$\varphi_{s,t}^{\prime}$ are:

  1. 1.

When $s=0$ and $t=0$, then $\varphi_{0,0}=0$ and $\varphi_{0,0}^{\prime}=0$.

  2. 2.

When $s=0$ and $t>0$, then $\varphi_{0,t}=\sqrt{t}$ and
$\varphi_{0,t}^{\prime}=-\sqrt{t}$.

  3. 3.

When $s\neq 0$ and $t=0$, then $\varphi_{s,0}=s$ and
$\varphi_{s,0}^{\prime}=0$.

  4. 4.

When $s=2$ and $t=-1$, then $\varphi_{2,-1}=1$ and
$\varphi_{2,-1}^{\prime}=1$.

  5. 5.

When $s=1$ and $t=1$, then $\varphi_{1,1}=\varphi=\frac{1+\sqrt{5}}{2}$ and
$\varphi_{1,1}^{\prime}=\varphi^{\prime}=\frac{1-\sqrt{5}}{2}$.

  6. 6.

When $s=2$ and $t=1$, then $\varphi_{2,1}=1+\sqrt{2}$ and
$\varphi_{2,1}^{\prime}=1-\sqrt{2}$.

  7. 7.

When $s=1$ and $t=2$, then $\varphi_{1,2}=2$ and $\varphi_{1,2}^{\prime}=-1$.

  8. 8.

When $s=p+q$ and $t=-pq$, then $\varphi_{p+q,-pq}=p$ and
$\varphi_{p+q,-pq}^{\prime}=q$.

  9. 9.

When $s=2t$ and $t=-1$, then $\varphi_{2t,-1}=\frac{t+\sqrt{t^{2}-1}}{2}$ and
$\varphi_{2t,-1}^{\prime}=\frac{t-\sqrt{t^{2}-1}}{2}$.

  10. 10.

When $s=3$ and $t=-2$, then $\varphi_{3,-2}=2$ and
$\varphi_{3,-2}^{\prime}=1$.

  11. 11.

When $s=b+1$ and $t=-b$, then $\varphi_{b+1,-b}=b$ and
$\varphi_{b+1,-b}^{\prime}=1$.

  12. 12.

When $s=P$ and $t=-Q$, then $\varphi_{P,-Q}=\frac{P+\sqrt{P^{2}-4Q}}{2}$ and
$\varphi_{P,-Q}^{\prime}=\frac{P-\sqrt{P^{2}-4Q}}{2}$.

In the remainder of the paper we will assume that $s\neq 0$ and $t\neq 0$. The
Binet’s $(s,t)$-identity is

$\brk[c]{n}_{s,t}=\begin{cases}\frac{\varphi_{s,t}^{n}-\varphi_{s,t}^{\prime
n}}{\varphi_{s,t}-\varphi_{s,t}^{\prime}},&\text{ if }s\neq\pm 2i\sqrt{t};\\\
n(\pm i\sqrt{t})^{n-1},&\text{ if }s=\pm 2i\sqrt{t}.\end{cases}$ (11)

As $\varphi_{us,u^{2}t}=u\varphi_{s,t}$ and
$\varphi_{us,u^{2}t}^{\prime}=u\varphi_{s,t}^{\prime}$, then follows that
$\brk[c]{n}_{us,u^{2}t}=u^{n-1}\brk[c]{n}_{s,t}$. Thus, we have the following
definition.

######  Definition 1.

For a non-zero complex number $u$ define the $u$-deformation of the Fibonacci
polynomials $\brk[c]{n}_{s,t}$ as
$\brk[c]{n}_{us,u^{2}t}=u^{n-1}\brk[c]{n}_{s,t}$, for all $n\geq 1$.

For example, with $u=\frac{1}{2}$ we can obtain the $1/2$-deformation of the
Fibonacci sequence $F_{n}$, i.e.,

$\brk[c]{n}_{1/2,1/4}=(0,1,1/2,1/2,3/8,5/16,1/4,13/64,\ldots)$

generated by the recurrence equation

$\brk[c]{n+2}_{1/2,1/4}=\frac{1}{2}\brk[c]{n+1}_{1/2,1/4}+\frac{1}{4}\brk[c]{n}_{1/2,1/4}.$

In general, the sequence $u$-deformed $\brk[c]{n}_{su,tu^{2}}$ satisfies the
recurrence relation

$\brk[c]{n+2}_{su,tu^{2}}=su\brk[c]{n+1}_{su,tu^{2}}+tu^{2}\brk[c]{n}_{su,tu^{2}},$

so the $u$-deformed Fibonacci polynomials $\brk[c]{n}_{su,tu^{2}}$ are
associated with the characteristic polynomial $p_{s,t}(x,u)=x^{2}-sux-tu^{2}$
whose ratios are $\varphi_{su,tu^{2}}=u\varphi_{s,t}$ and
$\varphi_{su,tu^{2}}^{\prime}=u\varphi_{s,t}^{\prime}$. Then we can think of a
$u$-deformation as a dilation of the $(s,t)$-Fibonacci constants
$\varphi_{s,t}$ and $\varphi_{s,t}^{\prime}$. Next we will give the definition
of deformed Fibotorial and Fibonomial polynomials.

######  Definition 2.

The definition of $u$-deformed $(s,t)$-Fibotorial is

$\brk[c]{n}_{us,u^{2}t}!=u^{\binom{n}{2}}\prod_{k=1}^{n}\brk[c]{k}_{s,t},\
n\geq 1,\ \brk[c]{0}_{s,t}!=1.$ (12)

Let us introduce also the $u$-deformed $(s,t)$-Fibonomial polynomials

$\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}_{us,u^{2}t}=u^{k(n-k)}\frac{\brk[c]{n}_{s,t}!}{\brk[c]{n-k}_{s,t}!\brk[c]{k}_{s,t}!}.$
(13)

Some $u$-deformed $(s,t)$-Fibonomial polynomials are

$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{n}{0}_{us,u^{2}t}$
$\displaystyle=\genfrac{\\{}{\\}}{0.0pt}{}{n}{n}_{us,u^{2}t}=1,$
$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{2}{1}_{us,u^{2}t}$
$\displaystyle=su,$
$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{3}{1}_{us,u^{2}t}$
$\displaystyle=\genfrac{\\{}{\\}}{0.0pt}{}{3}{2}_{us,u^{2}t}=(s^{2}+t)u^{2},$
$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{4}{1}_{us,u^{2}t}$
$\displaystyle=\genfrac{\\{}{\\}}{0.0pt}{}{4}{3}_{us,u^{2}t}=(s^{3}+2st)u^{3},\
\genfrac{\\{}{\\}}{0.0pt}{}{4}{2}_{us,u^{2}t}=(s^{2}+t)(s^{2}+2t)u^{4}.$

For extreme cases $(s,0)$ and $(0,t)$, $t>0$, we have

$\displaystyle\brk[c]{n}_{us,0}!$ $\displaystyle=(us)^{\binom{n}{2}},$
$\displaystyle\brk[c]{n}_{0,u^{2}t}!$
$\displaystyle=u^{\binom{n}{2}}\brk[c]{1}_{0,t}\brk[c]{2}_{0,t}\cdots\brk[c]{n}_{0,t}$
$\displaystyle=u^{\binom{n}{2}}(t^{0})(0)(t)(0)\cdots=0.$

and $\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}=s^{k(n-k)}$. Then the case $s=0$ and
$t\neq 0$ will be of no interest in this paper.

For $t\neq 0$, the negative $(s,t)$-Fibonacci polynomials are

$\brk[c]{-n}_{s,t}=-(-t)^{-n}\brk[c]{n}_{s,t}$ (14)

for all $n\in\mathbb{N}$. Some negative $(s,t)$-Fibonacci polynomials are

$\displaystyle\brk[c]{-1}_{s,t}$ $\displaystyle=\frac{1}{t},$
$\displaystyle\brk[c]{-2}_{s,t}$ $\displaystyle=-\frac{s}{t^{2}},$
$\displaystyle\brk[c]{-3}_{s,t}$ $\displaystyle=\frac{1}{t^{3}}(s^{2}+t),$
$\displaystyle\brk[c]{-4}_{s,t}$ $\displaystyle=-\frac{1}{t^{4}}(s^{3}+2st).$

Negative Fibonacci polynomials $\brk[c]{-n}_{s,t}$ have negative exponents in
the variable $t$. This follows easily by using the expansion for generalized
Fibonacci polynomials found by Amdeberhan et al. ([33], Prop. 2.1), i.e.,

$-(-t)^{-n}\brk[c]{n}_{s,t}=(-1)^{n+1}\sum_{k\geq
0}\binom{n-k-1}{k}s^{n-2k-1}t^{k-n}.$

For $t=0$, the negative $(s,0)$-Fibonacci polynomials are
$\brk[c]{-n}_{s,0}=s^{-n-1}$, for all $n\in\mathbb{N}$. We will now give a
formula for the factorial of a negative Fibonacci polynomial, which will be
used later.

######  Proposition 1.

For $t\neq 0$, the negative $(s,t)$-Fibotorial polynomial is

$\displaystyle\brk[c]{-n}_{s,t}!=(-1)^{-\binom{n}{2}}t^{-\binom{n+1}{2}}\brk[c]{n}_{s,t}!.$

The $(s,0)$-Fibotorial monomial is

$\brk[c]{-n}_{s,0}!=s^{-\binom{n+1}{2}-1}.$

###### Proof.

By using Proposition 14 we obtain that

$\displaystyle\brk[c]{-n}_{s,t}!$
$\displaystyle=\brk[c]{-n}_{s,t}\brk[c]{-n+1}_{s,t}\cdots\brk[c]{-1}_{s,t}$
$\displaystyle=(-(-t)^{-n}\brk[c]{n}_{s,t})(-(-t)^{-n+1}\brk[c]{n-1}_{s,t})\cdots(-(-1)^{-1}\brk[c]{1}_{s,t})$
$\displaystyle=(-1)^{n}(-t)^{-(n+n-1+\cdots+1)}\brk[c]{n}_{s,t}!$
$\displaystyle=(-1)^{n}(-1)^{-\binom{n+1}{2}}t^{-\binom{n+1}{2}}\brk[c]{n}_{s,t}$
$\displaystyle=(-1)^{-\binom{n}{2}}t^{-\binom{n+1}{2}}\brk[c]{n}_{s,t}!.$

The last identity follows directly from the definition of negative
$(s,0)$-Fibonacci polynomials. ∎

From Eq. (11) we obtain that

$\brk[c]{n}_{s,t}=\varphi_{s,t}^{n-1}\frac{1-\left(\frac{\varphi_{s,t}^{\prime}}{\varphi_{s,t}}\right)^{n}}{1-\left(\frac{\varphi_{s,t}^{\prime}}{\varphi_{s,t}}\right)}.$

If we set $Q=\frac{\varphi_{s,t}^{\prime}}{\varphi_{s,t}}$, then

$\brk[c]{n}_{s,t}=\varphi_{s,t}^{n-1}\frac{1-Q^{n}}{1-Q}=\varphi_{s,t}^{n-1}\brk[s]{n}_{Q},$
(15)

where $\brk[s]{n}_{Q}=\frac{1-Q^{n}}{1-Q}$, from which the relationship
between $(s,t)$-Fibonacci polynomials and the $Q$-numbers is clear. Throughout
this paper we use this value for $Q$. Eq. (15) implies the following
identities

$\displaystyle\brk[c]{n}_{s,t}!$ $\displaystyle=$
$\displaystyle\varphi_{s,t}^{\binom{n}{2}}\brk[s]{n}_{Q}!,=\varphi_{s,t}^{\prime\binom{n}{2}}\brk[s]{n}_{1/Q}!,$
(16) $\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}_{s,t}$ $\displaystyle=$
$\displaystyle\varphi_{s,t}^{k(n-k)}\binom{n}{k}_{Q}=\varphi_{s,t}^{\prime
k(n-k)}\binom{n}{k}_{1/Q}.$ (17)

When $u=\varphi_{s,t}$, then

$\displaystyle\varphi_{s,t}^{n-1}\brk[s]{n}_{Q}$
$\displaystyle=\varphi_{s,t}^{n-1}\brk[c]{n}_{1+Q,-Q}$
$\displaystyle=\brk[c]{n}_{(1+Q)\varphi,-Q\varphi^{2}}$
$\displaystyle=\brk[c]{n}_{\varphi+\varphi^{\prime},-\varphi\varphi^{\prime}}$
$\displaystyle=\brk[c]{n}_{s,t}$

and accordingly $\brk[c]{n}_{s,t}$ is a $\varphi_{s,t}$-deformation of
$\brk[s]{n}_{Q}$.

On the other hand, the Chebysheff polynomial of the second kind $U_{n}(t)$ is
a polynomial of degree $n$ in the variable $t$ defined by

$U_{n}(t)=\frac{\sin(n+1)\theta}{\sin\theta}$

where $t=\cos\theta$. The polynomials $U_{n}(t)$ satisfy the recurrence
relation

$U_{n}(t)=2tU_{n-1}(t)-U_{n-2}(t)$

together with the initial conditions $U_{0}(t)=1$ and $U_{1}(t)=2t$. Then the
polynomials $U_{n}(t)$ are $(2t,-1)$-Fibonacci polynomials with
$\brk[c]{n}_{2t,-1}=U_{n-1}(t)$, where $\brk[c]{0}_{2t,-1}=U_{-1}(t)=0$. Then
an explicit expression for Chebysheff polynomials of the second kind is by
means of Binet’s form

$U_{n-1}(t)=\frac{(t+\sqrt{t^{2}-1})^{n}-(t-\sqrt{t^{2}-1})^{n}}{2\sqrt{t^{2}-1}}.$
(18)

Then we can use this representation to express the polynomials
$\brk[c]{n}_{s,t}$ in terms of Chebysheff polynomials of the second kind. For
$t\neq 0$ and $s\neq\pm 2i\sqrt{t}$, the $(s,t)$-Fibonacci polynomials are
related to the Chebysheff polynomials in the following way [34]

$\displaystyle\brk[c]{n}_{s,t}$
$\displaystyle=(-i\sqrt{t})^{n-1}U_{n-1}\left(\frac{is}{2\sqrt{t}}\right).$
(19)

Let $\theta_{s,t}$ denote the function

$\theta_{s,t}=\arccos\left(\frac{is}{2\sqrt{t}}\right).$

Then a trigonometric expression for $(s,t)$-Fibonacci polynomials is

$\brk[c]{n}_{s,t}=(-i\sqrt{t})^{n-1}\frac{\sin(n\theta_{s,t})}{\sin(\theta_{s,t})}=\frac{2(-i)^{n-1}(\sqrt{t})^{n}}{\sqrt{s^{2}+4t}}\sin(n\theta_{s,t}).$

Thus the $(s,t)$-Fibotorial and the $(s,t)$-Fibonomial polynomials can be
expressed as

$\displaystyle\brk[c]{n}_{s,t}!$
$\displaystyle=(-i\sqrt{t})^{\binom{n}{2}}U_{n-1}(t)!=(-i\sqrt{t})^{\binom{n}{2}}\frac{\prod_{k=1}^{n}\sin(k\theta_{s,t})}{\sin^{n}(\theta_{s,t})}$

and

$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}_{s,t}$
$\displaystyle=(-i\sqrt{t})^{k(n-k)}\frac{\prod_{j=k+1}^{n}\sin(j\theta_{s,t})}{\prod_{j=1}^{n-k}\sin(j\theta_{s,t})}$

respectively. When $u=i/\sqrt{t}$, then

$\displaystyle(i/\sqrt{t})^{-n+1}U_{n-1}(is/2\sqrt{t})$
$\displaystyle=(i/\sqrt{t})^{-n+1}\brk[c]{n}_{2(is/2\sqrt{t}),-1}$
$\displaystyle=\brk[c]{n}_{2(is/2\sqrt{t})(i/\sqrt{t})^{-1},-(i/\sqrt{t})^{-2}}$
$\displaystyle=\brk[c]{n}_{s,t}$

and thus $\brk[c]{n}_{s,t}$ is an $(i/\sqrt{t})^{-1}$-deformation of
$U_{n-1}(is/2\sqrt{t})$. Moreover, it follows from Eq. (11) that
$\brk[c]{n}_{\pm 2i\sqrt{t},t}$ is a $(\pm i\sqrt{t})$-deformation of $n$.

##  3 The Ward ring of Fibonomial exponential generating functions

Now let $\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[x]]$ denote the set of
$(s,t)$-exponential generating functions of the form
$\sum_{n=0}^{\infty}a_{n}(x^{n}/\brk[c]{n}_{s,t}!)$ with coefficients in
$\mathbb{C}$. It is clear that
$(\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[x]],+,\cdot)$ is a ring with
sum and product ordinary of series, that is,

$f(x)+g(x)=\sum_{n=0}^{\infty}(a_{n}+b_{n})\frac{x^{n}}{\brk[c]{n}_{s,t}!}$

and

$f(x)\cdot
g(x)=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}_{s,t}a_{k}b_{n-k}\frac{x^{n}}{\brk[c]{n}_{s,t}!},$

where $f(x)=\sum_{n=0}^{\infty}a_{n}(x^{n}/\brk[c]{n}_{s,t}!)$,
$g(x)=\sum_{n=0}^{\infty}b_{n}(x^{n}/\brk[c]{n}_{s,t}!)\in\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[x]]$.

######  Definition 3.

The ring $\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ will be called
generalized Fibonomial ring of Ward of $(s,t)$-exponential generating
functions, or $(s,t)$-Ward ring,

$\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{\brk[c]{n}_{s,t}!}$

with coefficient in $\mathbb{C}$.

In particular, we will obtain $(s,t)$-Ward rings according to each major
specialization of $s$ and $t$.

######  Definition 4.

$(\mathop{\mbox{{W}}}\nolimits_{F,\mathbb{C}}[[z]],+,\cdot)$ is the Ward-
Fibonacci ring of $F$-exponential generating functions with coefficients in
$\mathbb{C}$

$\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{F_{n}!}.$

######  Definition 5.

$(\mathop{\mbox{{W}}}\nolimits_{P,\mathbb{C}}[[z]],+,\cdot)$ is the Ward-Pell
ring of $P$-exponential generating functions with coefficients in $\mathbb{C}$

$\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{P_{n}!}.$

######  Definition 6.

$(\mathop{\mbox{{W}}}\nolimits_{J,\mathbb{C}}[[z]],+,\cdot)$ is the Ward-
Jacobsthal ring of $J$-exponential generating functions with coefficients in
$\mathbb{C}$

$\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{J_{n}!}.$

######  Definition 7.

$(\mathop{\mbox{{W}}}\nolimits_{p,q,\mathbb{C}}[[z]],+,\cdot)$ is the
$(p,q)$-Ward ring of $(p,q)$-exponential generating functions with
coefficients in $\mathbb{C}$

$\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{\brk[s]{n}_{p,q}!}.$

Finally, we have the following definition.

######  Definition 8.

$(\mathop{\mbox{{W}}}\nolimits_{U,\mathbb{C}}[[z]],+,\cdot)$ is the Ward-
Chebysheff ring of $U$-exponential generating functions of second kind with
coefficients in $\mathbb{C}$

$a_{0}+\sum_{n=1}^{\infty}a_{n}\frac{\sin^{n}(\theta)}{\sin(\theta)\sin(2\theta)\cdots\sin(n\theta)}z^{n}.$

Other specializations will not be taken into account for the remainder of this
paper. When $s\rightarrow\pm 2i\sqrt{t}$, the Ward Fibonomial ring reduces to
the Hurwitz ring [14] of series

$f_{\pm 2i\sqrt{t},t}(z)=\sum_{n=0}^{\infty}a_{n}(1/\pm
i\sqrt{t})^{\binom{n}{2}}\frac{z^{n}}{n!}.$

Accordingly, the Hurwitz ring of exponential generating functions is the
degenerate Ward Fibonomial ring. On the other hand, the series
$\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{\brk[s]{n}_{q}!}$ is an Eulerian
generating functions. Then Definition 3 is a generalization of the exponential
and Eulerian generating functions and we can, therefore, think of applications
in enumerative combinatorics. Next, we introduce the positive and negative
deformed functions in the ring
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$.

######  Definition 9.

Suppose $s\neq 0$ and $t\neq 0$. A positive deformed function $f_{s,t}(z,u)$,
or simply deformed, in $\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ is
a function of the form

$f_{s,t}(z,u)=\begin{cases}\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,t}!},&\text{
if }u\neq 0;\\\ a_{0}+a_{1}z,&\text{ if }u=0.\end{cases}$

A negative deformed function $f_{s,t}^{-}(z,u)$ is a function of the form

$f_{s,t}^{-}(z,u)=\begin{cases}\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{-n}_{s,t}!},&\text{
if }u\neq 0;\\\ a_{0}+ta_{1}z,&\text{ if }u=0.\end{cases}$

When $s\neq 0$ and $t=0$, then

$\displaystyle f_{s,0}(z,1)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{\brk[c]{n}_{s,0}!}=\sum_{n=0}^{\infty}a_{n}\frac{z^{n}}{s^{\binom{n}{2}}}=f_{1,0}(z,s^{-1}).$
$\displaystyle f_{s,0}(z,u)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,0}!}=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{s^{\binom{n}{2}}}=f_{1,0}(z,us^{-1}).$
$\displaystyle f_{s,0}^{-}(z,u)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{-n}_{s,0}!}=s\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{s^{-\binom{n+1}{2}}}=sf_{1,0}(sz,us).$

In the following result we establish the relationship between the positive and
negative deformed functions.

######  Proposition 2.

For all $u\in\mathbb{C}$ and for $t\neq 0$ the positive and negative deformed
functions are related in the following way

$\displaystyle f_{s,t}(z,-u)$ $\displaystyle=f_{s,t}^{-}(z/t,u/t).$

###### Proof.

When $u=0$, done. If $u\neq 0$, we will use Proposition 1. We have that

$\displaystyle f_{s,t}(z,-u)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}(-1)^{\binom{n}{2}}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,t}!}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{t^{-\binom{n+1}{2}}}{(-1)^{-\binom{n}{2}}t^{-\binom{n+1}{2}}}\frac{z^{n}}{\brk[c]{n}_{s,t}!}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\left(-\frac{u}{t}\right)^{\binom{n}{2}}t^{-n}\frac{z^{n}}{\brk[c]{-n}_{s,t}!}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}\left(-\frac{u}{t}\right)^{\binom{n}{2}}\frac{(z/t)^{n}}{\brk[c]{-n}_{s,t}!}=f_{s,t}^{-}(z/t,u/t)$

and we obtain the desired result. ∎

From the above proposition it follows that
$f_{s,t}^{-}(z,u)\in\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$. For
$u=1$, suppose that there does not exist $v\in\mathbb{C}$ such that
$v^{\binom{n}{2}}\mid a_{n}$ for all $n\geq 0$ and $v\neq 0$. Then we will
make $f_{s,t}(z)=f_{s,t}(z,1)$.

######  Proposition 3.

For all $v\in\mathbb{C}$ non-zero

$f_{sv,tv^{2}}(z,u)=f_{s,t}(z,u/v).$

Thus,

$\displaystyle f_{su,tu^{2}}(z,u)$ $\displaystyle=f_{s,t}(z).$

###### Proof.

The proof consists of $v$-deforming the polynomials $\brk[c]{n}_{s,t}$. This
is

$\displaystyle f_{sv,tv^{2}}(z,u)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{sv,tv^{2}}!}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{v^{\binom{n}{2}}\brk[c]{n}_{s,t}!}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}(u/v)^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,t}!}$
$\displaystyle=f_{sv,tv^{2}}(z,u/v)$

as expected. ∎

For example, if $s\neq\pm 2i\sqrt{t}$ and
$Q=\varphi_{s,t}^{\prime}/\varphi_{s,t}$, we can express every function in
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ as a $Q$-exponential
generating function

$\displaystyle f_{s,t}(z,u)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,t}!}$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}(u\varphi_{s,t}^{-1})^{\binom{n}{2}}\frac{z^{n}}{\brk[s]{n}_{q}!}$
$\displaystyle=f_{s\varphi^{-1},t\varphi^{-2}}(z,u\varphi_{s,t}^{-1})$
$\displaystyle=f_{1+Q,-Q}(z,u\varphi_{s,t}^{-1})$
$\displaystyle=f_{q}(z,u\varphi_{s,t}^{-1})$

and for $t=is/2\sqrt{r}$ we can express every function in
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ as a $U(t)$-exponential
generating function

$\displaystyle f_{s,r}(z,u)$
$\displaystyle=\sum_{n=0}^{\infty}a_{n}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,r}!}$
$\displaystyle=a_{0}+\sum_{n=1}^{\infty}a_{n}u^{\binom{n}{2}}(-i\sqrt{r})^{-\binom{n}{2}}\frac{\sin^{n}(\theta_{s,r})}{\sin(\theta_{s,r})\sin(2\theta_{s,r})\cdots\sin(n\theta_{s,r})}z^{n}$
$\displaystyle=f_{s(-i\sqrt{r})^{-1},r(-i\sqrt{r})^{-2}}(z,u(-i\sqrt{r})^{-1})$
$\displaystyle=f_{is/\sqrt{r},-1}(z,iu/\sqrt{r})$
$\displaystyle=f_{2(is/2\sqrt{r}),-1}(z,iu/\sqrt{r}).$

Then
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]=\mathop{\mbox{{W}}}\nolimits_{Q,\mathbb{C}}[[z]]=\mathop{\mbox{{W}}}\nolimits_{U,\mathbb{C}}[[z]]$.

##  4 The deformed $(s,t)$-derivative

In this section we introduce the deformed differential operator. In addition,
we introduce the set of $\varphi_{s,t}$-periodic functions and it will be
shown that these functions are invariant by $u$-deformations.

######  Definition 10.

For all $u\in\mathbb{C}$ and for all $s\neq\pm 2i\sqrt{t}$, define the
$u$-deformed $(s,t)$-derivative $\mathbf{D}_{su,tu^{2}}$ of function $f(x)$ as

$(\mathbf{D}_{su,tu^{2}}f)(x)=\frac{f(u\varphi_{s,t}x)-f(u\varphi_{s,t}^{\prime}x)}{u(\varphi_{s,t}-\varphi_{s,t}^{\prime})x}$

for all $x\neq 0$ and $(\mathbf{D}_{us,u^{2}t}f)(0)=f^{\prime}(0)$, provided
$f^{\prime}(0)$ exists.

The $u$-deformed $(s,t)$-derivative is a particular case of the
$(p,q)$-derivative with $p=u\varphi_{s,t}$ and $q=u\varphi_{s,t}^{\prime}$.
Next, we will give the definitions of $u$-deformed derivative according to
each specialization.

######  Definition 11.

When $s=1,\ t=1$, we get the $u$-deformed Fibonacci derivative

$\mathbf{D}_{u,u^{2}}f(x)=\mathbf{D}_{F(u)}f(x)=\begin{cases}\frac{f\left(u\varphi
x\right)-f\left(u\varphi^{\prime}x\right)}{u\sqrt{5}x},&\text{ if }x\neq 0;\\\
f^{\prime}(0),&\text{ if }x=0.\end{cases}$

######  Definition 12.

When $s=2,\ t=1$, we get the $u$-deformed Pell derivative

$\mathbf{D}_{2u,u^{2}}f(x)=\mathbf{D}_{P(u)}f(x)=\begin{cases}\frac{f\left(u(1+\sqrt{2})x\right)-f\left(u(1-\sqrt{2})x\right)}{2u\sqrt{2}x},&\text{
if }x\neq 0;\\\ f^{\prime}(0),&\text{ if }x=0.\end{cases}$

######  Definition 13.

When $s=1,\ t=2$, we get the $u$-deformed Jacobsthal derivative

$\mathbf{D}_{u,2u^{2}}f(x)=\mathbf{D}_{J(u)}f(x)=\begin{cases}\frac{f\left(2ux\right)-f\left(-ux\right)}{3ux},&\text{
if }x\neq 0;\\\ f^{\prime}(0),&\text{ if }x=0.\end{cases}$

######  Definition 14.

When $s=p+q,\ t=-pq$, we get the $u$-deformed $(p,q)$-derivative

$\mathbf{D}_{u(p+q),-pqu^{2}}f(x)=\mathbf{D}_{Q(u)}f(x)=\begin{cases}\frac{f\left(upx\right)-f\left(uqx\right)}{u(p-q)x},&\text{
if }x\neq 0;\\\ f^{\prime}(0),&\text{ if }x=0.\end{cases}$

######  Definition 15.

When $s=2r,\lvert r\rvert>1,\ t=-1$, we get the $u$-deformed Chebysheff of
second kind derivative

$\mathbf{D}_{2ru,-u^{2}}f(x)=\mathbf{D}_{U(u)}f(x)=\begin{cases}\frac{f\left(u(r+\sqrt{r^{2}-1})x\right)-f\left(u(r-\sqrt{r^{2}-1})x\right)}{2u\sqrt{r^{2}-1}x},&\text{
if }x\neq 0;\\\ f^{\prime}(0),&\text{ if }x=0.\end{cases}$

For two fixed complex numbers $s,t$ the ring
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ is equipped with the
family of derivatives $\mathbf{D}_{su,tu^{2}}$, for every nonzero complex
number $u$. In particular, in the ring
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$, we define the following
derivatives

$\mathbf{D}_{q}=\mathbf{D}_{1+Q,-Q}=\mathbf{D}_{s(1/\varphi_{s,t}),t(1/\varphi_{s,t})^{2}}$

and

$\mathbf{D}_{U}=\mathbf{D}_{2(is/2\sqrt{t}),-1}=\mathbf{D}_{s(i/\sqrt{t}),t(i/\sqrt{t})^{2}}.$

The following results on the $u$-deformed $(s,t)$-derivative are standard.

  1. 1.

$\mathbf{D}_{su,tu^{2}}(f(x)+g(x))=(\mathbf{D}_{su,tu^{2}}f)(x)+(\mathbf{D}_{su,tu^{2}}g)(x)$.

  2. 2.

$\mathbf{D}_{su,tu^{2}}(\alpha f(x))=\alpha(\mathbf{D}_{su,tu^{2}}f)(x)$, for
all $\alpha\in\mathbb{C}$.

  3. 3.

$\displaystyle\mathbf{D}_{su,tu^{2}}(f(x)g(x))$
$\displaystyle=f(u\varphi_{s,t}x)(\mathbf{D}_{su,tu^{2}}g)(x)+g(u\varphi_{s,t}^{\prime}x)(\mathbf{D}_{su,tu^{2}}f)(x),$
$\displaystyle=f(u\varphi_{s,t}^{\prime}x)(\mathbf{D}_{su,tu^{2}}g)(x)+g(u\varphi_{s,t}x)(\mathbf{D}_{su,tu^{2}}f)(x).$

######  Definition 16.

We will say that the function $f(x)$ is $Q$-periodic if $f(y)=f(Qy)$, with
$y=\varphi_{s,t}x$. Let $\mathbb{P}_{s,t}$ denote the set of $Q$-periodic
functions. The $Q$-periodic functions form the kernel of the operator
$\mathbf{D}_{s,t}$.

Set $s\neq 0$ and $t<0$. From the condition of $Q$-periodicity of $f(x)$ it
follows that $f(y)=f(Qy)$, with $Q>0$. Then

$\displaystyle f(Q^{y})$ $\displaystyle=f(Q^{y+1})$ $\displaystyle G(y)$
$\displaystyle=G(y+1)$

where $G$ is an arbitrary periodic function with period one and
$f(x)=G(\log_{Q}(x))$, $x>0$. Thus,

$\displaystyle\mathbf{D}_{s,t}f(x)$
$\displaystyle=\frac{G(\log_{Q}(\varphi_{s,t}x))-G(\log_{Q}(\varphi_{s,t}^{\prime}x))}{(\varphi_{s,t}-\varphi_{s,t}^{\prime})x}$
$\displaystyle=\frac{G(\log_{Q}(\varphi_{s,t}x))-G(\log_{Q}(Q\varphi_{s,t}x))}{(\varphi_{s,t}-\varphi_{s,t}^{\prime})x}$
$\displaystyle=\frac{G(\log_{Q}(\varphi_{s,t}x))-G(\log_{Q}(\varphi_{s,t}x))}{(\varphi_{s,t}-\varphi_{s,t}^{\prime})x}=0$

and $f(x)\in\mathbb{P}_{s,t}$. If $t>0$, then $Q<0$ and
$f(x)=G(\log(x)/(\log(-Q)+i\pi))$, so that $x\in\mathbb{C}/\\{0\\}$ and
$f(x)\in\mathbb{P}_{s,t}$. As
$Q=\frac{\varphi_{su,tu^{2}}^{\prime}}{\varphi_{su,tu^{2}}}=\frac{\varphi_{s,t}^{\prime}}{\varphi_{s,t}}$,
every function in $\mathbb{P}_{s,t}$ is invariant by $u$-deformations. Thus
$\mathbb{P}_{s,t}=\mathbb{P}_{su,tu^{2}}$. Now, set $s\neq 0$ and $t=0$. Then
$\mathbb{P}_{s,0}=\mathbb{C}$.

In the following theorem we will show that the set of $\varphi_{s,t}$-periodic
functions is an algebra.

######  Theorem 1.

The set $\mathbb{P}_{s,t}$ is a $\mathbb{C}$-algebra.

###### Proof.

If $f(x),g(x)\in\mathbb{P}_{s,t}$, then

$\displaystyle\mathbf{D}_{s,t}(f(x)+g(x))$ $\displaystyle=0$
$\displaystyle\mathbf{D}_{s,t}(f(x)g(x))$ $\displaystyle=0$
$\displaystyle\mathbf{D}_{s,t}(\alpha f(x))$ $\displaystyle=0$

for all $\alpha\in\mathbb{C}$. Then $\mathbb{P}_{s,t}$ is closed with respect
to sum, product of functions and product by scalar. Now the properties of
$\mathbb{C}$-algebra follow easily. ∎

If $f(x)\in\mathbb{P}_{s,t}$, then

$\mathbf{D}_{s,t}(f(x)g(x))=f(\varphi_{s,t}x)(\mathbf{D}_{s,t}g)(x)=f(\varphi_{s,t}^{\prime}x)(\mathbf{D}_{s,t}g)(x).$

In $\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[x]]$ the only function
satisfying $(\mathbf{D}_{s,t}f)(x)=0$ is the constant function $f(x)=C$. Then
$\mathbb{P}_{s,t}\cap\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[x]]=\mathbb{C}$.

##  5 Deformed $(s,t)$-exponential function

######  Definition 17.

Set $s\neq 0$. For all $u\in\mathbb{C}$, we define the $u$-deformed
$(s,t)$-exponential function in
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ as

$\exp_{s,t}(z,u)=\begin{cases}\sum_{n=0}^{\infty}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{n}_{s,t}!}&\text{
if }u\neq 0;\\\ 1+z&\text{ if }u=0.\end{cases}$

Also, we define

$\displaystyle\exp_{s,t}(z)$ $\displaystyle=\exp_{s,t}(z,1),$
$\displaystyle\exp_{s,t}^{\prime}(z)$ $\displaystyle=\exp_{s,t}(z,-t),$
$\displaystyle\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z)$
$\displaystyle=\exp_{s,t}(z,\varphi_{s,t}),$
$\displaystyle\mathop{\mbox{{Exp}}}\nolimits^{\prime}_{s,t}(z)$
$\displaystyle=\exp_{s,t}(z,\varphi^{\prime}_{s,t}).$

When $s\neq 0$ and $t=0$,

$\exp_{s,0}(z,u)=\sum_{n=0}^{\infty}\left(\frac{u}{s}\right)^{\binom{n}{2}}z^{n}.$
(20)

If $u=s$

$\mathop{\mbox{{Exp}}}\nolimits_{s,0}(z)=\frac{1}{1-z}\text{ and
}\mathop{\mbox{{Exp}}}\nolimits_{s,0}^{\prime}(z)=1+z.$ (21)

######  Theorem 2.

For $s\neq 0,t\neq 0$ and $|-t/\varphi_{s,t}^{2}|<1$, the function
$\exp_{s,t}(z,u)$ is

  1. 1.

an entire function if $|u|<|\varphi_{s,t}|$,

  2. 2.

convergent in the disk $|z|<|\varphi_{s,t}|/|\sqrt{s^{2}+4t}|$ when
$|u|=|\varphi_{s,t}|$,

  3. 3.

convergent in $z=0$ when $|u|>|\varphi_{s,t}|$.

Suppose that $|-t/\varphi_{s,t}^{2}|>1$.

  1. 4.

If $|u|\leq|\varphi_{s,t}|$, then $\exp_{s,t}(z,u)$ is entire.

  2. 5.

If $|u|>|\varphi_{s,t}|$, then $\exp_{s,t}(z,u)$ converge in $z=0$.

###### Proof.

If $t\neq 0$ and $|-t|/|\varphi_{s,t}^{2}|<1$, then

$\displaystyle\lim_{n\rightarrow\infty}\bigg{|}\frac{u^{n}}{\brk[c]{n+1}_{s,t}}\bigg{|}$
$\displaystyle=\lim_{n\rightarrow\infty}\bigg{|}\frac{\sqrt{s^{2}+4t}u^{n}}{\varphi_{s,t}^{n+1}-(-t)^{n+1}\varphi_{s,t}^{-n-1}}\bigg{|}$
$\displaystyle=\bigg{|}\frac{\sqrt{s^{2}+4t}}{\varphi_{s,t}}\Bigg{|}\lim_{n\rightarrow\infty}\bigg{|}\frac{1}{1-(-t)^{n+1}\varphi_{s,t}^{-2n-2}}\bigg{|}\bigg{|}\frac{u^{n}}{\varphi_{s,t}^{n}}\bigg{|}$
$\displaystyle=\Bigg{|}\frac{\sqrt{s^{2}+4t}}{\varphi_{s,t}}\Bigg{|}\lim_{n\rightarrow\infty}\bigg{|}\frac{u^{n}}{\varphi_{s,t}^{n}}\bigg{|}.$

Then follow the first 3 statements. Now suppose that
$|-t/\varphi_{s,t}^{2}|>1$ and set

$\displaystyle\lambda_{n}=\bigg{|}\frac{1}{1-(-t)^{n+1}/\varphi_{s,t}^{2n+2}}\bigg{|}\bigg{|}\frac{u^{n}}{\varphi_{s,t}^{n}}\bigg{|}.$

If $|u|\leq|\varphi_{s,t}|$, then $\lim_{n\rightarrow\infty}\lambda_{n}=0$.
Thus follows the fourth statement. If $|u|>|\varphi_{s,t}|$, then
$\lim_{n\rightarrow\infty}\lambda_{n}=\infty$ and $\exp_{s,t}(z,u)$ has radius
of convergence 0. ∎

It is very straightforward to obtain the following result.

######  Theorem 3.

For $s\neq 0,t=0$ the function $\exp_{s,0}(az,u)$ is

  1. 1.

an entire function if $|u|<|s|$,

  2. 2.

convergent in the disk $|z|<1/|a|$ when $|u|=|s|$,

  3. 3.

convergent in $z=0$ when $|u|>|s|$.

######  Definition 18.

For all $t\neq 0$, and all complex $u$, define the negative deformed
$(s,t)$-exponential function in
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$ as

$\exp_{s,t}^{-}(z,u)\begin{cases}\sum_{n=0}^{\infty}u^{\binom{n}{2}}\frac{z^{n}}{\brk[c]{-n}_{s,t}!},&\text{
if }u\neq 0;\\\ 1+tz,&\text{ if }u=0.\end{cases}$

######  Proposition 4.

For all $u\in\mathbb{C}$ and $t\neq 0$ the deformed $(s,t)$-exponential
functions are related in the following way

$\displaystyle\exp_{s,t}(z,-u)$ $\displaystyle=\exp_{s,t}^{-}(z/t,u/t).$

###### Proof.

Follows from Proposition 2. ∎

######  Proposition 5.

For $u\neq 0$ we have

$\displaystyle\exp_{s,t}(z,u)$ $\displaystyle=\exp_{s/u,t/u^{2}}(z),$
$\displaystyle\exp_{s,t}^{-}(z,-u)$ $\displaystyle=\exp_{s/tu,1/tu^{2}}(tz).$

Then $\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z)=e_{q}^{z}$ and
$\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(z)=E_{q}^{z}$.

###### Proof.

Follows from Proposition 3. ∎

######  Theorem 4.

The deformed derivatives of the positive and negative deformed
$(s,t)$-exponential functions are

  1. 1.

$\mathbf{D}_{sv,tv^{2}}(\exp_{s,t}(z,u))=\exp_{s,t}(uvz,u)$ and

  2. 2.

$\mathbf{D}_{sv,tv^{2}}(\exp_{s,t}^{-}(z,-u))=t\exp_{s,t}^{-}(tuvz,-u)$.

###### Proof.

We prove 1 and then we use this result to prove 2. We have that

$\displaystyle\mathbf{D}_{sv,tv^{2}}(\exp_{s,t}(z,u))$
$\displaystyle=\mathbf{D}_{sv,tv^{2}}\left(\sum_{n=0}^{\infty}u^{\binom{n}{2}}\frac{z^{n}}{v^{-\binom{n}{2}}\brk[c]{n}_{sv,tv^{2}}!}\right)$
$\displaystyle=\sum_{n=0}^{\infty}u^{\binom{n}{2}}\frac{\mathbf{D}_{sv,tv^{2}}\left(z^{n}\right)}{v^{-\binom{n}{2}}\brk[c]{n}_{sv,tv^{2}}!}$
$\displaystyle=\sum_{n=1}^{\infty}u^{\binom{n}{2}}\frac{\brk[c]{n}_{sv,tv^{2}}z^{n-1}}{v^{-\binom{n}{2}}\brk[c]{n}_{sv,tv^{2}}!}$
$\displaystyle=\sum_{n=1}^{\infty}u^{\binom{n}{2}}\frac{z^{n-1}}{v^{-\binom{n}{2}}\brk[c]{n-1}_{sv,tv^{2}}!}$
$\displaystyle=\sum_{n=0}^{\infty}u^{\binom{n}{2}}u^{n}\frac{z^{n}}{v^{-\binom{n}{2}}v^{-n}\brk[c]{n}_{sv,tv^{2}}!}=\exp_{s,t}(uvz,u).$

The proof of 2 is as follows:

$\displaystyle\mathbf{D}_{sv,tv^{2}}(\exp_{s,t}^{-}(z,-u))$
$\displaystyle=\mathbf{D}_{sv,tv^{2}}(\exp_{s,t}(tz,tu))$
$\displaystyle=t\exp_{s,t}(t^{2}uvz,tu)$
$\displaystyle=t\exp_{s,t}^{-}(tuvz,-u).$

∎

######  Corollary 1.

The $Q$-derivative of the deformed $(s,t)$-exponential function is

$\displaystyle\mathbf{D}_{q}(\exp_{s,t}(z,u))$
$\displaystyle=\exp_{s,t}(uz/\varphi_{s,t},u).$

In particular

$\displaystyle\mathbf{D}_{q}(\exp_{s,t}(z))$
$\displaystyle=\exp_{s,t}(z/\varphi_{s,t}),$
$\displaystyle\mathbf{D}_{q}(\exp_{s,t}^{\prime}(z))$
$\displaystyle=\exp_{s,t}^{\prime}(-tz/\varphi_{s,t}),$
$\displaystyle\mathbf{D}_{q}(\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z))$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z)$
$\displaystyle\mathbf{D}_{q}(\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(z))$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits^{\prime}_{s,t}(\varphi^{\prime}_{s,t}z/\varphi_{s,t}).$

######  Corollary 2.

The $U$-derivative of the deformed $(s,t)$-exponential functions is

$\displaystyle\mathbf{D}_{U}(\exp_{s,t}(z,u))$
$\displaystyle=\exp_{s,t}(iuz/\sqrt{b},u).$

In particular

$\displaystyle\mathbf{D}_{U}(\exp_{s,t}(z))$
$\displaystyle=\exp_{s,t}(iz/\sqrt{t}),$
$\displaystyle\mathbf{D}_{U}(\exp_{s,t}^{\prime}(z))$
$\displaystyle=\exp_{s,t}(-i\sqrt{t}z),$
$\displaystyle\mathbf{D}_{U}(\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z))$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits_{s,t}(i\varphi_{s,t}z/\sqrt{t})$
$\displaystyle\mathbf{D}_{U}(\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(z))$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits^{\prime}_{s,t}(i\varphi^{\prime}_{s,t}z/\sqrt{t}).$

######  Theorem 5.

Set $s\neq 0$ and $t\neq 0$. If $|Q|<1$, then

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z)=\prod_{k=0}^{\infty}\frac{1}{1-(1-Q)Q^{k}z}.$

If $|Q|>1$, then

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z)=\prod_{k=0}^{\infty}(1-(Q^{-1}-1)Q^{-k}z).$

###### Proof.

As $\mathbf{D}_{s,t}\exp_{s,t}(z,u)=\exp_{s,t}(\varphi_{s,t}z,u)$, then

$\frac{\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}z)-\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}^{\prime}z)}{(\varphi_{s,t}-\varphi_{s,t}^{\prime})x}=\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}z).$

If $|Q|<1$,

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}z)=[1-(\varphi_{s,t}-\varphi_{s,t}^{\prime})z]^{-1}\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}^{\prime}z)$
(22)

and if $|Q|>1$,

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}^{\prime}z)=[1-(\varphi_{s,t}-\varphi_{s,t}^{\prime})z]\mathop{\mbox{{Exp}}}\nolimits_{s,t}(\varphi_{s,t}z).$
(23)

Iterating Eqs. (22) and (23) yields desired results. ∎

The following are the specializations for the infinite product representation
of the $(s,t)$-exponential function:

  1. 1.

$\mathop{\mbox{{Exp}}}\nolimits_{1,1}(z)=\prod_{k=0}^{\infty}\frac{(1+\sqrt{5})^{k+1}}{(1+\sqrt{5})^{k+1}-2\sqrt{5}(1-\sqrt{5})^{k}z}.$

  2. 2.

$\mathop{\mbox{{Exp}}}\nolimits_{2,1}(z)=\prod_{k=0}^{\infty}\frac{(1+\sqrt{2})^{k+1}}{(1+\sqrt{2})^{k+1}-2\sqrt{2}(1-\sqrt{2})^{k}z}.$

  3. 3.

$\mathop{\mbox{{Exp}}}\nolimits_{1,2}(z)=\prod_{k=0}^{\infty}\frac{2^{k+1}}{2^{k+1}-3(-1)^{k}z}.$

  4. 4.

$\mathop{\mbox{{Exp}}}\nolimits_{3,-2}(z)=\prod_{k=0}^{\infty}\frac{1}{1-(z/2^{k+1})}.$

  5. 5.

$\mathop{\mbox{{Exp}}}\nolimits_{2t,-1}(z)=\prod_{k=0}^{\infty}\frac{(t+\sqrt{t^{2}-1})^{k+1}}{(t+\sqrt{t^{2}-1})^{k+1}-2\sqrt{t^{2}-1}(t-\sqrt{t^{2}-1})^{k}z}.$

  6. 6.

$\mathrm{e}_{p,q}(z)=\prod_{k=0}^{\infty}\frac{p^{k+1}}{p^{k+1}-(p-q)q^{k}z}.$

When $s\neq 0$ and $t=0$, then

$\mathop{\mbox{{Exp}}}\nolimits_{s,0}(z)=\frac{1}{1-z}.$

Analogously the following theorem is proved.

######  Theorem 6.

Set $s\neq 0$ and $t\neq 0$. If $|Q|<1$, then

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(z)=\prod_{k=0}^{\infty}(1+(1-Q)Q^{k}z).$
(24)

If $|Q|>1$, then

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(z)=\prod_{k=0}^{\infty}\frac{1}{1+(Q^{-1}-1)Q^{-k}z}.$
(25)

From Theorem 5 and 6 we have the following result.

######  Corollary 3.

Set $s\neq 0$. For all $Q$ with $|Q|<1$ and $|Q|>1$

$\mathop{\mbox{{Exp}}}\nolimits_{s,t}(z)\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-z)=1.$
(26)

##  6 Functional-difference equation $\mathbf{D}_{s,t}f(z)=af(uz)$

######  Theorem 7.

Let $a$ be a complex number and set $s\neq 0$, $t\neq 0$. Then the equation in
difference with proportional delay

$\mathbf{D}_{s,t}f(z)=af(\varphi_{s,t}z).$ (27)

have solution $f(z)=G(\log_{Q}(z))\mathop{\mbox{{Exp}}}\nolimits_{s,t}(az)$
for every $G(\log_{Q}(z))$ in $\mathbb{P}_{s,t}$ and $|z|>0$. If $G(z)\equiv
c\in\mathbb{C}$, then the solutions exists for all $z\in\mathbb{C}$.

###### Proof.

Let $(p_{n}(z))_{n=0}^{\infty}$ be a sequence of functions in
$\mathbb{P}_{s,t}$ and define $f(z)=\sum_{n=0}^{\infty}p_{n}(z)z^{n}$. As
$\mathbf{D}_{s,t}f(z)=af(\varphi_{s,t}z)$ is equivalent to

$\sum_{n=0}^{\infty}p_{n+1}(\varphi_{s,t}z)\brk[c]{n+1}_{s,t}z^{n}=a\sum_{n=0}^{\infty}p_{n}(\varphi_{s,t}z)\varphi_{s,t}^{n}z^{n}$

it follows that

$p_{n+1}(\varphi_{s,t}z)=\frac{a\varphi_{s,t}^{n}p_{n}(\varphi_{s,t}z)}{\brk[c]{n+1}_{s,t}}=\frac{a^{2}\varphi_{s,t}^{2n-1}p_{n-1}(\varphi_{s,t}z)}{\brk[c]{n+1}_{s,t}\brk[c]{n}_{s,t}}=\ldots=\frac{a^{n+1}\varphi_{s,t}^{\binom{n+1}{2}}p_{0}(\varphi_{s,t}z)}{\brk[c]{n+1}_{s,t}!}.$

From which we obtain

$f(z)=p_{0}(z)\sum_{n=0}^{\infty}\varphi_{s,t}^{\binom{n}{2}}a^{n}\frac{z^{n}}{\brk[c]{n}_{s,t}!}=p(z)\mathop{\mbox{{Exp}}}\nolimits_{s,t}(az)$

with $p(z)=p_{0}(z)$. The solution is not unique because it exists for every
function $p\in\mathbb{P}_{s,t}$. However, a solution of the form
$y(z)=q(z)\mathop{\mbox{{Exp}}}\nolimits_{s,t}(az)$, with
$(\mathbf{D}_{s,t}q)(z)\neq 0$, does not exist. To prove this, suppose $y(z)$
is another solution. Then taking $(s,t)$-derivative

$\displaystyle\mathbf{D}_{s,t}(\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-az)y(z))$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-a\varphi_{s,t}^{\prime}z)(\mathbf{D}_{s,t}y)(z)-a\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-a\varphi_{s,t}^{\prime}z)y(\varphi_{s,t}z)$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-a\varphi_{s,t}^{\prime}z)ay(\varphi_{s,t}z)-a\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-a\varphi_{s,t}^{\prime}z)y(\varphi_{s,t}z)$
$\displaystyle=\left(\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-a\varphi_{s,t}^{\prime}z)-a\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-a\varphi_{s,t}^{\prime}z)\right)y(\varphi_{s,t}z)$
$\displaystyle=0.$

Therefore, $\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(-az)y(z)=q(z)$, with
$q(z)\in\mathbb{P}_{s,t}$, and thus
$y(z)=q(z)\mathop{\mbox{{Exp}}}\nolimits_{s,t}(az)$. Here, we have used the
Corollary 3. ∎

######  Theorem 8.

Let $a$ be a complex number and set $s\neq 0$, $t\neq 0$. Then the equation in
difference with proportional delay

$\mathbf{D}_{s,t}f(z)=af(\varphi_{s,t}^{\prime}z)$ (28)

have solution
$f(z)=G(\log_{Q}(z))\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(az)$ for
every $G(\log_{Q}(z))$ in $\mathbb{P}_{s,t}$ and $|z|>0$. If
$G(z)=c\in\mathbb{C}$, then the solutions exists for all $z\in\mathbb{C}$.

######  Theorem 9.

Set $s\neq 0$, $t\neq 0$. For every non-zero complex number $u$ the solutions
of the equation in difference with proportional delay

$\mathbf{D}_{s,t}f(z)=af(uz)$ (29)

are of the form

$\mathrm{e}_{s,t}(a,z,u,G)=\sum_{n=0}^{\infty}u^{\binom{n}{2}}a^{n}G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n}}z\right)\right)\frac{z^{n}}{\brk[c]{n}_{s,t}!}$
(30)

with $G$ a periodic function with period one and $|z|>0$. If $G(z)=c$, then
$f(z)=c\exp_{s,t}(az,u)$.

###### Proof.

We first show that Eq.(30) is a solution of Eq.(29). Taking $(s,t)$-derivative
to Eq.(30), we obtain

$\displaystyle\mathbf{D}_{s,t}f(z)$
$\displaystyle=\mathbf{D}_{s,t}\left(\sum_{n=0}^{\infty}u^{\binom{n}{2}}a^{n}G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n}}z\right)\right)\frac{z^{n}}{\brk[c]{n}_{s,t}!}\right)$
$\displaystyle=\sum_{n=1}^{\infty}u^{\binom{n}{2}}a^{n}G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n-1}}z\right)\right)\frac{z^{n-1}}{\brk[c]{n-1}_{s,t}!}$
$\displaystyle=\sum_{n=0}^{\infty}u^{\binom{n}{2}}u^{n}a^{n+1}G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n}}uz\right)\right)\frac{z^{n}}{\brk[c]{n}_{s,t}!}=af(uz).$

As

$G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n}}z\right)\right)=G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{\prime
n}}z\right)\right)$

for all periodic function $G$ with period one and all $n\in\mathbb{N}$, then

$\sum_{n=0}^{\infty}u^{\binom{n}{2}}a^{n}G\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{\prime
n}}z\right)\right)\frac{z^{n}}{\brk[c]{n}_{s,t}!}$ (31)

are also solutions of Eq.(29). Let $(p_{n}(z))_{n\in\mathbb{N}}$ be a sequence
of functions in $\mathbb{P}_{s,t}$ and suppose that
$f(z)=\sum_{n=0}^{\infty}p_{n}(z)z^{n}$ is solution of Eq. (29). Then
$\mathbf{D}_{s,t}f(z)=af(uz)$ is equivalent to

$\sum_{n=0}^{\infty}p_{n+1}(\varphi_{s,t}z)\brk[c]{n+1}_{s,t}z^{n}=a\sum_{n=0}^{\infty}p_{n}(uz)u^{n}z^{n}$

and it follows that

$p_{n+1}(\varphi_{s,t}z)=\frac{au^{n}p_{n}(uz)}{\brk[c]{n+1}_{s,t}}=\frac{a^{2}u^{2n-1}p_{n-1}(u^{2}z/\varphi_{s,t})}{\brk[c]{n+1}_{s,t}\brk[c]{n}_{s,t}}=\cdots=\frac{a^{n+1}u^{\binom{n+1}{2}}p_{0}(u^{n+1}z/\varphi_{s,t}^{n})}{\brk[c]{n+1}_{s,t}!}.$

From which we obtain

$p_{n}(z)=\frac{a^{n}u^{\binom{n}{2}}p_{0}(u^{n}z/\varphi_{s,t}^{n})}{\brk[c]{n}_{s,t}!}$

and

$f(z)=\sum_{n=0}^{\infty}u^{\binom{n}{2}}a^{n}p_{0}(u^{n}z/\varphi_{s,t}^{n})\frac{z^{n}}{\brk[c]{n}_{s,t}!}.$

∎

Next we will give some properties of the functions
$\mathrm{e}_{s,t}(a,z,u,p)$.

######  Proposition 6.

Set $s\neq 0$ and $t\neq 0$. For all $a,u\in\mathbb{C}$ and
$p,q\in\mathbb{P}_{s,t}$

  1. 1.

$\mathrm{e}_{s,t}(a,z,u,0)=0$.

  2. 2.

$\mathrm{e}_{s,t}(a,z,u,c)=c\exp_{s,t}(az,u)$.

  3. 3.

$\mathrm{e}_{s,t}(a,z,\varphi_{s,t},p)=p(z)\mathop{\mbox{{Exp}}}\nolimits_{s,t}(az)$.

  4. 4.

$\mathrm{e}_{s,t}(a,z,\varphi_{s,t}^{\prime},p)=p(z)\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(az)$.

  5. 5.

$\mathrm{e}_{s,t}(a,z,u,p+q)=\mathrm{e}_{s,t}(a,z,u,p)+\mathrm{e}_{s,t}(a,z,u,q)$.

  6. 6.

$\mathrm{e}_{s,t}(a,z,Q^{-m}u,p)=e_{Q^{m}s,Q^{2m}t}(a,z,u,p)$, for all
$m\in\mathbb{Z}$.

###### Proof.

By direct application of Eq. (30). ∎

From the previous proposition it follows that the function
$\mathrm{e}_{s,t}(a,z,u,p)$ generalizes to the $(s,t)$-exponential functions
$\mathop{\mbox{{Exp}}}\nolimits_{s,t}(az,u)$ and
$\mathop{\mbox{{Exp}}}\nolimits_{s,t}^{\prime}(az,u)$. In the following
theorem analytic properties of $\mathrm{e}_{s,t}(a,z,u,p)$ are shown.

######  Theorem 10.

Suppose $p$ is a continuous periodic function with period one. Set $s,t$ such
that $\brk[c]{n}_{s,t}>0$ for all $n>0$ and suppose that
$|-t/\varphi_{s,t}^{2}|<1$. Then the function $\mathrm{e}_{s,t}(a,z,u,p)$ is

  1. 1.

an entire function if $|u|<|\varphi_{s,t}|$,

  2. 2.

convergent in the disk $|z|<|\varphi_{s,t}|/|\sqrt{s^{2}+4t}|$ when
$|u|=|\varphi_{s,t}|$,

  3. 3.

convergent in $z=0$ when $|u|>|\varphi_{s,t}|$.

Suppose that $|-t/\varphi_{s,t}^{2}|>1$.

  1. 4.

If $|u|\leq|\varphi_{s,t}|$, then $\mathrm{e}_{s,t}(a,z,u,p)$ is entire.

  2. 5.

If $|u|>|\varphi_{s,t}|$, then $\mathrm{e}_{s,t}(a,z,u,p)$ converge in $z=0$.

###### Proof.

Since $p$ is continuous and periodical, then there exists a number $M>0$ such
that $p$ is bounded with $|p(z)|<M$ for all $z$. Then

$\displaystyle|\mathrm{e}_{s,t}(a,z,u,p)|$
$\displaystyle=\Bigg{|}\sum_{n=0}^{\infty}u^{\binom{n}{2}}a^{n}p\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n}}z\right)\right)\frac{z^{n}}{\brk[c]{n}_{s,t}!}\Bigg{|}$
$\displaystyle<\sum_{n=0}^{\infty}|u|^{\binom{n}{2}}|a|^{n}\Bigg{|}p\left(\log_{Q}\left(\frac{u^{n}}{\varphi_{s,t}^{n}}z\right)\right)\Bigg{|}\frac{|z|^{n}}{\brk[c]{n}_{s,t}!}$
$\displaystyle<M\sum_{n=0}^{\infty}|u|^{\binom{n}{2}}|a|^{n}\frac{|z|^{n}}{\brk[c]{n}_{s,t}!}=M\exp_{s,t}(|az|,|u|)$

and $\exp_{s,t}(|az|,|u|)$ is an upper bound of $|\mathrm{e}_{s,t}(a,z,u,p)|$.
All statements follow from Theorem 2. ∎

######  Corollary 4.

For all $s\neq 0$ and $t\neq 0$, the functions $\mathrm{e}_{s,t}(a,tz,tu,p)$
are solutions of functional-difference equation with proportional delay

$\mathbf{D}_{s,t}f(z)=atf(tuz).$ (32)

######  Theorem 11.

Let $a$ be a complex number and set $s\neq 0$, $t=0$. Then the equation in
difference with proportional delay

$\mathbf{D}_{s,0}f(z)=af(uz).$ (33)

have solution $f(z)=c\exp_{s,0}(az,u)$.

##  7 Degenerate case $s^{2}+4t=0$

When $s^{2}+4t=0$ we obtain the degenerate case of the $(s,t)$-Fibonacci
polynomials. When $\varphi_{s,t}\rightarrow\varphi_{s,t}^{\prime}$, we obtain

$\displaystyle\lim_{\varphi_{s,t}\rightarrow\varphi_{s,t}^{\prime}}\frac{\varphi_{s,t}^{n}-\varphi^{\prime
n}_{a,b}}{\varphi_{s,t}-\varphi^{\prime}_{a,b}}=n\varphi_{s,t}^{\prime(n-1)}.$

Likewise, when $\varphi_{s,t}^{\prime}\rightarrow\varphi_{s,t}$, then
$\brk[c]{n}_{s,t}\rightarrow n\varphi_{s,t}^{n-1}$. Therefore, this implies
that $s\rightarrow\pm 2i\sqrt{t}$, and that
$\varphi_{s,t}=\varphi_{s,t}^{\prime}=\pm i\sqrt{t}$. In this way we obtain
the $(\pm 2i\sqrt{t},t)$-Fibonacci function

$\displaystyle\brk[c]{n}_{\pm 2i\sqrt{t},t}=n(\pm i\sqrt{t})^{n-1}$ (34)

for all $t\in\mathbb{C}$. When $t=-1$, then $\brk[c]{n}_{\pm
2i\sqrt{t},t}=\brk[c]{n}_{\mp 2,-1}=n(\mp 1)^{n-1}$. On the other hand, in the
$q$-calculus the degenerate case is obtained when $q\mapsto 1$. In this
situation, the $q$-numbers $\brk[s]{n}_{q}$ tend to the integers $n$. Then
$\frac{\varphi_{s,t}^{\prime}}{\varphi_{s,t}}\mapsto 1$ implies that
$\varphi_{s,t}\mapsto\sqrt{-t}$ and $\varphi_{s,t}^{\prime}\mapsto\sqrt{-t}$.
Therefore, if $t=-1$, then

$\displaystyle\lim_{\varphi_{s,-1}\mapsto
1}\frac{\varphi_{s,-1}^{n}-\varphi_{s,-1}^{\prime
n}}{\varphi_{s,-1}-\varphi_{s,-1}^{\prime}}=n.$

Then, on the Riemann surface $\pm 2i\sqrt{z}$ we obtain the extreme case of
the generalized Fibonacci calculus.

For $t\neq 0$, if $s\rightarrow 2i\sqrt{t}$, then $\theta_{s,t}\rightarrow\pi$
and thus

$\brk[c]{n}_{2i\sqrt{t},t}=\lim_{\theta_{s,t}\rightarrow
0}(-i\sqrt{t})^{n-1}\frac{\sin(n\theta_{s,t})}{\sin(\theta_{s,t})}=(i\sqrt{t})^{n-1}n.$

If $s\rightarrow-2i\sqrt{t}$, then $\theta_{s,t}\rightarrow 0$ and therefore

$\brk[c]{n}_{-2i\sqrt{t},t}=\lim_{\theta_{s,t}\rightarrow\pi}(-i\sqrt{t})^{n-1}\frac{\sin(n\theta_{s,t})}{\sin(\theta_{s,t})}=(-i\sqrt{t})^{n-1}n$

for all $n\in\mathbb{N}$. Thus

$\brk[c]{n}_{\pm 2i\sqrt{t},t}=n(\pm i\sqrt{t})^{n-1}$

and accordingly we can express the $(\pm 2i\sqrt{t},t)$-Fibotorial and the
$(\pm 2i\sqrt{t},t)$-Fibonomial functions as

$\displaystyle\brk[c]{n}_{\pm 2i\sqrt{t},t}!$ $\displaystyle=(\pm
i\sqrt{t})^{\binom{n}{2}}n!$

and

$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}_{\pm 2i\sqrt{t},t}$
$\displaystyle=(\pm i\sqrt{t})^{k(n-k)}\binom{n}{k},$

respectively.

Finally, taking $s\rightarrow\pm 2i\sqrt{t}$, then the deformed
$(s,t)$-exponential functions reduce to the following deformed $(\pm
2i\sqrt{t},t)$-exponential functions:

$\displaystyle\exp_{\pm 2i\sqrt{t},t}(z,u)$
$\displaystyle=\sum_{n=0}^{\infty}(u/\pm
i\sqrt{t})^{\binom{n}{2}}\frac{z^{n}}{n!},$ $\displaystyle\exp_{\pm
2i\sqrt{t},t}(z)$ $\displaystyle=\sum_{n=0}^{\infty}(\pm
i\sqrt{t})^{-\binom{n}{2}}\frac{z^{n}}{n!},$ $\displaystyle\exp_{\pm
2i\sqrt{t},t}^{\prime}(z)$ $\displaystyle=\sum_{n=0}^{\infty}(\pm
i\sqrt{t})^{\binom{n}{2}}\frac{z^{n}}{n!},$
$\displaystyle\mathop{\mbox{{Exp}}}\nolimits_{\pm 2i\sqrt{t},t}(z)$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits_{F_{\pm
2i\sqrt{t},t}}^{\prime}(z)=e^{z}.$

When $t=-1$, then

$\displaystyle\exp_{\mp 2,-1}(z,u)$ $\displaystyle=\sum_{n=0}^{\infty}(\mp
u)^{\binom{n}{2}}\frac{z^{n}}{n!},$ $\displaystyle\exp_{\mp 2,-1}(z)$
$\displaystyle=\exp^{\prime}_{\mp
2,-1}(z)=\sum_{n=0}^{\infty}(\mp)^{\binom{n}{2}}\frac{z^{n}}{n!},$
$\displaystyle\mathop{\mbox{{Exp}}}\nolimits_{2,-1}(z)$
$\displaystyle=\mathop{\mbox{{Exp}}}\nolimits_{2,-1}^{\prime}(z)=e^{z}.$

Thus

$\exp_{2,-1}(x,u)=\sum_{n=0}^{\infty}u^{\binom{n}{2}}\frac{x^{n}}{n!}$

and therefore $\exp_{2,-1}(x,u)$ is the deformed exponential function
$\mathop{\mbox{{Exp}}}\nolimits(x,u)$ and

$\exp_{2,-1}(x)=\exp^{\prime}_{2,-1}(x)=\mathop{\mbox{{Exp}}}\nolimits_{2,-1}(x)=\mathop{\mbox{{Exp}}}\nolimits^{\prime}_{2,-1}(x)=e^{x}.$

######  Theorem 12.

For all non-zero complex number $t$ the function $\exp_{\pm
2i\sqrt{t},t}(z,u)$

  1. 1.

is entire if $|u|\leq|\sqrt{t}|$.

  2. 2.

Converge in $z=0$ when $|u|>|\sqrt{t}|$.

##  8 Conclusions and Perspectives

Despite the vast existing literature on $q$-calculus and the growing
literature on golden calculus, each of them special cases of the
$(p,q)$-calculus, none of those treatments involve the calculus developed in
the present paper, indicating the importance of the results obtained above.
This leads to several theoretical and applied problems and some directions we
can follow with a calculus on deformed generalized Fibonacci polynomials are
the following. Just as exponential generating functions are a very useful tool
for counting objects with labels, we also expect that functions on the ring
$\mathop{\mbox{{W}}}\nolimits_{s,t,\mathbb{C}}[[z]]$, especially the deformed
$(s,t)$-exponential generating functions, will be useful for counting other
combinatorial objects. On the other hand, it is possible to construct a
$(s,t)$-analytic number theory to obtain results on the distribution of
Fibonacci primes, Pell primes, Jacobsthal primes, Mersenne primes and Repunit
primes. To achieve the latter it is necessary to construct an integral
calculus on generalized Fibonacci polynomials and furthermore to define the
$(s,t)$-analogues of the Gamma and Zeta functions. Finally, it will be
possible to develop a theory of difference equations based on the difference
operator $\mathbf{D}_{s,t}$ together with problems of existence, uniqueness,
approximation and asymptotic analysis of its solutions. An outstanding feature
of this paper is the existence of a calculus on Chebysheff polynomials, which
could be very useful in approximation theory.

## References

  * [1] A.T. Benjamin, S.S. Plott, A combinatorial approach to Fibonomial coefficients, Fibonacci Quart., 46/47(2008/2009), 7–9
  * [2] F.H. Jackson, On $q$-functions and a certain difference operator. Trans. Roy. Soc. Edin., 46,253–281 (1908)
  * [3] F.H. Jackson, A $q$-form of Taylor’s theorem. Messenger 38, 62–64 (1909)
  * [4] F.H. Jackson, On $q$-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)
  * [5] F.H. Jackson, On $q$-difference equations. Am. J. Math. 32, 305–314 (1910)
  * [6] F.H. Jackson, Transformations of $q$-series. Messenger 39, 145–153 (1910)
  * [7] F.H. Jackson, Summation of $q$-hypergeometric functions. Messenger Math. 50, 101–112 (1921)
  * [8] F.H. Jackson, Certain $q$-identities. Q. J. Math. Oxf. Ser 12, 167–172 (1941)
  * [9] F.H. Jackson, On basic double hypergeometric functions. Q. J. Math. 13, 69–82 (1942)
  * [10] G. Fontené, Générralisation d’une formule connue, Nouv. Ann. Math., (4) 15 (1915) 112–112.
  * [11] H.W. Gould, The bracket function and Fontené-Ward generalized binomial coefficients with application to Fibonomial coefficients, Fibonacci Quart., 7(1969) 23–40
  * [12] L. Euler, Introductio in Analysin Infinitorum, T. 1, Chapter XVI, p.259, Lausanne, 1748
  * [13] V. Kac, C. Pokman, Quantum Calculus, Universitext, Spinger-Verlag, 2002
  * [14] W.F. Keigher, On the ring of Hurwitz series, Comm. Algebra, 25:6(1997) 1845–1859.
  * [15] M. Ward, A calculus of sequences, Am. J. Math. 58(1936) 255–266.
  * [16] P. Trojovský, On some identities for the Fibonomial coefficients via generating function, Discrete Appl. Math., 155(15)(2007), 2017–2024
  * [17] N. Tuglu, F. Yesil, E. Kocer, and M. Dziemiańczuk, The $F$-Analogue of Riordan Representation of Pascal Matrices via Fibonomial Coefficients, J. Appl. Math., 2014(2014) 1–6
  * [18] A.F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quart., 3:3 (1965), 161–176.
  * [19] C.F. Gauss, Werke 2, (1876), 9–45.
  * [20] C.F. Gauss, Werke 3, (1876).
  * [21] A. Scott, A. Sokal, The repulsive lattice gas, the independent-set polynomial, and the lovsz local lemma, J. Stat. Phys. 118(5-6) (2005), 1151–1261
  * [22] G.R. Morris, A. Feldstein, and E.W. Bowen, The Phragmén-Lindelof principle and a class of functional differential equations, in ’Ordinary Differential Equations’ 513–540, Academic Press, San Diego, 1972
  * [23] O.K Pashaev, S. Nalci, Golden Quantum Oscillator and Binet-Fibonacci Calculus, J. Phys. A: Math.Theor., 45, 015303 (2012)
  * [24] O.K. Pashaev, Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators, Int. J. Geom. Methods Mod. Phys., 18 05, 2150075 (2021)
  * [25] M. Ozvatan, Generalized golden-Fibonacci calculus and applications. Ph.D. thesis, Izmir Institute of Technology (2018)
  * [26] R. Chakrabarti, R. Jagannathan, A $(p,q)$-oscillator realization of two-parameter quantum algebras. J. Phys. A: Math. Gen. 24, L711 (1991)
  * [27] G. Brodimas, A. Jannussis, and R. Mignani, Two-parameter quantum group. Universita di Roma Preprint. 820, 1–16 (1991)
  * [28] M. Wahcs, D. White, $p,q$-Stirling numbers and set partition statistics. J. Combin. Theory Ser. A 56, 27–46 (1991)
  * [29] M. Arik, E. Demircan, T. Turgut, L. Ekinci, and M. Mungan, Fibonacci oscillators. Z. Phys. C 55, 89–95 (1991)
  * [30] I.M. Gessel, A $q$-analog of the exponential formula. Discrete Math. 40, 69–80 (1982)
  * [31] A. Iserles, On the generalized pantograph functional-differential equation. European J. Appl. Math. 4, (1993), 1–38
  * [32] A. Iserles, Y. Liu, On functional-differential equation with proportional delays, J. Math. Anal. Appl. 207, 73–95 (1997)
  * [33] T. Amdeberhan, X. Chen, V.H. Moll, Generalized Fibonacci Polynomials and Fibonomial Coefficients. Ann. Comb. 18, 541–562 (2014). https://doi.org/10.1007/s00026-014-0242-9
  * [34] G. Udrea, A note on the sequence $(W_{n})_{n\geq 0}$ of A.F. Horadam, Port. Math. 53 2, 143–155 (1996)