# Convolutions and Mixtures of Gamma, Stable and Mittag-Leffler Distributions

Nomvelo Karabo Sibisi
<EMAIL_ADDRESS>
###### Abstract

This paper uses convolutions of the gamma density and the one-sided stable
density to construct higher level densities. The approach is applied to
constructing a 4-parameter Mittag-Leffler density, whose Laplace transform is
a corresponding Mittag-Leffler function, which is completely monotone (CM) by
construction. Laplace transforms of mixtures of the stable densities with
respect to the 4-parameter Mittag-Leffler distribution are compositions of the
Mittag-Leffler functions with Bernstein functions, thereby generating a rich
family of CM variants of the base CM Mittag-Leffler functions, including known
instances as special cases.

Keywords— Gamma, beta, stable, Mittag-Leffler distributions; Mittag-Leffler
functions;
convolution, mixing; Bernstein functions; complete monotonicity.

##  1 Introduction

The goal of this paper is to use convolutions and mixtures involving the
familiar gamma and one-sided stable distributions to construct higher level
distributions. We apply this approach to constructing Mittag-Leffler
distributions, whose Laplace-Stieltjes transforms are Mittag-Leffler functions
in various guises. This is a ‘forward problem’ that involves constructing
distributions and evaluating their transforms, as opposed to an ‘inverse
problem’ that seeks to infer densities from given functions through complex
analytic inversion (Laplace or Stieltjes inversion, depending on the transform
linking the desired density to the function at hand). Such inversion cannot,
by itself, ensure that the result will be nonnegative, supplementary
conditions typically need to be imposed. Common though inversion methods may
be, our primary objective strictly remains one of direct construction of
distributions. In any event, Mittag-Leffler distributions have applications in
their own right to model, for example, random partitions of the integers or
the growth of random trees and graphs. In such context, their property as
generators of Mittag-Leffler functions through Laplace-Stieltjes transforms
may only be of passing interest.

It is well-known that the Mittag-Leffler function $E_{\alpha}(-x^{\alpha})$
for $0<\alpha<1$ is the Laplace-Stieltjes transform of a distribution with a
straightforward density (equivalently, $E_{\alpha}(-x^{\alpha})$ is the
ordinary Laplace transform of said density). It is also known (if perhaps less
widely so) that $E_{\alpha}(-x)$ for $0<\alpha<1$ is the Laplace-Stieltjes
transform of a distribution known as the Mittag-Leffler distribution, which
has a known density too. The primary question is whether the relationship
might also hold for other values of $\sigma$, at least for the same
restriction $0<\alpha<1$ suggested by the two examples cited. To be precise:
a) Is $E_{\alpha}(-x^{\sigma})$ for certain $\sigma\notin\\{1,\alpha\\}$ also
the Laplace-Stieltjes transform of a distribution?
b) If (a) holds, what is the explicit form of the distribution associated with
$E_{\alpha}(-x^{\sigma})$?

Part (a) of the question has a simple answer, which may be looked upon as
arising from the concept of completely monotone functions. A function is said
to be completely monotone (CM) if it and all its derivatives of alternating
sign are nonnegative. By a theorem due to Bernstein, a function is CM if and
only if it is the Laplace-Stieltjes transform of a distribution.
$E_{\alpha}(-x)$ and $E_{\alpha}(-x^{\alpha})$ are thus examples of CM
functions. Furthermore, by a property of CM functions, if $E_{\alpha}(-x)$ is
CM so is the composition $E_{\alpha}(-\lambda x^{\sigma})$ for $\lambda>0$ and
$0<\sigma<1$. Hence we have the answer to (a): there exists a distribution
whose Laplace-Stieltjes transform is $E_{\alpha}(-x^{\sigma})$ for
$0<\alpha<1$ and $0<\sigma<1$. This clearly includes the case $\sigma=\alpha$,
i.e. $E_{\alpha}(-x^{\alpha})$ is CM because $E_{\alpha}(-x)$ is CM. It is
also true that $E_{\alpha}(-(x^{\alpha})^{\sigma})$ is CM because
$E_{\alpha}(-x^{\alpha})$ is CM, but this is narrower since we cannot deduce
that $E_{\alpha}(-x)$ is CM by using the CM nature of
$E_{\alpha}(-x^{\alpha})$ as a starting point.

Part (b) seeks to go beyond the existence statement and determine the explicit
distribution associated with $E_{\alpha}(-\lambda x^{\sigma})$. This is core
to what this paper seeks to achieve: explicit construction of distributions
that mirror corresponding CM functions. In particular, since
$E_{\alpha}(-\lambda x^{\sigma})$ is CM by virtue of the fact that
$E_{\alpha}(-x)$ is CM, we shall establish that the distribution whose
Laplace-Stieltjes transform is $E_{\alpha}(-\lambda x^{\sigma})$ arises from
the Mittag-Leffler distribution, whose Laplace-Stieltjes transform is
$E_{\alpha}(-x)$. This is despite the fact that the well-known case
$E_{\alpha}(-\lambda x^{\alpha})$ has been studied directly and independently
of the $E_{\alpha}(-x)$ case in the literature.

###  1.1 Contribution

In this paper, convolutions and mixtures of gamma and stable distributions are
the fundamental concepts that underpin all higher distributions. As a starting
point, we will use this methodology to construct the Mittag-Leffler
distribution, which involves one parameter $0<\alpha<1$ (we can include the
ends of this interval with an extended definition of the stable density, but
this is not a key conceptual point here). It then becomes evident how to
generalise to the construction of multi-parameter Mittag-Leffler
distributions.

For a ‘quick read’, Tables 1, 2, 3, 4 summarise the results derived in the
paper, including known instances as special cases. However, the methodology
presented in the paper, based on convolution and mixing of basic
distributions, is more broadly applicable than the illustrative examples shown
in the Tables.

##  2 Background

###  2.1 Mittag-Leffler Functions

Gorenflo et al. [18] is a comprehensive treatise on Mittag-Leffler functions
and their applications. The 3-parameter Mittag-Leffler function, also known as
the Prabhakar function, has an infinite series representation for
$z\in\mathbb{C}$

$\displaystyle E^{\gamma}_{\alpha,\beta}(z)$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\sum_{k=0}^{\infty}\frac{\Gamma(\gamma+k)}{k!\,\Gamma(\alpha
k+\beta)}\,z^{k}\qquad{\rm Re}(\alpha)>0,\,{\rm Re}(\beta)>0,\,\gamma>0$ (1)

The termwise Laplace transform of
$x^{\beta-1}E^{\gamma}_{\alpha,\beta}(-\lambda x^{\alpha})$ $(\lambda>0)$ sums
to

$\displaystyle\int_{0}^{\infty}e^{-sx}\,x^{\beta-1}E^{\gamma}_{\alpha,\beta}(-\lambda
x^{\alpha})\,dx$
$\displaystyle=\frac{s^{\alpha\gamma-\beta}}{(\lambda+s^{\alpha})^{\gamma}}$
(2)

As discussed in Gorenflo et al. [18] and numerous other references such as the
survey by Giusti et al. [15], there are many other attributes of Mittag-
Leffler functions, often related to fractional calculus and its applications,
but these will not be of direct interest here.

What is key in this paper is that we shall often interpret
$E_{\alpha}(-yx^{\alpha})$ as a function of two variables $x$ and $y$, not
merely as a function of $x$ for some fixed parameter $y$. Thus, we shall
typically work with $E_{\alpha}(-yx^{\alpha})$ as a function of $x$ given $y$
or $y$ given $x$.

###  2.2 Mittag-Leffler Distributions

Pollard [47] used complex analytic methods to derive the distribution
$P_{\alpha}(t)$ $(t>0)$ for $0<\alpha<1$, with Laplace-Stieltjes transform
$E_{\alpha}(-x)$, $x\geq 0$. In a probabilistic derivation, Feller [11,
XIII.8] considered the two-dimensional Laplace transform of the one-sided
$\alpha$-stable distribution with an associated scale factor. $P_{\alpha}$ has
become known as the (1-parameter) Mittag-Leffler distribution.

In the context of studying random partitions of the integers, Pitman [44,
Theorem 3.8] derived the 2-parameter Mittag-Leffler distribution
$P_{\alpha,\theta}$ where $\alpha$ is again the index of the one-sided
$\alpha$-stable distribution $(0<\alpha<1)$ and $\theta>-\alpha$ is associated
with “polynomial tilting” of the $\alpha$-stable density. $P_{\alpha,\theta}$
is also referred to as the generalized Mittag-Leffler distribution and
alternatively denoted by ${\rm ML(\alpha,\theta})$ (Goldschmidt and Haas [17],
Ho et al. [22]). The Mittag-Leffler distribution is the $\theta=0$ case
$P_{\alpha}\equiv{\rm ML}(\alpha,0)$. A derivation of ${\rm
ML(\alpha,\theta})$ due to Janson [26] takes the form of a limiting
distribution of a Pólya urn scheme (also see Flajolet et al. [12]). The same
limiting distribution arises from the Chinese restaurant process (Pitman [44,
3.1]), which is intimately associated with random partitions and the Poisson-
Dirichlet process ${\rm PD(\alpha,\theta})$ of Pitman and Yor [45]. Pursuing
the Chinese restaurant metaphor, Möhle [40] added a cocktail bar in order to
construct a 3-parameter Mittag-Leffler distribution ${\rm
ML(\alpha,\beta,\gamma})$.

Mittag-Leffler distributions have finite moments of all order, thereby
uniquely determining the distributions. Hence the distributions are often
described by specification of their moments. By studying such moments, Möhle
[40, Corollary 3] proved that if $X$ has the 3-parameter Mittag-Leffler
distribution, then $X$ is equal in distribution to the product $YZ$ where $Y$
is beta distributed and $Z$ has the 2-parameter Mittag-Leffler distribution.
Pursuing the same observation, Banderier et al. [2] discussed a 3-parameter
beta-Mittag-Leffler distribution ${\rm BML(\alpha,\theta,\beta})$.

In another context, Mittag-Leffler distributions play a fundamental role in
the probabilistic modelling of the evolution of random trees and graphs, often
used as models of real-world network behaviour. In the study of stable trees
(Goldschmidt and Haas [17], Rembart and Winkel [48, 49]) and stable graphs
(Goldschmidt et al. [16]), the distributions that feature as building blocks
are beta, generalised Mittag-Leffler ${\rm ML(\alpha,\theta})$, Dirichlet and
Poisson-Dirichlet ${\rm PD(\alpha,\theta})$. James [24], Sénizergues [54]
discussed the appearance of ${\rm ML(\alpha,\theta})$ as limit distributions
in the growth of random graphs inspired by the preferential attachment model
due to Barabási and Albert [3]. ${\rm ML(\alpha,\theta})$ also arises in a
random walk model of network formation by Bloem-Reddy and Orbanz [8].

In yet another context, the complete memory elephant random walk with delays
converges to the Mittag-Leffler distribution (Bercu [6], Janson [29, Example
14.10], Miyazaki and Takei [39]).

More generally, Mittag-Leffler distributions belong to a class that Janson
[28] referred to as distributions with “moments of Gamma type”. In an
addendum, Janson [27] added that the densities of this class were $H$
functions (Fox [13], Mathai et al. [37]), with Meijer $G$ functions as a
subclass. The distributions are often described in terms of their moments,
thereby facilitating the use of “Mellin transform techniques to obtain
expansions or asymptotics of the density function” [28]. Pursuing a similar
theme, Beghin et al. [5] explored Fox-$H$ densities with moments of all order,
whose Laplace transforms are generalised Wright functions. Under a particular
choice of parameters, Beghin et al. [5, Example 5.5] identified a Fox-$H$
density whose Laplace transform is
$\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-x)$ for $0<\alpha<1$ and
$\beta-\alpha\gamma=1-\alpha$.

It thus seems worthwhile to study Mittag-Leffler distributions in their own
right – in the case of this paper, from a perspective of convolutions and
mixtures of distributions rather than as limiting distributions of random
selection processes. In addition, the paper explores in some detail the
duality, mediated by the Laplace-Stieltjes transform, between Mittag-Leffler
distributions and corresponding Mittag-Leffler functions in various guises.
Notably, the Laplace-Stieltjes transform of ${\rm ML(\alpha,\theta})$ is
$\Gamma(1+\theta)E^{1+\theta/\alpha}_{\alpha,1+\theta}(-x)$ (Ho et al. [22],
James [23]).

A key highlight of this paper is the derivation of Mittag-Leffler densities
using straightforward properties of convolutions, without any need to appeal
to complex analytic inversion or special function representation.

####  2.2.1 4-parameter Mittag-Leffler Distribution

Pursuing the approach of mixtures and convolutions of stable and gamma
distributions, this paper derives a 4-parameter Mittag-Leffler distribution
$P_{\alpha,\beta,\gamma,\theta}\equiv{\rm ML(\alpha,\beta,\gamma,\theta})$ for
$0<\alpha<1,\beta>\alpha\gamma,\theta>-\alpha\gamma$, whose Laplace-Stieltjes
transform is
$\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)$
$(x\geq 0)$. The difference $\beta-\alpha\gamma>0$ plays an important role. As
we shall demonstrate, for $\beta-\alpha\gamma=1-\alpha$ the convolution
essentially simplifies to the $\alpha$-stable density (Property 3 below gives
the precise statement). ${\rm ML(\alpha,\theta})\equiv{\rm
ML(\alpha,1,1,\theta})$ is one instance of $\beta-\alpha\gamma=1-\alpha$.

With the parameter restrictions stated,
$E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)$ is CM. Hence
$E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda x^{\sigma})$
$(\lambda>0,0<\sigma<1)$ is also CM. We derive the explicit form of the
associated density, a mixture of a $\sigma$-stable density with respect to
${\rm ML}(\alpha,\beta,\gamma,\theta)$, with Laplace transform
$\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda
x^{\sigma})$. Through multiplication with other CM functions and iterative
composition, we may generate a cascade of CM functions as Laplace transforms
of associated explicit densities. We discuss the general construction
principle but demonstrate only a limited selection of such possibilities.

###  2.3 Other Perspectives

####  2.3.1 Other Definition of Mittag-Leffler Distributions

Pillai [43] used the term ‘Mittag-Leffler distribution’ to refer to
$1-E_{\alpha}(-\lambda x^{\alpha})$ $(\lambda>0,0<\alpha\leq 1)$ (Pillai
defined it with $\lambda=1$), with Laplace-Stieltjes transform

$\displaystyle\int_{0}^{\infty}e^{-sx}\,d(1-E_{\alpha}(-\lambda x^{\alpha}))$
$\displaystyle=1-\frac{s^{\alpha}}{\lambda+s^{\alpha}}=\frac{\lambda}{\lambda+s^{\alpha}}$
(3)

In this context, ‘generalized Mittag-Leffler distribution’ is taken to refer
to the distribution with Laplace-Stieltjes transform
$\lambda^{\gamma}/(\lambda+s^{\alpha})^{\gamma}$, $\gamma>0$ (Barabesi et al.
[4], Fujii [14], Haubold et al. [21], Jose et al. [31], Korolev et al. [33],
Korolev and Zeifman [34])). $\lambda^{\gamma}/(\lambda+s^{\alpha})^{\gamma}$,
$\gamma>0,0<\alpha<1$ is known to be the Laplace transform of a stable-gamma
mixture density.

Several authors have commented on this regrettable clash of nomenclature. We
shall reserve the term ‘Mittag-Leffler distributions’ exclusively for the
distributions discussed in 2.2 above.

####  2.3.2 Complex Analytic Inversion

There is a body of literature that places emphasis on exploring the CM
character of Mittag-Leffler functions, with the fact that they are Laplace
transforms of nonnegative functions seeming to arise as a means to that end
rather than as the primary objective. Such literature might thus be described
as more analytic than probabilistic.

Pollard [47] took the complex analytic Laplace inverse of the 1-parameter
instance of (2) as the starting point to establish the CM character of
$E_{\alpha}(-x)$. Inspired by this, Górska et al. [19] proved that
$E^{\gamma}_{\alpha,\beta}(-x)$ is CM, i.e. that it is the Laplace transform
of a nonnegative function for $0<\alpha\leq 1,\beta>\alpha\gamma,\gamma>0$
(also see Tomovski et al. [57, Theorem 2]). Górska et al. [19, Remark 2.2]
noted that the CM nature of $E^{\gamma}_{\alpha,\beta}(-x)$ implies that: (a)
$E^{\gamma}_{\alpha,\beta}(-x^{\alpha})$ is CM by the composition property of
CM functions and (b) $x^{\beta-1}E^{\gamma}_{\alpha,\beta}(-x^{\alpha})$ is CM
by the product property, with the additional constraint $\beta\leq 1$.

The CM character of (b) had previously been investigated by Capelas de
Oliveira et al. [10] using Stieltjes inversion of the Laplace transform (2) of
$x^{\beta-1}E^{\gamma}_{\alpha,\beta}(-x^{\alpha})$. Stieltjes inversion by
itself (Titchmarsh [56, 11.8 (p318)], Widder [58, VIII.7 (p341)]) does not
shed light on whether the inverse is a nonnegative function, the paper
determined the required parameter restrictions $0<\alpha\leq 1$,
$0<\alpha\gamma\leq\beta\leq 1$ using a theorem due to Gripenberg [20].
Mainardi and Garrappa [36] pursued the same inversion approach, but with a
variation on the argument leading to the same parameter restrictions.

The condition $\beta\leq 1$ arises from treating $x^{\beta-1}$ and
$E^{\gamma}_{\alpha,\beta}(-x^{\alpha})$ as separate CM functions, so that the
product is also CM. Amongst other things, we will show that $\beta\leq 1$ is,
in fact, redundant for $x^{\beta-1}E^{\gamma}_{\alpha,\beta}(-x^{\alpha})$ to
be the Laplace transform of a density, i.e. for it to be CM.

##  3 Preliminaries

Let $\psi(x),\,x>0$ be the Laplace-Stieltjes transform of a distribution
function $F(t)$

$\displaystyle\psi(x)$ $\displaystyle=\int_{0}^{\infty}e^{-xt}\,dF(t)$ (4)

If $dF(t)=f(t)dt$ for a density $f(t)$, then $\psi(x)$ is equivalently the
(ordinary) Laplace transform of $f(t)$. $\psi(x)$ of the form (4) is said to
be completely monotone (CM), i.e. its derivatives of alternating sign are
nonnegative: $(-1)^{n}\psi^{(n)}(x)\geq 0,n\geq 0$. In fact, by a theorem due
to Bernstein (Feller [11, XIII.4], Widder [58, p160]), a function is CM if and
only if it is of the form (4). The topic of completely monotone functions is
well-studied (e.g. Feller [11, XIII.4], Schilling et al. [52], Widder [58,
IV]). Properties of interest here are:

######  Property 1 (Product).

If $\psi_{1}(x)$ and $\psi_{2}(x)$ are CM, so is their product
$\psi_{1}(x)\psi_{2}(x)$.

######  Property 2 (Composition).

If $\psi(x)$ is CM, so is the composition $\psi(\eta(x))$ where $\eta(x)$ is a
nonnegative function with a CM derivative, also known as a Bernstein function.

We study properties of distributions corresponding to the properties above for
CM functions. The product case is straightforward. If the CM functions
$\psi_{1}(x),\psi_{2}(x)$ are Laplace transforms of densities
$f_{1}(t),f_{2}(t)$ respectively, then the convolution theorem states that the
product $\psi_{1}(x)\psi_{2}(x)$ is the Laplace transform of the convolution
density $\\{f_{1}\star f_{2}\\}(t)$

$\displaystyle\\{f_{1}\star f_{2}\\}(t)$
$\displaystyle=\int_{0}^{t}f_{1}(t-u)f_{2}(u)\,du=\int_{0}^{t}f_{2}(t-u)f_{1}(u)\,du$
(5)

The composition case is more subtle. If the CM function $\psi(x)$ is the
Laplace-Stieltes transform of a known distribution, what can be said about the
distribution whose Laplace-Stieltes transform is $\psi(\eta(x))$ for a
Bernstein function $\eta(x)$? Rather than address the question for the very
broad class of Bernstein functions $\eta(x)$, we consider the case $\lambda
x^{\alpha}$ for $\lambda>0$ and $0<\alpha<1$.

###  3.1 Gamma Distribution

The gamma distribution $G(\gamma,\lambda)$ with density
$\rho_{\gamma,\lambda}$ with shape and scale parameters $\gamma,\lambda>0$ is

$\displaystyle dG(t|\gamma,\lambda)$
$\displaystyle\equiv\rho_{\gamma,\lambda}(t)\,dt=\frac{\lambda^{\gamma}}{\Gamma(\gamma)}\,t^{\gamma-1}\,e^{-\lambda
t}dt$ (6)

We shall make frequent reference to the unnormalised power density
$\rho_{\gamma}(t)=t^{\gamma-1}/\Gamma(\gamma)$. The Laplace-Stieltjes
transform of $G(\gamma,\lambda)$ (or the Laplace transform of its density
$\rho_{\gamma,\lambda}$) and the Laplace transform of $\rho_{\gamma}(t)$ are
respectively

$\displaystyle\int_{0}^{\infty}e^{-xt}\,dG(t|\gamma,\lambda)\equiv\int_{0}^{\infty}e^{-xt}\,\rho_{\gamma,\lambda}(t)\,dt$
$\displaystyle=\frac{\lambda^{\gamma}}{(\lambda+x)^{\gamma}}$
$\displaystyle\quad(x\geq 0)$ (7)
$\displaystyle\int_{0}^{\infty}e^{-xt}\,\rho_{\gamma}(t)\,dt$
$\displaystyle=x^{-\gamma}$ $\displaystyle\quad(x>0)$ (8)

We also let $\rho_{\gamma=0}(t)\equiv\delta(t)$ with Laplace transform 1
($\delta(t)$ is the Dirac delta function).

###  3.2 Beta Distribution

The beta distribution ${\rm Beta}(\alpha,\beta)$, for given parameters
$\alpha,\beta>0$, is defined on $(0,1)$ with density

$\displaystyle{\rm beta}(t|\alpha,\beta)$
$\displaystyle=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\,t^{\alpha-1}(1-t)^{\beta-1}$
(9) $\displaystyle\int_{0}^{1}t^{k}\,{\rm beta}(t|\alpha,\beta)\,dt$
$\displaystyle=\frac{\Gamma(\alpha+\beta)\Gamma(\alpha+k)}{\Gamma(\alpha)\Gamma(\alpha+\beta+k)}\quad({\rm
moments})$ (10)

In anticipation of later discussion, let $U,V>0$ be independent random
variables whose distributions have densities $f_{U}(u),f_{V}(v)$ respectively.
Then it is a standard result that the product $T=VU$ has density $f_{T}(t)$
given by

$\displaystyle f_{T}(t)$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\delta(t-uv)f_{U}(u)f_{V}(v)\,du\,dv=\int_{0}^{\infty}f_{V}(t/u)f_{U}(u)\frac{du}{u}$
(11)

In particular, if $f_{U}(u)\equiv{\rm beta}(u|\alpha,\beta)$ on $(0,1)$ and 0
on $u\geq 1$, then

$\displaystyle f_{T}(t)$ $\displaystyle=\int_{0}^{1}f_{V}(t/u)\,{\rm
beta}(u|\alpha,\beta)\,\frac{du}{u}$
$\displaystyle=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{1}f_{V}(t/u)\,u^{\alpha-1}(1-u)^{\beta-1}\,\frac{du}{u}$
(12)

###  3.3 Stable Distribution

The one-sided stable distribution $F_{\alpha}(t|\lambda),t>0$ ($0<\alpha<1$),
conditioned on a scale parameter $\lambda>0$, is indirectly defined by its
Laplace-Stieltjes transform $($equivalently, the ordinary Laplace transform of
its density $f_{\alpha}(t|\lambda)$)

$\displaystyle e^{-\lambda x^{\alpha}}$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,dF_{\alpha}(t|\lambda)=\int_{0}^{\infty}e^{-xt}f_{\alpha}(t|\lambda)\,dt$
(13)

$f_{\alpha}(t|\lambda)$ may also be written as
$f_{\alpha}(t\lambda^{-1/\alpha})\,\lambda^{-1/\alpha}$, where
$f_{\alpha}(t)\equiv f_{\alpha}(t|\lambda=1)$.

$e^{-\lambda x^{\alpha}}$ is a CM function by construction. In particular,
minus its first derivative is

$\displaystyle\lambda\alpha x^{\alpha-1}e^{-\lambda x^{\alpha}}$
$\displaystyle=\int_{0}^{\infty}e^{-xt}tf_{\alpha}(t|\lambda)\,dt=\lambda\alpha\int_{0}^{\infty}e^{-xt}\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|\lambda)\\}(t)\,dt$ (14)

where $\rho_{1-\alpha}(t)=t^{-\alpha}/\Gamma(1-\alpha)$ is the power density
$\rho_{\gamma}(t)$ of Section 3.1 for $\gamma=1-\alpha$. The second equality
arises because $x^{\alpha-1}e^{-\lambda x^{\alpha}}$ is the Laplace transform
of the convolution $\\{\rho_{1-\alpha}\star f_{\alpha}(\cdot|\lambda\\}(t)$.
Uniqueness of Laplace transforms implies the following known property of the
stable density

######  Property 3.

Let $f_{\alpha}(t|\lambda)$ $(0<\alpha<1,t>0)$ be the stable density with
scale factor $\lambda>0$. Then the product $tf_{\alpha}(t|\lambda)$ is
proportional to the convolution $\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|\lambda)\\}(t)$ of the power density
$\rho_{1-\alpha}(t)=t^{-\alpha}/\Gamma(1-\alpha)$ and $f_{\alpha}(t|\lambda)$

$\displaystyle tf_{\alpha}(t|\lambda)$
$\displaystyle\equiv\lambda\alpha\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|\lambda)\\}(t)=\frac{\lambda\alpha}{\Gamma(1-\alpha)}\int_{0}^{t}(t-u)^{-\alpha}f_{\alpha}(u|\lambda)\,du$
(15)

We will refer to Property 3 in discussion of the Mittag-Leffler distribution.

Property 3 may be viewed as a particular instance of the representation of
infinitely divisible distributions on the nonnegative half-line
$\mathbb{R}_{+}=[0,\infty)$, a class that includes the stable distribution,
Feller [11, XIII.7], Steutel and van Harn [55, III (Theorem 4.17)] (also see
Sato [51]). There is also an intimate link to the generalised gamma
convolutions studied by Bondesson [9].

We note that $\\{\rho_{1-\alpha}\star f_{\alpha}(\cdot|\lambda\\}(t)$ may be
regarded as the Riemann-Liouville fractional integral
$\\{I_{+}^{1-\alpha}\,f_{\alpha}(\cdot|\lambda\\}(t)$ of the stable density
$f_{\alpha}(\cdot|\lambda)$, where $\\{I_{+}^{\nu}\,f\\}(t)$ for a function
$f(t)$ ($t\geq 0,\nu>0$) is the convolution $\\{\rho_{\nu}\star f\\}(t)$,
$\rho_{\nu}(t)=t^{\nu-1}/\Gamma(\nu)$ (Kiryakova [32], Miller and Ross [38],
Samko et al. [50]). We shall continue to use the general language of
convolutions than that of fractional integrals. We further note that Property
3 has inspired the study of generalised stable densities $f_{\alpha,m}(t)$
defined by $t^{m}f_{\alpha,m}(t)=\alpha\\{\rho_{1-\alpha}\star
f_{\alpha,m}\\}(t)$ for $m>0$ (Jedidi et al. [30], Pakes [41], Schneider
[53]). We shall not have cause for such generalisation here.

The stable density $f_{\alpha}(t|\lambda)$ is only known in closed form for
selected values of $\alpha$, the simplest being for $\alpha=1/2$. For the
general rational case $\alpha=l/k$ for positive integers $l,k$ with $k>l$,
Penson and Górska [42] expressed $f_{\alpha}$ in terms of the Meijer $G$
function (also see James [23, 8]).

Pollard [46] (also see Feller [11, XVII.6]) proved that
$f_{\alpha}(t|\lambda)$ has the infinite series representation (with
modification here to include conditioning on the scale factor $\lambda>0$)

$\displaystyle f_{\alpha}(t|\lambda)$
$\displaystyle=\phantom{-}\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,e^{-\lambda(e^{-i\pi}u)^{\alpha}}\,du$ (16)
$\displaystyle=\phantom{-}\frac{1}{\pi}\,{\rm
Im}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}e^{-i\pi\alpha
k}\int_{0}^{\infty}e^{-tu}\,u^{\alpha k}\,du$ (17)
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\sin(\pi\alpha
k)\,\frac{\Gamma(\alpha k+1)}{t^{\alpha k+1}}$ (18)

We may equivalently sum from $k=1$ since the $k=0$ term is identically 0.

##  4 Main Contribution

######  Theorem 1.

Let $\psi(x),x\geq 0$ be the Laplace-Stieltjes transform of a distribution
$F(t)$

$\displaystyle\psi(x)$ $\displaystyle=\int_{0}^{\infty}e^{-xt}\,dF(t)\quad
x\geq 0$ (19)

Then, for $0<\alpha<1$ and $\lambda>0$, the composition $\psi(\lambda
x^{\alpha})$ is the Laplace transform of a mixture density
$w_{\alpha}(t|\lambda)$

$\displaystyle\psi(\lambda x^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\alpha}(t|\lambda)\,dt$ (20)
$\displaystyle\textrm{where}\quad w_{\alpha}(t|\lambda)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dF(u)$ (21)

$f_{\alpha}(t|\lambda u)$ being the $\alpha$-stable density conditioned on the
scale factor $\lambda u$.

###### Proof of Theorem 1.

$e^{-\lambda ux^{\alpha}}$ ($0<\alpha<1$) is the Laplace transform of the
stable density $f_{\alpha}(\cdot|\lambda u)$. Hence

$\displaystyle\psi(\lambda x^{\alpha})=\int_{0}^{\infty}e^{-\lambda
x^{\alpha}u}\,dF(u)$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}e^{-xt}f_{\alpha}(t|\lambda
u)dt\,dF(u)$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\left[\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\,dF(u)\right]\,dt$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\alpha}(t|\lambda)\,dt$

where $w_{\alpha}(t|\lambda)$ is the integral in square braces, thereby
proving (21). ∎

For clarity, we may explicitly condition on $F$ or its density $f$: e.g.
$w_{\alpha}(t|\lambda)\to w_{\alpha}(t|\lambda,F)\equiv
w_{\alpha}(t|\lambda,f)$. Equivalently, we may condition on any parameters
that are known to characterise $F$ uniquely, e.g.
$w_{\alpha}(t|\lambda,\gamma,y)$ where $\gamma,y$ are understood to be the
shape and scale parameters of the gamma distribution $G(\gamma,y)$.

######  Corollary 1.1.

Let $\rho_{\nu}(t)=t^{\nu-1}/\Gamma(\nu)$ $(\nu>0)$ and let $f(t)$ be the
density of $F(t)$. Then the following holds

$\displaystyle\\{\rho_{\nu}\star w_{\alpha}(\cdot|\lambda,F)\\}(t)$
$\displaystyle\equiv\int_{0}^{\infty}\\{\rho_{\nu}\star
f_{\alpha}(\cdot|\lambda u)\\}(t)\,dF(u)$
$\displaystyle=\lambda^{\nu/\alpha}\,w_{\alpha}(t|\lambda,\rho_{\nu/\alpha}\star
f)$ (22) $\displaystyle{\rm where}\quad
w_{\alpha}(t|\lambda,\rho_{\nu/\alpha}\star f)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\,\\{\rho_{\nu/\alpha}\star f\\}(u)du$

###### Proof of Corollary 1.1.

By the convolution theorem, the Laplace transform of the convolution density
$\\{\rho_{\nu}\star w_{\alpha}(\cdot|\lambda,F)\\}(t)$ is the product
$x^{-\nu}\psi(\lambda x^{\alpha}|F)=\lambda^{\nu/\alpha}\left[(\lambda
x^{\alpha})^{-\nu/\alpha}\psi(\lambda x^{\alpha}|F)\right]$. The product in
square braces is also the Laplace transform of
$w_{\alpha}(t|\lambda,\\{\rho_{\nu/\alpha}\star f\\})$. Hence the validity of
(22) by uniqueness of Laplace transforms. ∎

The point of Corollary 1.1 is that it gives two different ways to express the
convolution of the power density with the mixture density
$w_{\alpha}(t|\lambda,F)$ – one involving convolution with the stable density
and the other involving convolution with the density of $F$. This will have a
bearing on our approach to the derivation of the Mittag-Leffler distribution
below.

######  Corollary 1.2.

The mixture density $w_{\alpha}(t|\lambda,F)$ admits the representation

$\displaystyle w_{\alpha}(t|\lambda,F)$ $\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\psi(\lambda e^{-i\pi\alpha}u^{\alpha}|F)\,du$
(23) In particular, for the convolution $\rho_{\nu}\star f$ $(\nu>0)$
$\displaystyle w_{\alpha}(t|\lambda,\rho_{\nu}\star f)$
$\displaystyle=\frac{1}{\pi}\,{\rm Im}\int_{0}^{\infty}e^{-tu}\,(\lambda
e^{-i\pi\alpha}u^{\alpha})^{-\nu}\psi(\lambda
e^{-i\pi\alpha}u^{\alpha}|f)\,du$ (24)

###### Proof of Corollary 1.2.

Using the Pollard representation (16) for $f_{\alpha}(t|\lambda u)$ gives

$\displaystyle w_{\alpha}(t|\lambda,F)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dF(u)$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}\int_{0}^{\infty}e^{-tv}\,e^{-\lambda
u(e^{-i\pi}v)^{\alpha}}\,dv\,dF(u)$ $\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tv}\int_{0}^{\infty}\,e^{-\lambda
u(e^{-i\pi}v)^{\alpha}}\,dF(u)\,dv$ $\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tv}\psi(\lambda e^{-i\pi\alpha}v^{\alpha}|F)\,dv$

thereby proving (23). The convolution theorem for $dF(u)=\\{\rho_{\nu}\star
f\\}(u)du$ leads to (24). ∎

######  Corollary 1.3.

Let $F(t)$ have finite moments $\\{M_{k}\\}$ of all order

$\displaystyle M_{k}$
$\displaystyle=\int_{0}^{\infty}u^{k}\,dF(u)\equiv(-1)^{k}\psi^{(k)}(0)<\infty\qquad
k\geq 0$ (25)

Then the mixture density $w_{\alpha}(t|\lambda)$ has the infinite series
representation

$\displaystyle w_{\alpha}(t|\lambda)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dF(u)$
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\sin(\pi\alpha
k)\,M_{k}\int_{0}^{\infty}e^{-tv}\,v^{\alpha k}\,dv$ (26)
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\sin(\pi\alpha
k)\,M_{k}\frac{\Gamma(\alpha k+1)}{t^{\alpha k+1}}$ (27)

###### Proof of Corollary 1.3.

Using the infinite series representation (17) of the stable density
$f_{\alpha}(t|\lambda u)$ in (21) gives

$\displaystyle w_{\alpha}(t|\lambda)$
$\displaystyle=-\frac{1}{\pi}\int_{0}^{\infty}\left[\sum_{k=0}^{\infty}\frac{(-\lambda
u)^{k}}{k!}\sin(\pi\alpha k)\int_{0}^{\infty}e^{-tv}\,v^{\alpha
k}\,dv\right]dF(u)$
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\sin(\pi\alpha
k)\,\left[\int_{0}^{\infty}e^{-tv}\,v^{\alpha
k}\,dv\right]\left[\int_{0}^{\infty}u^{k}\,dF(u)\right]$

The rightmost term is $M_{k}$, thereby proving (26) and hence (27). ∎

We may now turn to a simple step function as our initial choice of $F$.

##  5 Unit Step $F$

Let $F(t)$ be 0 on $0\leq t<1$ with a jump of size 1 at $t=1$, i.e.
$F(t)=\mathbbm{1}_{[1,\infty)}$, where the indicator function
$\mathbbm{1}_{[c,\infty)}$ is 0 on $[0,c)$ and 1 on $[c,\infty)$. The density
of $F(t)$ is $f(t)=\delta(t-1)$. The Laplace-Stieltjes transform of $F(t)$
(Laplace transform of $f(t)$) is

$\displaystyle\psi(x)$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,dF(t)=e^{-x}\implies\psi(\lambda
x^{\alpha})=e^{-\lambda x^{\alpha}}\;(\lambda>0)$ (28)

For $0<\alpha<1$, Theorem 1 states that $\psi(\lambda x^{\alpha})$ is the
Laplace transform of a mixture density

$\displaystyle\psi(\lambda x^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\alpha}(t|\lambda)\,dt=e^{-\lambda
x^{\alpha}}\quad 0<\alpha<1$ (29) $\displaystyle\textrm{where}\quad
w_{\alpha}(t|\lambda)$ $\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\,dF(u)=f_{\alpha}(t|\lambda)$ (30)

For $\rho_{\nu}(t)=t^{\nu-1}/\Gamma(\nu)$ $(\nu>0)$, the convolution density
$\\{\rho_{\nu}\star f\\}(t)$ has the explicit form

$\displaystyle\\{\rho_{\nu}\star f\\}(t)$
$\displaystyle=\int_{0}^{t}\rho_{\nu}(t-u)\,dF(u)=\rho_{\nu}(t-1)\mathbbm{1}_{[1,\infty)}$
(31)

with Laplace transform $x^{-\nu}\psi(x)=x^{-\nu}e^{-x}$.

######  Theorem 2.

For $0<\alpha<1$, $\rho_{\nu}(t)=t^{\nu-1}/\Gamma(\nu)$ $(\nu>0)$,
$\rho_{0}(t)=\delta(t)$ and $w_{\alpha}(t|\lambda)$ as defined in
$(\ref{eq:wstep})$, the convolution density $\\{\rho_{\nu}\star
w_{\alpha}(\cdot|\lambda)\\}(t)\equiv\\{\rho_{\nu}\star
f_{\alpha}(\cdot|\lambda)\\}(t)$ $(\nu>0)$ has the following equivalent forms

$\displaystyle\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$
$\displaystyle=\lambda^{\nu/\alpha}\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\,\\{\rho_{\nu/\alpha}\star f\\}(u)du$ (32)
$\displaystyle=\lambda^{\nu/\alpha}\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\rho_{\nu/\alpha}(u-1)\mathbbm{1}_{[1,\infty)}\,du$ (33)
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\;e^{i\pi\nu}\int_{0}^{\infty}e^{-tu}\,u^{-\nu}\,e^{-\lambda(e^{-i\pi}u)^{\alpha}}\,du$
(34)
$\displaystyle=\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\sin\pi(\nu-\alpha
k)\frac{\Gamma(\alpha k-\nu+1)}{t^{\alpha k-\nu+1}}$ (35)
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\frac{t^{\nu-\alpha
k-1}}{\Gamma(\nu-\alpha
k)}\equiv\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\rho_{\nu-\alpha k}(t)$
(36)

$\nu=0$ reproduces the representation (18) of the stable density
$f_{\alpha}(t|\lambda)\equiv\\{\rho_{0}\star f_{\alpha}(\cdot|\lambda\\}(t)$.

###### Proof of Theorem 2.

Corollary 1.1 gives (32). Substituting (31) in (32) gives (33). Given that the
Laplace transform of (31) is $x^{-\nu}\psi(x)=x^{-\nu}e^{-x}$, Corollary 1.2
gives (34). Explicitly

$\displaystyle\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda\\}(t)$
$\displaystyle=\lambda^{\nu/\alpha}\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\rho_{\nu/\alpha}(u-1)\mathbbm{1}_{[1,\infty)}\,du$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,(e^{-i\pi\alpha}u^{\alpha})^{-\nu/\alpha}\,e^{-\lambda(e^{-i\pi}u)^{\alpha}}\,du$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}e^{i\pi(\nu-\alpha
k)}\int_{0}^{\infty}e^{-tu}u^{\alpha k-\nu}\,du$ The integral may be evaluated
for $\alpha k-\nu+1>0$, which only holds for all $k$ if $\nu<1$, to give
$\displaystyle=\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\sin\pi(\nu-\alpha
k)\frac{\Gamma(\alpha k-\nu+1)}{t^{\alpha k-\nu+1}}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\frac{t^{\nu-\alpha
k-1}}{\Gamma(\nu-\alpha k)}$

The last step follows from the Euler reflection formula $\sin(\pi
z)=\pi/\Gamma(z)\Gamma(1-z)$, where $1/\Gamma(z)$ is defined on the entire
complex plane by standard analytic continuation. Hence, in the end,
1/$\Gamma(\nu-\alpha k)$ is well-defined for $\nu\geq 0$, without the $\nu<1$
restriction imposed to evaluate the intermediate integral. ∎

This shows that the convolution density $\\{\rho_{\nu}\star
f_{\alpha}(\cdot|\lambda)\\}(t)$ may be replaced by a mixture involving the
simple convolution between $\rho_{\nu/\alpha}$ and a $\delta$ density. In
turn, the mixture may optionally be represented as an infinite sum of
(analytically continued) copies $\\{\rho_{\nu-\alpha k}(t)\\}$ of the power
density $\rho_{\nu}(t)$. $\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$
is the basic building block for all other densities discussed in the balance
of this paper. In the case of $\nu=1-\alpha$, we have already seen from
Property 3 that the convolution has a simple form $\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|\lambda)\\}(t)=tf_{\alpha}(t|\lambda)/\lambda\alpha$.

In addition to the forms
$(\ref{eq:rhoconvstable1})-(\ref{eq:rhoconvstableinfsum1})$,
$\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$ has an equivalent
representation involving the beta distribution.

######  Corollary 2.1.

The convolution density $\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$
can be written as

$\displaystyle\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$
$\displaystyle=\frac{\lambda^{\nu/\alpha}}{\Gamma(\nu/\alpha)}\int_{0}^{1}f_{\alpha}(t|\lambda/u)\,u^{-\nu/\alpha}(1-u)^{\nu/\alpha-1}\,\frac{du}{u}$
(37)
$\displaystyle\equiv\Gamma\left(1-\tfrac{\nu}{\alpha}\right)\lambda^{\nu/\alpha}\int_{0}^{1}f_{\alpha}(t|\lambda/u)\,{\rm
beta}\left(u|1-\tfrac{\nu}{\alpha},\tfrac{\nu}{\alpha}\right)\frac{du}{u}$
(38)

This has the form of (12), the density of a product of two variables where one
of them is beta distributed. We shall pursue this fundamental result further
later. It suffices to note for now that there are various ways to represent
the convolution $\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$.

###### Proof of Corollary 2.1.

By (33)

$\displaystyle\\{\rho_{\nu}\star f_{\alpha}(\cdot|\lambda)\\}(t)$
$\displaystyle=\lambda^{\nu/\alpha}\int_{1}^{\infty}f_{\alpha}(t|\lambda
u)\rho_{\nu/\alpha}(u-1)\,du$ $\displaystyle(u\to 1/u)\quad$
$\displaystyle=\lambda^{\nu/\alpha}\int_{0}^{1}f_{\alpha}(t|\lambda/u)\rho_{\nu/\alpha}(1/u-1)\,\frac{du}{u^{2}}$
$\displaystyle=\frac{\lambda^{\nu/\alpha}}{\Gamma(\nu/\alpha)}\int_{0}^{1}f_{\alpha}(t|\lambda/u)\,u^{-\nu/\alpha}(1-u)^{\nu/\alpha-1}\,\frac{du}{u}$

thereby proving (37), hence (38). ∎

For $\lambda=1$, (37) is equivalent to Ho et al. [22, Proposition 2.2(iii)],
derived by different reasoning (with the convolution $\rho_{\nu}\star
f_{\alpha}$ expressed as the Riemann-Liouville fractional integral
$I_{+}^{\nu}\,f_{\alpha}$).

We may now turn to other choices of $F$, starting with the exponential
distribution. The results presented for this case are almost entirely well-
known, the point here is the manner of their derivation using convolutions and
mixtures and the insights arising therefrom.

##  6 Exponentially Distributed $F$

Let $F\equiv G(1,y)$ with density $\rho_{1,y}$ be the exponential
distribution, a particular case of the gamma distribution defined in Section
3.1, with shape parameter $\gamma=1$ and scale parameter $y>0$. Explicitly,
$dF(t|y)\equiv\rho_{1,y}(t)dt=ye^{-yt}dt$ with Laplace-Stieltjes transform

$\displaystyle\psi(x|y)$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,dF(t|y)=\frac{y}{y+x}\quad\implies\psi(x^{\alpha}|y)=\frac{y}{y+x^{\alpha}}$
(39)

For $0<\alpha<1$, Theorem 1 states that $\psi(x^{\alpha}|y)$ is the Laplace
transform of a mixture density

$\displaystyle\psi(x^{\alpha}|y)$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\alpha}(t|y)\,dt$ (40)
$\displaystyle\textrm{where}\quad w_{\alpha}(t|y)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\,dF(u|y)$ (41)

######  Theorem 3.

Let $0<\alpha<1$, $w_{\alpha}(t|y)$ given by $(\ref{eq:wexp})$ and
$\rho_{1-\alpha}(t)=t^{-\alpha}/\Gamma(1-\alpha)$. Then the Mittag-Leffler
function $E_{\alpha}(-yt^{\alpha})$ has the convolution representation

$\displaystyle yE_{\alpha}(-yt^{\alpha})$
$\displaystyle=\\{\rho_{1-\alpha}\star w_{\alpha}(\cdot|y)\\}(t)\quad t>0,y>0$
(42) $\displaystyle\equiv\int_{0}^{\infty}\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|u)\\}(t)\,dF(u|y)$ (43)
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{1/\alpha-1}\star\rho_{1,y}\\}(u)\,du$
(44)

where $dF(u|y)\equiv\rho_{1,y}(u)du=ye^{-yu}du$.

###### Proof of Theorem 3.

The convolution $\\{\rho_{1-\alpha}\star w_{\alpha}(\cdot|y)\\}(t)$ of
$\rho_{1-\alpha}(t)$ (Laplace transform $x^{\alpha-1}$) and $w_{\alpha}(t|y)$
gives the following

$\displaystyle
x^{\alpha-1}\psi(x^{\alpha}|y)=y\,\frac{x^{\alpha-1}}{y+x^{\alpha}}$
$\displaystyle=y\int_{0}^{\infty}e^{-xt}\,E_{\alpha}(-yt^{\alpha})\,dt$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,\\{\rho_{1-\alpha}\star
w_{\alpha}(\cdot|y)\\}(t)\,dt$

where the Laplace transform of the Mittag-Leffler function
$E_{\alpha}(-yt^{\alpha})$ is read from (2). Hence
$yE_{\alpha}(-yt^{\alpha})=\\{\rho_{1-\alpha}\star w_{\alpha}(\cdot|y)\\}(t)$,
which is identical to (43) by the definition (41) of $w_{\alpha}(t|y)$. The
equality of (43) and (44) follows from Corollary 1.1 of Theorem 1. ∎

While (43) and (44) are formally equivalent, they lead to different outcomes.
We interpret $E_{\alpha}(-yt^{\alpha})$ as a function of two variables $t$ and
$y$. Corollary 3.1 treats (43) as a function of $y$ given $t$. This leads to
the Mittag-Leffler distribution, expressed as a convolution. Corollary 3.2
gives a consequence thereof. Corollary 3.3, on the other hand, treats (44) as
a function of $t$ given $y$, leading directly to the result of Corollary 3.2,
but without the ability to reproduce the underlying results of Corollary 3.1
concerning the Mittag-Leffler distribution.

######  Corollary 3.1 (Mittag-Leffler distribution as a convolution).

For $0<\alpha<1$, $E_{\alpha}(-yt^{\alpha})$ as a function of $y$ given $t>0$
is the Laplace-Stieltjes transform of a distribution $P_{\alpha}(u|t)$

$\displaystyle E_{\alpha}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-yu}\,dP_{\alpha}(u|t)\quad(y\geq 0)$ (45)
$\displaystyle{\rm where}\quad dP_{\alpha}(u|t)$
$\displaystyle=\\{\rho_{1-\alpha}\star f_{\alpha}(\cdot|u)\\}(t)\,du$ (46)
$\displaystyle=\frac{t}{\alpha}f_{\alpha}(t|u)\,u^{-1}\,du\equiv\frac{t}{\alpha}f_{\alpha}(tu^{-1/\alpha})\,u^{-1/\alpha-1}\,du$
(47) For $t=1$, $P_{\alpha}(u)\equiv P_{\alpha}(u|1)$ with density
$p_{\alpha}(t)$ is known as the Mittag-Leffler distribution with Laplace-
Stieltjes transform $E_{\alpha}(-y)$ $\displaystyle E_{\alpha}(-y)$
$\displaystyle=\int_{0}^{\infty}e^{-yu}\,dP_{\alpha}(u)=\int_{0}^{\infty}e^{-yu}\,p_{\alpha}(u)\,du$
(48) $\displaystyle{\rm where}\quad p_{\alpha}(u)$
$\displaystyle=\\{\rho_{1-\alpha}\star f_{\alpha}(\cdot|u)\\}(1)$ (49)
$\displaystyle=\frac{1}{\alpha}f_{\alpha}(1|u)\,u^{-1}\equiv\frac{1}{\alpha}f_{\alpha}(u^{-1/\alpha})\,u^{-1/\alpha-1}\,$
(50) $\displaystyle=\frac{1}{\pi\alpha u}\,{\rm
Im}\int_{0}^{\infty}e^{-v}\,e^{-u(e^{-i\pi}v)^{\alpha}}\,dv$ (51)
$\displaystyle=\frac{1}{\pi\alpha}\sum_{k=1}^{\infty}\frac{(-u)^{k-1}}{k!}\sin(\pi\alpha
k)\,\Gamma(\alpha k+1)$ (52)

The moments of $P_{\alpha}$ are

$\displaystyle M_{\alpha;k}$
$\displaystyle=\int_{0}^{\infty}u^{k}\,dP_{\alpha}(u)\equiv(-1)^{k}E^{(k)}_{\alpha}(0)=\frac{k!}{\Gamma(\alpha
k+1)}\quad k\geq 0$ (53)

###### Proof of Corollary 3.1.

Given that $dF(u|y)=ye^{-yu}du$, $(\ref{eq:MLconv1})$ gives

$\displaystyle E_{\alpha}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-yu}\,\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|u)\\}(t)\,du\equiv\int_{0}^{\infty}e^{-yu}\,dP_{\alpha}(u|t)\quad(t>0)$
where $dP_{\alpha}(u|t)=\\{\rho_{1-\alpha}\star f_{\alpha}(\cdot|u)\\}(t)du$.
By Property 3, $\alpha u\,\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|u)\\}(t)=tf_{\alpha}(t|u)$. Hence $\displaystyle
dP_{\alpha}(u|t)$ $\displaystyle=\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|u)\\}(t)\,du=\frac{t}{\alpha}f_{\alpha}(t|u)\,u^{-1}\,du\equiv\frac{t}{\alpha}f_{\alpha}(tu^{-1/\alpha})\,u^{-1/\alpha-1}\,du$

The Mittag-Leffler distribution is $P_{\alpha}(u)\equiv P_{\alpha}(u|1)$. The
forms of the density given by (51) and (52) follow from (16) and (18)
respectively. The moments of $P_{\alpha}(u)$ follow easily. ∎

As discussed in Section 2.2, Pollard [47] used complex analysis to derive
$P_{\alpha}$. In a probabilistic derivation, Feller [11, XIII.8] used the two-
dimensional Laplace transform of the one-sided $\alpha$-stable distribution
with an associated scale factor. Our approach here is inspired by Feller’s
approach, with convolution as a running theme throughout the paper. Hence the
density takes a convolution form equivalent to the familiar form given by
(50).

######  Corollary 3.2 (Composition).

For $0<\alpha<1$ and $0<\sigma<1$, the composition $E_{\alpha}(-\lambda
x^{\sigma})$ as a function of $x$ given $\lambda>0$ is the Laplace transform
of a mixture density $w_{\sigma,\alpha}(t|\lambda)$

$\displaystyle E_{\alpha}(-\lambda x^{\sigma})$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\sigma,\alpha}(t|\lambda)\,dt\quad(x\geq
0)$ (54) $\displaystyle\textrm{where}\quad w_{\sigma,\alpha}(t|\lambda)$
$\displaystyle=\int_{0}^{\infty}f_{\sigma}(t|\lambda u)\,dP_{\alpha}(u)$ (55)
$\displaystyle=\phantom{-}\frac{1}{\pi}{\rm
Im}\int_{0}^{\infty}e^{-tu}\,E_{\alpha}(-\lambda
e^{-i\pi\sigma}u^{\sigma})\,du$ (56)
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\sigma
k)\,\frac{\Gamma(\sigma k+1)}{\Gamma(\alpha k+1)}\,t^{-\sigma k-1}$ (57)

For $\sigma=\alpha$, $E_{\alpha}(-\lambda x^{\alpha})$ is the Laplace
transform of the mixture density $w_{\alpha}(t|\lambda)\equiv
w_{\alpha,\alpha}(t|\lambda)$

$\displaystyle w_{\alpha}(t|\lambda)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda
u)\,dP_{\alpha}(u)=\frac{\sin\pi\alpha}{\pi}\,\frac{\lambda\,t^{\alpha-1}}{\lambda^{2}+2\lambda\,t^{\alpha}\cos\pi\alpha+t^{2\alpha}}$
(58)

###### Proof of Corollary 3.2.

By Theorem 1 with $F(t)\equiv P_{\alpha}(t)$ and $\psi(x)\equiv
E_{\alpha}(-x)$, $\psi(\lambda x^{\sigma})\equiv E_{\alpha}(-\lambda
x^{\sigma})$ is the Laplace transform of a mixture density given by
$(\ref{eq:wML})$. The equivalent forms $(\ref{eq:wMLPollard})$ and
$(\ref{eq:wMLPollardseries})$ follow from Corollaries 1.2 and 1.3
respectively. For $\sigma=\alpha$, (55) becomes

$\displaystyle w_{\alpha}(t|\lambda)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dP_{\alpha}(u)$
$\displaystyle=\phantom{-}\frac{1}{\pi}{\rm
Im}\int_{0}^{\infty}e^{-tu}\,E_{\alpha}(-\lambda
e^{-i\pi\alpha}u^{\alpha})\,du$ $\displaystyle=\frac{1}{\pi}\,{\rm
Im}\,\frac{t^{\alpha-1}}{\lambda e^{-i\pi\alpha}+t^{\alpha}}$ or
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\alpha
k)\,t^{-\alpha k-1}$ $\displaystyle=\frac{1}{\pi t}\,{\rm
Im}\,\frac{1}{1+\lambda e^{-i\pi\alpha}t^{-\alpha}}$
$\displaystyle=\frac{\sin\pi\alpha}{\pi}\,\frac{\lambda\,t^{\alpha-1}}{\lambda^{2}+2\lambda\,t^{\alpha}\cos\pi\alpha+t^{2\alpha}}$

with the Laplace transform and the geometric sum leading to the same result. ∎

The fundamental role of the Mittag-Leffler distribution $P_{\alpha}$ is thus
evident. $E_{\alpha}(-x)$ and the compositions $E_{\alpha}(-\lambda
x^{\sigma}),E_{\alpha}(-\lambda x^{\alpha})$ arise from $P_{\alpha}$ and
mixtures of $f_{\sigma},f_{\alpha}$ with respect to $P_{\alpha}$.

The $\sigma=\alpha$ density (58) is well-known in the probabilistic
literature. It arises as the distribution of the ratio of two $\alpha$-stable
random variables (e.g. James [23], James et al. [25]). It has an intimate
association with occupation times in Brownian motion (Bertoin et al. [7],
Lamperti [35])). It is also the Thorin measure (in the language of the family
of infinitely divisible distributions known as generalised gamma convolutions)
induced by a product involving gamma and stable random variables (Bondesson
[9, 3.2.4 (p38)]).

Theorem 3 also enables an alternative and direct construction of
$E_{\alpha}(-\lambda x^{\alpha})$ as the Laplace transform of the density
(58), as shown in Corollary 3.3 below. However, $P_{\alpha}$ cannot be
inferred from this.

######  Corollary 3.3.

For $0<\alpha<1$, the representation $(\ref{eq:MLconv2})$ directly implies
that $E_{\alpha}(-yt^{\alpha})$ as a function of $t$ given $y>0$ is the
Laplace transform of a density $q_{\alpha}(u|y)$

$\displaystyle yE_{\alpha}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{1/\alpha-1}\star\rho_{1,y}\\}(u)\,du$
$\displaystyle\implies E_{\alpha}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-tu}\,q_{\alpha}(u|y)\,du\quad(t\geq 0)$
(59) $\displaystyle{\rm where}\quad q_{\alpha}(u|y)$
$\displaystyle=\frac{1}{y\pi}\,{\rm
Im}\left\\{\widetilde{\rho}_{1/\alpha-1}(e^{-i\pi\alpha}u^{\alpha})\,\widetilde{\rho}_{1,y}(e^{-i\pi\alpha}u^{\alpha})\right\\}$
(60)
$\displaystyle=\frac{\sin\pi\alpha}{\pi}\,\frac{y\,u^{\alpha-1}}{y^{2}+2y\,u^{\alpha}\cos\pi\alpha+u^{2\alpha}}\,du$
(61)

###### Proof of Corollary 3.3.

The Laplace transforms of $\rho_{1/\alpha-1}(t)$ and $\rho_{1,y}(t)$ are
$\widetilde{\rho}_{1/\alpha-1}(x)=x^{1-1/\alpha}$ and
$\widetilde{\rho}_{1,y}(x)=y/(y+x)$ respectively. Hence, using the
representation (24) of Corollary 1.2 in $(\ref{eq:MLconv2})$ gives

$\displaystyle yE_{\alpha}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{1/\alpha-1}\star\rho_{1,y}\\}(u)\,du$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\widetilde{\rho}_{1/\alpha-1}(e^{-i\pi\alpha}u^{\alpha})\,\widetilde{\rho}_{1,y}(e^{-i\pi\alpha}u^{\alpha})\,du$
$\displaystyle=\frac{y}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\frac{(e^{-i\pi\alpha}u^{\alpha})^{1-1/\alpha}}{y+e^{-i\pi\alpha}u^{\alpha}}\,du$
$\displaystyle\implies E_{\alpha}(-yt^{\alpha})$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\frac{(e^{-i\pi}u)^{\alpha-1}}{y+(e^{-i\pi}u)^{\alpha}}\,du$
$\displaystyle=\frac{\sin\pi\alpha}{\pi}\int_{0}^{\infty}e^{-tu}\,\frac{y\,u^{\alpha-1}}{y^{2}+2y\,u^{\alpha}\cos\pi\alpha+u^{2\alpha}}\,du$

thereby proving that $E_{\alpha}(-yt^{\alpha})$ is the Laplace transform of
the density (61). ∎

We summarise the results of Theorem 3 in Table 1. We may now readily
generalise the foregoing construction, based on the exponentially distributed
$F$, to that based on the gamma distribution with general shape parameter.

##  7 Gamma Distributed $F$

Let $F\equiv G(\gamma,y)$ with density $\rho_{\gamma,y}$ be the gamma
distribution as defined in Section 3.1, with shape parameter $\gamma>0$ and
scale parameter $y>0$. Explicitly,
$dG(t|\gamma,y)=y^{\gamma}t^{\gamma-1}/\Gamma(\gamma)e^{-yt}dt$ with Laplace-
Stieltjes transform

$\displaystyle\psi(x|\gamma,y)$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,dG(t|\gamma,y)=\frac{y^{\gamma}}{(y+x)^{\gamma}}\implies\psi(x^{\alpha}|\gamma,y)=\frac{y^{\gamma}}{(y+x^{\alpha})^{\gamma}}$
(62)

For $0<\alpha<1$, Theorem 1 states that (62) is the Laplace transform of a
mixture density

$\displaystyle\psi(x^{\alpha}|\gamma,y)$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\alpha}(t|\gamma,y)\,dt$ (63)
$\displaystyle\textrm{where}\quad w_{\alpha}(t|\gamma,y)$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\,dG(u|\gamma,y)$ (64)

######  Theorem 4.

Let $0<\alpha<1$, $w_{\alpha}(t|\gamma,y)$ $(\gamma>0)$ given by
$(\ref{eq:wgamma})$ and
$\rho_{\beta-\alpha\gamma}(t)=t^{\beta-\alpha\gamma-1}/\Gamma(\beta-\alpha\gamma)$
for $\beta-\alpha\gamma>0$. More compactly, $0<\alpha<1$,
$0<\alpha\gamma<\beta$. Then
$t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$, where
$E^{\gamma}_{\alpha,\beta}(\cdot)$ is the 3-parameter Mittag-Leffler or
Prabhakar function, has the convolution representation

$\displaystyle y^{\gamma}\,t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$
$\displaystyle=\\{\rho_{\beta-\alpha\gamma}\star
w_{\alpha}(\cdot|\gamma,y)\\}(t)$ (65)
$\displaystyle\equiv\int_{0}^{\infty}\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(t)\,dG(u|\gamma,y)$ (66)
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{\beta/\alpha-\gamma}\star\rho_{\gamma,y}\\}(u)\,du$
(67)

###### Proof of Theorem 4.

The convolution $\\{\rho_{\beta-\alpha\gamma}\star
w_{\alpha}(\cdot|\gamma,y)\\}(t)$ of $w_{\alpha}(t|\gamma,y)$ and
$\rho_{\beta-\alpha\gamma}(t)$ (Laplace transform $x^{\alpha\gamma-\beta}$)
gives the following

$\displaystyle
x^{\alpha\gamma-\beta}\psi(x^{\alpha}|\gamma,y)=y^{\gamma}\,\frac{x^{\alpha\gamma-\beta}}{(y+x^{\alpha})^{\gamma}}$
$\displaystyle=y^{\gamma}\int_{0}^{\infty}e^{-xt}\,t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})\,dt$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,\\{\rho_{\beta-\alpha\gamma}\star
w_{\alpha}(\cdot|\gamma,y)\\}(t)\,dt$

where the Laplace transform of
$t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$ is read from (2). Hence
$y^{\gamma}t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})=\\{\rho_{\beta-\alpha\gamma}\star
w_{\alpha}(\cdot|y)\\}(t)$, which is identical to (66) by the definition (64)
of $w_{\alpha}(t|\gamma,y)$. The equality of (66) and (67) follows from
Corollary 1.1 of Theorem 1. ∎

As in Theorem 3 above, the equivalent representations (66) and (67) have
different outcomes. We start with (66) which gives a definition of the
3-parameter Mittag-Leffler distribution as a generalisation of the single
parameter case.

######  Corollary 4.1 (3-parameter Mittag-Leffler distribution).

For $0<\alpha<1$, $0<\alpha\gamma<\beta$,
$\Gamma(\beta)t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-xt^{\alpha})$ as a
function of $x$ given $t>0$ is the Laplace-Stieltjes transform of a
3-parameter distribution $P_{\alpha,\beta,\gamma}(u|t)$ with density
$p_{\alpha,\beta,\gamma}(u|t)$

$\displaystyle\Gamma(\beta)t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-xt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-xu}\,dP_{\alpha,\beta,\gamma}(u|t)\quad(x\geq
0)$ (68) $\displaystyle{\rm where}\quad p_{\alpha,\beta,\gamma}(u|t)$
$\displaystyle=\Gamma(\beta)\,\rho_{\gamma}(u)\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(t)$ (69)
$\displaystyle=\Gamma(\beta)\,\rho_{\gamma}(u)u^{\beta/\alpha-\gamma}\int_{0}^{\infty}f_{\alpha}(t|uv)\rho_{\beta/\alpha-\gamma}(v-1)\mathbbm{1}_{[1,\infty)}dv$
(70) $\displaystyle=\frac{\Gamma(\beta)\rho_{\gamma}(u)}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tv}\,(e^{-i\pi}v)^{\alpha\gamma-\beta}\,e^{-u(e^{-i\pi}v)^{\alpha}}\,dv$
(71)
$\displaystyle=\frac{\Gamma(\beta)}{\Gamma(\gamma)}\,u^{\gamma-1}\sum_{k=0}^{\infty}\frac{(-u)^{k}}{k!}\frac{t^{\beta-\alpha\gamma-\alpha
k-1}}{\Gamma(\beta-\alpha\gamma-\alpha k)}$ (72)

For $t=1$, $P_{\alpha,\beta,\gamma}(u)\equiv P_{\alpha,\beta,\gamma}(u|1)$
with density $p_{\alpha,\beta,\gamma}(u)\equiv p_{\alpha,\beta,\gamma}(u|1)$
is the 3-parameter Mittag-Leffler distribution with Laplace-Stieltjes
transform $\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-x)$

$\displaystyle\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-x)$
$\displaystyle=\int_{0}^{\infty}e^{-xu}\,dP_{\alpha,\beta,\gamma}(u)=\int_{0}^{\infty}e^{-xu}\,p_{\alpha,\beta,\gamma}(u)\,du$
(73) $\displaystyle{\rm where}\quad p_{\alpha,\beta,\gamma}(u)$
$\displaystyle=\Gamma(\beta)\,\rho_{\gamma}(u)\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(1)$ (74)
$\displaystyle=\Gamma(\beta)\rho_{\gamma}(u)u^{\beta/\alpha-\gamma}\int_{0}^{\infty}f_{\alpha}(1|uv)\rho_{\beta/\alpha-\gamma}(v-1)\mathbbm{1}_{[1,\infty)}dv$
(75) $\displaystyle=\frac{\Gamma(\beta)\rho_{\gamma}(u)}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-v}\,(e^{-i\pi}v)^{\alpha\gamma-\beta}\,e^{-u(e^{-i\pi}v)^{\alpha}}\,dv$
(76)
$\displaystyle=\frac{\Gamma(\beta)}{\Gamma(\gamma)}\,u^{\gamma-1}\sum_{k=0}^{\infty}\frac{(-u)^{k}}{k!}\frac{1}{\Gamma(\beta-\alpha\gamma-\alpha
k)}$ (77)

The moments $M_{\alpha,\beta,\gamma;k}$ $k\geq 0$ of $P_{\alpha,\beta,\gamma}$
are

$\displaystyle M_{\alpha,\beta,\gamma;k}$
$\displaystyle\equiv\int_{0}^{\infty}u^{k}\,dP_{\alpha,\beta,\gamma}(u)=\frac{\Gamma(\beta)\Gamma(\gamma+k)}{\Gamma(\gamma)\Gamma(\alpha
k+\beta)}$ (78)

###### Proof of Corollary 4.1.

Using $dG(u|\gamma,y)=y^{\gamma}\rho_{\gamma}(u)e^{-yu}\,du$ in (66) and
scaling by $\Gamma(\beta)$ gives (68) with the density
$p_{\alpha,\beta,\gamma}(u|t)$ given by (69). The equivalent forms (70), 71)
and (72) follow from expressing the convolution density
$\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|u)\\}(t)$ in the
respective forms (33), (34) and (35) of Theorem 2. Setting $t=1$ gives (73)
with the density $p_{\alpha,\beta,\gamma}(u)$ given by (74) and corresponding
equivalent forms. The moments (78) follow as
$\Gamma(\beta)(-1)^{k}E^{\gamma}_{\alpha,\beta}{}^{(k)}(0)$. ∎

Möhle [40, Lemma 2] proved (77) by reference to a representation based on the
Wright function by Tomovski et al. [57, Theorem 2]. Möhle also cites Flajolet
et al. [12, (80)] for the same result, obtained by complex analytic inversion.

######  Corollary 4.2 (Composition).

For $0<\alpha<1,0<\sigma<1$ and $0<\alpha\gamma<\beta$, the composition
$\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-\lambda x^{\sigma})$ as a function of
$x$ given $\lambda>0$ is the Laplace transform of a mixture density
$w_{\sigma,\alpha}(t|\lambda,\gamma,\beta)$

$\displaystyle\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-\lambda x^{\sigma})$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\sigma,\alpha}(t|\lambda,\gamma,\beta)\,dt\quad(x\geq
0)$ (79) $\displaystyle\textrm{where}\quad
w_{\sigma,\alpha}(t|\lambda,\gamma,\beta)$
$\displaystyle=\int_{0}^{\infty}f_{\sigma}(t|\lambda
u)\,dP_{\alpha,\beta,\gamma}(u)$ (80)
$\displaystyle=\phantom{-}\frac{\Gamma(\beta)}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,E^{\gamma}_{\alpha,\beta}(-\lambda
e^{-i\pi\sigma}u^{\sigma})\,du$ (81)
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\sigma
k)\,M_{\alpha,\beta,\gamma;k}\frac{\Gamma(\sigma k+1)}{t^{\sigma k+1}}$ (82)

###### Proof of Corollary 4.2.

By Theorem 1 with $F(t)\equiv P_{\alpha,\beta,\gamma}(t)$ and
$\psi(x)\equiv\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-x)$, the composition
$\psi(\lambda x^{\sigma})\equiv\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-\lambda
x^{\sigma})$ is the Laplace transform of a mixture density given by
$(\ref{eq:wML3par})$. The equivalent forms $(\ref{eq:wML3parPollard})$ and
$(\ref{eq:wML3parPollardseries})$ follow from Corollaries 1.2 and 1.3
respectively. ∎

######  Corollary 4.3.

For $0<\alpha<1$ and $0<\alpha\gamma<\beta$, the representation
$(\ref{eq:ML3parconv2})$ implies that
$t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$ as a function of $t$
given $y>0$ is the Laplace transform of a density
$q_{\alpha,\beta,\gamma}(u|y)$

$\displaystyle y^{\gamma}\,t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{\beta/\alpha-\gamma}\star\rho_{\gamma,y}\\}(u)\,du$
$\displaystyle\implies t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-tu}\,q_{\alpha,\beta,\gamma}(u|y)\,du\quad(t\geq
0)$ (83) $\displaystyle{\rm where}\quad q_{\alpha,\beta,\gamma}(u|y)$
$\displaystyle=\frac{1}{y^{\gamma}\pi}\,{\rm
Im}\left\\{\widetilde{\rho}_{\beta/\alpha-\gamma}(e^{-i\pi\alpha}u^{\alpha})\,\widetilde{\rho}_{\gamma,y}(e^{-i\pi\alpha}u^{\alpha})\right\\}$
(84) $\displaystyle=\frac{1}{\pi}\,{\rm
Im}\frac{(e^{-i\pi}u)^{\alpha\gamma-\beta}}{(y+e^{-i\pi\alpha}u^{\alpha})^{\gamma}}$
(85)

###### Proof of Corollary 4.3.

The Laplace transforms of $\rho_{\beta/\alpha-\gamma}(t)$ and
$\rho_{\gamma,y}(t)$ are respectively $x^{\gamma-\beta/\alpha}$ and
$y^{\gamma}/(y+x)^{\gamma}$. Hence, using the representation (24) of Corollary
1.2 in $(\ref{eq:ML3parconv2})$ gives

$\displaystyle y^{\gamma}\,t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{\beta/\alpha-\gamma}\star\rho_{\gamma,y}\\}(u)\,du$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\widetilde{\rho}_{\beta/\alpha-\gamma}(e^{-i\pi\alpha}u^{\alpha})\,\widetilde{\rho}_{\gamma,y}(e^{-i\pi\alpha}u^{\alpha})\,du$
$\displaystyle=\frac{y^{\gamma}}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\frac{(e^{-i\pi\alpha}u^{\alpha})^{\gamma-\beta/\alpha}}{(y+e^{-i\pi\alpha}u^{\alpha})^{\gamma}}\,du$
$\displaystyle\implies t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$
$\displaystyle=\frac{1}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,\frac{(e^{-i\pi}u)^{\alpha\gamma-\beta}}{(y+e^{-i\pi\alpha}u^{\alpha})^{\gamma}}\,du$

thereby completing the proof. ∎

At face value, the same result can be obtained from the product of
$t^{\beta-1}$ and the composition $E^{\gamma}_{\alpha,\beta}(-\lambda
t^{\sigma})$ of Corollary 4.2 for $\sigma=\alpha$. However, $t^{\beta-1}$ then
has to be the Laplace transform of the density
$\rho_{1-\beta}(t)=t^{-\beta}/\Gamma(1-\beta)$ for $\beta<1$, so that
$t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-\lambda t^{\alpha})$ is the Laplace
transform of the convolution of two densities. By contrast, the additional
condition $\beta<1$ is not needed in Corollary 4.3 since it does not rely on
looking upon $t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$ as a product
of 2 separate completely monotone functions $t^{\beta-1}$ and
$E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$.

As noted in Section 2.3.2, Capelas de Oliveira et al. [10] and Mainardi and
Garrappa [36] used a complex analytic approach to prove that
$t^{\beta-1}E^{\gamma}_{\alpha,\beta}(-yt^{\alpha})$ is the Laplace transform
of a nonnegative function for $0<\alpha\leq 1$ and
$0<\alpha\gamma\leq\beta\leq 1$. However, as argued above, the additional
condition $\beta\leq 1$ can be avoided.

It is easy to verify that all results of the single parameter case are special
cases of the 3-parameter case. Corollary 3.3 is the special case
$\beta=\gamma=1$.

We summarise the results of Theorem 4 in Table 2. It is straightforward to
generalise the 3-parameter case to a 4-parameter case, as discussed next.

###  7.1 A 4-parameter Mittag-Leffler distribution

We note that $\beta-\alpha\gamma$ is invariant under $\beta\to\beta+\theta$
and $\gamma\to\gamma+\theta/\alpha$ for an additional parameter $\theta$.
$\gamma+\theta/\alpha$ is the modified shape parameter of the Gamma
distribution considered in Theorem 4 and it must satisfy
$\gamma+\theta/\alpha>0$ or $\theta>-\alpha\gamma$. Hence we may state an
extended form of Theorem 4, omitting repetition of proofs

######  Theorem 5.

Consider Theorem 4 with $\beta\to\beta+\theta$ and
$\gamma\to\gamma+\theta/\alpha$ so that $\beta-\alpha\gamma$ remains
invariant. Then $(\ref{eq:wgamma})$ becomes
$w_{\alpha}(t|\gamma+\theta/\alpha,y)$ with $\gamma+\theta/\alpha>0$. More
compactly, $0<\alpha<1$, $-\theta<\alpha\gamma<\beta$. Then
$t^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-yt^{\alpha})$,
has the convolution representation

$\displaystyle
y^{\gamma+\theta/\alpha}\,t^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-yt^{\alpha})$
$\displaystyle=\\{\rho_{\beta-\alpha\gamma}\star
w_{\alpha}(\cdot|\gamma+\theta/\alpha,y)\\}(t)$ (86)
$\displaystyle\equiv\int_{0}^{\infty}\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(t)\,dG(u|\gamma+\theta/\alpha,y)$ (87)
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{\beta/\alpha-\gamma}\star\rho_{\gamma+\theta/\alpha,y}\\}(u)\,du$
(88)

######  Corollary 5.1 (4-parameter Mittag-Leffler distribution).

For $0<\alpha<1$ and $-\theta<\alpha\gamma<\beta$,
$\Gamma(\beta+\theta)t^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-xt^{\alpha})$
as a function of $x$ given $t>0$ is the Laplace-Stieltjes transform of a
4-parameter distribution $P_{\alpha,\beta,\gamma,\theta}(\cdot|t)$ with
density $p_{\alpha,\beta,\gamma,\theta}(u|t)$

$\displaystyle\Gamma(\beta+\theta)t^{\beta+\theta-1}$ $\displaystyle
E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-xt^{\alpha})=\int_{0}^{\infty}e^{-xu}\,dP_{\alpha,\beta,\gamma,\theta}(u|t)\quad(x\geq
0)$ (89) $\displaystyle{\rm where}\;p_{\alpha,\beta,\gamma,\theta}(u|t)$
$\displaystyle=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(u)\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(t)$ (90)
$\displaystyle=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(u)u^{\beta/\alpha-\gamma}\int_{0}^{\infty}f_{\alpha}(t|uv)\rho_{\beta/\alpha-\gamma}(v-1)\mathbbm{1}_{[1,\infty)}dv$
(91)
$\displaystyle=\frac{\Gamma(\beta+\theta)\rho_{\gamma+\theta/\alpha}(u)}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tv}\,(e^{-i\pi}v)^{\alpha\gamma-\beta}\,e^{-u(e^{-i\pi}v)^{\alpha}}\,dv$
(92)
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\,u^{{\gamma+\theta/\alpha}-1}\sum_{k=0}^{\infty}\frac{(-u)^{k}}{k!}\frac{t^{\beta-\alpha\gamma-\alpha
k-1}}{\Gamma(\beta-\alpha\gamma-\alpha k)}$ (93)

For $t=1$, $P_{\alpha,\beta,\gamma,\theta}(u)\equiv
P_{\alpha,\beta,\gamma,\theta}(u|1)$ with density
$p_{\alpha,\beta,\gamma,\theta}(u)\equiv p_{\alpha,\beta,\gamma,\theta}(u|1)$
is the 4-parameter Mittag-Leffler distribution with Laplace-Stieltjes
transform
$\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)$

$\displaystyle\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)$
$\displaystyle=\int_{0}^{\infty}e^{-xu}\,dP_{\alpha,\beta,\gamma,\theta}(u)=\int_{0}^{\infty}e^{-xu}\,p_{\alpha,\beta,\gamma,\theta}(u)\,du$
(94) $\displaystyle{\rm where}\quad p_{\alpha,\beta,\gamma,\theta}(u)$
$\displaystyle=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(u)\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(1)$ (95)
$\displaystyle=\Gamma(\beta+\theta)\rho_{\gamma+\theta/\alpha}(u)u^{\beta/\alpha-\gamma}\int_{0}^{\infty}f_{\alpha}(1|uv)\rho_{\beta/\alpha-\gamma}(v-1)\mathbbm{1}_{[1,\infty)}dv$
(96)
$\displaystyle=\frac{\Gamma(\beta+\theta)\rho_{\gamma+\theta/\alpha}(u)}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-v}\,(e^{-i\pi}v)^{\alpha\gamma-\beta}\,e^{-u(e^{-i\pi}v)^{\alpha}}\,dv$
(97)
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\,u^{\gamma+\theta/\alpha-1}\sum_{k=0}^{\infty}\frac{(-u)^{k}}{k!}\frac{1}{\Gamma(\beta-\alpha\gamma-\alpha
k)}$ (98)

The moments $M_{\alpha,\beta,\gamma,\theta;k}$ $k\geq 0$ of
$P_{\alpha,\beta,\gamma,\theta}$ are

$\displaystyle
M_{\alpha,\beta,\gamma,\theta;k}=\int_{0}^{\infty}u^{k}\,dP_{\alpha,\beta,\gamma,\theta}(u)$
$\displaystyle=\frac{\Gamma(\beta+\theta)\Gamma(\gamma+\theta/\alpha+k)}{\Gamma(\gamma+\theta/\alpha)\Gamma(\alpha
k+\beta+\theta)}$ (99)

We note in passing that, for general exponent $r>-\gamma-\theta/\alpha$, it
can readily be shown that

$\displaystyle\int_{0}^{\infty}u^{r}\,dP_{\alpha,\beta,\gamma,\theta}(u)$
$\displaystyle=\frac{\Gamma(\beta+\theta)\Gamma(\gamma+\theta/\alpha+r)}{\Gamma(\gamma+\theta/\alpha)\Gamma(\alpha
r+\beta+\theta)}$ (100)

The choice $\gamma=0\implies\beta>0,\theta>0$ gives the distribution that
Banderier et al. [2] termed the beta-Mittag-Leffler distribution ${\rm
BML(\alpha,\theta,\beta})$. In this case, the density (95) reduces to

$\displaystyle p_{\alpha,\beta,\theta}(u)$
$\displaystyle=\Gamma(\beta+\theta)\,\rho_{\theta/\alpha}(u)\\{\rho_{\beta}\star
f_{\alpha}(\cdot|u)\\}(1)$ (101)
$\displaystyle=\Gamma(\beta+\theta)\rho_{\theta/\alpha}(u)u^{\beta/\alpha}\int_{0}^{\infty}f_{\alpha}(1|uv)\rho_{\beta/\alpha}(v-1)\mathbbm{1}_{[1,\infty)}dv$
(102)
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\theta/\alpha)}\,u^{\theta/\alpha-1}\sum_{k=0}^{\infty}\frac{(-u)^{k}}{k!}\frac{1}{\Gamma(\beta-\alpha
k)}$ (103)

The latter sum reproduces [2, (25)], obtained in that paper by the inverse
Mellin transform. The conceptual argument is similar to the proof in Flajolet
et al. [12, Proposition 16].

######  Corollary 5.2 (2-parameter Mittag-Leffler distribution).

For $\theta>-\alpha\gamma$ and $\beta-\alpha\gamma=1-\alpha$,

$\displaystyle dP_{\alpha,\beta,\gamma,\theta}(t)$
$\displaystyle=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(t)\,dP_{\alpha}(t)=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\,t^{\gamma+\theta/\alpha-1}\,dP_{\alpha}(t)$
(104)

where $P_{\alpha}\equiv P_{\alpha,1,1,0}$ is the Mittag-Leffler distribution.
In particular, for $\beta=\gamma=1$, $P_{\alpha,\theta}\equiv
P_{\alpha,1,1,\theta}$ is the 2-parameter or generalized Mittag-Leffler
distribution ${\rm ML}(\alpha,\theta)$

$\displaystyle d\,{\rm ML}(t|\alpha,\theta)\equiv dP_{\alpha,\theta}(t)$
$\displaystyle=\Gamma(1+\theta)\,\rho_{1+\theta/\alpha}(t)\,dP_{\alpha}(t)$
$\displaystyle=\frac{\Gamma(1+\theta)}{\Gamma(1+\theta/\alpha)}\,t^{\theta/\alpha}\,dP_{\alpha}(t)$
(105)

###### Proof of Corollary 5.2.

For $\beta-\alpha\gamma=1-\alpha$, $\\{\rho_{1-\alpha}\star
f_{\alpha}(\cdot|t)\\}(1)\equiv(t\alpha)^{-1}f_{\alpha}(1|t)$ by Property 3,
so that $(\ref{eq:ML4pardistribution})$ simplifies to

$\displaystyle dP_{\alpha,\beta,\gamma,\theta}(t)$
$\displaystyle=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(t)\,(t\alpha)^{-1}f_{\alpha}(1|t)\,dt$
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\,t^{\gamma+\theta/\alpha-1}\left[\frac{1}{\alpha}f_{\alpha}(t^{-1/\alpha})\,t^{-1/\alpha-1}\right]dt$
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\,t^{\gamma+\theta/\alpha-1}\,dP_{\alpha}(t)$

where we have recalled that $f_{\alpha}(1|t)\equiv
f_{\alpha}(t^{-1/\alpha})\,t^{-1/\alpha}$. $\beta=\gamma=1$ gives ${\rm
ML}(\alpha,\theta)\equiv P_{\alpha,\theta}$. ∎

The well-studied generalised Mittag-Leffler distribution ${\rm
ML}(\alpha,\theta)$ was discussed in Section 2.2. Using the same notation, we
also refer to $P_{\alpha,\beta,\gamma,\theta}$ as ${\rm
ML}(\alpha,\beta,\gamma,\theta)$. The general condition $\beta>\alpha\gamma$
means that the convolution $\\{\rho_{\beta-\alpha\gamma}\star
f_{\alpha}(\cdot|u)\\}(t)$ is not reducible to a form proportional to
$f_{\alpha}(t|u)$ as it is in the case $\beta-\alpha\gamma=1-\alpha$ for ${\rm
ML}(\alpha,\theta)$, including $P_{\alpha}\equiv{\rm ML}(\alpha,0)$.

######  Corollary 5.3 (Composition).

For $0<\alpha<1$, $0<\sigma<1$ and $-\theta<\alpha\gamma<\beta$, the
composition $E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda
x^{\sigma})$ as a function of $x$ given $\lambda>0$ is the Laplace transform
of a mixture density $w_{\sigma,\alpha}(t|\lambda,\gamma,\beta,\theta)$

$\displaystyle\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda
x^{\sigma})$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\,w_{\sigma,\alpha}(t|\lambda,\gamma,\beta,\theta)\,dt\quad(x\geq
0)$ (106) $\displaystyle\textrm{where}\quad
w_{\sigma,\alpha}(t|\lambda,\gamma,\beta,\theta)$
$\displaystyle=\int_{0}^{\infty}f_{\sigma}(t|\lambda
u)\,dP_{\alpha,\beta,\gamma,\theta}(u)$ (107)
$\displaystyle=\phantom{-}\frac{\Gamma(\beta+\theta)}{\pi}\,{\rm
Im}\int_{0}^{\infty}e^{-tu}\,E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda
e^{-i\pi\sigma}u^{\sigma})\,du$ (108)
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\sigma
k)\,M_{\alpha,\beta,\gamma,\theta;k}\frac{\Gamma(\sigma k+1)}{t^{\sigma k+1}}$
(109)

######  Corollary 5.4.

For $0<\alpha<1$ and $-\theta<\alpha\gamma<\beta$, the representation
$(\ref{eq:ML4parconv2})$ implies that
$t^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-yt^{\alpha})$
as a function of $t$ given $y>0$ is the Laplace transform of a density
$q_{\alpha,\beta,\gamma,\theta}(u|y)$

$\displaystyle
y^{\gamma+\theta/\alpha}\,t^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}f_{\alpha}(t|u)\\{\rho_{\beta/\alpha-\gamma}\star\rho_{\gamma+\theta/\alpha,y}\\}(u)\,du$
$\displaystyle\implies
t^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-yt^{\alpha})$
$\displaystyle=\int_{0}^{\infty}e^{-tu}\,q_{\alpha,\beta,\gamma,\theta}(u|y)\,du\quad(t\geq
0)$ (110) $\displaystyle{\rm where}\quad q_{\alpha,\beta,\gamma,\theta}(u|y)$
$\displaystyle=\frac{y^{-\gamma-\theta/\alpha}}{\pi}\,{\rm
Im}\left\\{\widetilde{\rho}_{\beta/\alpha-\gamma}(e^{-i\pi\alpha}u^{\alpha})\,\widetilde{\rho}_{\gamma+\theta/\alpha,y}(e^{-i\pi\alpha}u^{\alpha})\right\\}$
(111) $\displaystyle=\frac{1}{\pi}\,{\rm
Im}\frac{(e^{-i\pi}u)^{\alpha\gamma-\beta}}{(y+e^{-i\pi\alpha}u^{\alpha})^{\gamma+\theta/\alpha}}$
(112)

We summarise the results of Theorem 5 in Table 3. Formally, this completes our
narrative on Mittag-Leffler densities, mixtures involving such densities and
the stable density, as well as the Laplace transforms of all such objects,
featuring Mittag-Leffler functions. For completenesses though, Table 5
presents densities that have arisen in the course of discussion that are not
explicitly in the family of Mittag-Leffler densities but whose Laplace
transforms nonetheless involve Mittag-Leffler functions. Unlike all previous
instances, these densities are not normalised (with the exception of the first
one).

##  8 An Alternative Representation

We now turn to a representation of Mittag-Leffler distributions that builds on
Corollary 2.1, involving the beta distribution. By contrast to the foregoing
discussion, this approach does not rely on Mittag-Leffler functions as the
point of departure.

######  Theorem 6.

For $0<\alpha<1$ and $-\theta<\alpha\gamma<\beta$, the 4-parameter Mittag-
Leffler distribution $P_{\alpha,\beta,\gamma,\theta}\equiv{\rm
ML}(\alpha,\beta,\gamma,\theta)$ has a density
$p_{\alpha,\beta,\gamma,\theta}(t)$ given by

$\displaystyle p_{\alpha,\beta,\gamma,\theta}(t)$
$\displaystyle=C_{\alpha,\beta,\theta}\int_{0}^{1}p_{\alpha,\beta+\theta}(t/u)\,{\rm
beta}(u|\theta/\alpha+\gamma,\beta/\alpha-\gamma)\frac{du}{u}$ (113) where
$C_{\alpha,\beta,\theta}$ is the normalising constant $\displaystyle
C_{\alpha,\beta,\theta}$
$\displaystyle=\frac{\Gamma(1+\beta+\theta)}{\Gamma(1+(\beta+\theta)/\alpha)}\times\frac{\Gamma(1+\theta/\alpha)}{\Gamma(1+\theta)}$
(114)

Equivalently, if $U$ and $V$ are independent random variables, where $U$ has
the beta distribution ${\rm Beta}(\theta/\alpha+\gamma,\beta/\alpha-\gamma)$
and $V$ has the 2-parameter Mittag-Leffler distribution
${\rm}ML(\alpha,\beta+\theta)$, then $T=VU$ has the 4-parameter Mittag-Leffler
distribution ${\rm ML}(\alpha,\beta,\gamma,\theta)$ with density
$(\ref{eq:altML4pardistribution})$.

###### Proof of Theorem 6.

By (96) of Corollary 5.1, together with Corollary 2.1

$\displaystyle p_{\alpha,\beta,\gamma,\theta}(t)$
$\displaystyle=\Gamma(\beta+\theta)\rho_{\gamma+\theta/\alpha}(t)t^{\beta/\alpha-\gamma}\int_{1}^{\infty}f_{\alpha}(1|tu)\rho_{\beta/\alpha-\gamma}(u-1)du$
$\displaystyle(u\to 1/u)\;$
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\frac{t^{(\beta+\theta)/\alpha-1}}{\Gamma(\beta/\alpha-\gamma)}\int_{0}^{1}f_{\alpha}(1|t/u)(1/u-1)^{\beta/\alpha-\gamma-1}\,\frac{du}{u^{2}}$
$\displaystyle=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\frac{1}{\Gamma(\beta/\alpha-\gamma)}\,\times$
$\displaystyle\phantom{\propto}\int_{0}^{1}(t/u)^{(\beta+\theta)/\alpha-1}f_{\alpha}(1|t/u)\,u^{\theta/\alpha+\gamma-1}(1-u)^{\beta/\alpha-\gamma-1}\,\frac{du}{u}$
$\displaystyle=C_{\alpha,\beta,\gamma,\theta}\int_{0}^{1}p_{\alpha,\beta+\theta}(t/u)\,{\rm
beta}(u|\theta/\alpha+\gamma,\beta/\alpha-\gamma)\frac{du}{u}$

where $p_{\alpha,\beta+\theta}(t)$ is read from the definition of
$p_{\alpha,\theta}(t)$ in Corollary 5.2. By (12), this is the density of
$T=VU$ as described in Theorem 6. ∎

The particular case $\beta=\gamma=1$ reproduces ${\rm
ML}(\alpha,\theta)\equiv{\rm ML}(\alpha,1,1,\theta)$. This defines the Mittag-
Leffler Markov Chain (MLMC) with recursion of the form $T_{n-1}=T_{n}U_{n}$
([16, 17, 22, 48, 49, 54]). MLMC features in models of the growth of random
trees.

The representation of Theorem 6 is shown in Table 4 with various choices of
parameters. In keeping with the rest of the tables, a column of corresponding
Laplace transforms is reproduced.

##  9 Discussion

The 4-parameter Mittag-Leffler distribution
$P_{\alpha,\beta,\gamma,\theta}\equiv{\rm ML}(\alpha,\beta,\gamma,\theta)$ is
the general case, subsuming all other variants in the literature as special
cases:

  1. 1.

Mittag-Leffler Distribution: $P_{\alpha}\equiv{\rm ML}(\alpha,1,1,0)$ [11, 47]

  2. 2.

3-parameter Mittag-Leffler Distribution: ${\rm
ML}(\alpha,\beta,\gamma)\equiv{\rm ML}(\alpha,\beta,\gamma,0)$ [5, 12, 19, 40,
57]

  3. 3.

Beta-Mittag-Leffler Distribution: ${\rm BML(\alpha,\theta,\beta})\equiv{\rm
ML}(\alpha,\beta,0,\theta>0)$ [2, 12, 40, 16]

  4. 4.

2-parameter/Generalized Mittag-Leffler Distribution: ${\rm
ML}(\alpha,\theta)\equiv{\rm ML}(\alpha,1,1,\theta>-\alpha)$, equivalently
${\rm ML}(\alpha,0,0,\theta>0)$ [2, 8, 16, 17, 22, 23, 24, 28, 44]

  5. 5.

Mittag-Leffler Markov Chain defined by $T_{n-1}=T_{n}U_{n}$ where $U$ is a
beta variable and $T$ is a 2-parameter/Generalized Mittag-Leffler variable
[16, 17, 22, 48, 49, 54]

Logically, we might have constructed the general 4-parameter case from the
outset. We opted instead to start from the single parameter case for clarity
of exposition, despite the inevitable duplication of earlier results that
emerge as special cases of later more general formulation.

The primary point though is one of methodology. All distributions arise from
convolutions and mixtures of gamma and stable densities as the basic
construct. The Laplace-Stietljes transforms of all distributions thus
constructed are Mittag-Leffler functions and their compositions with a
Bernstein function of specific form. There has been no need for explicit
complex analysis. It has sufficed to refer to the representation (18) of the
stable density, which is an outcome of complex analytic inversion due to
Pollard [46].

There has also been no need to appeal to special functions. For completeness
though, we comment on how special functions might be seen to arise from
convolutions of gamma densities.

###  9.1 Convolutions and Special Functions

We have seen that the Mittag-Leffler distribution involves the convolution
$\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)$. By Theorem 2,
this may be expressed as

$\displaystyle\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)$
$\displaystyle=t^{\beta/\alpha-\gamma}\int_{0}^{\infty}f_{\alpha}(1|tu)\rho_{\beta/\alpha-\gamma}(u-1)\mathbbm{1}_{[1,\infty)}\,du$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-t)^{k}}{k!}\frac{1}{\Gamma(\beta-\alpha\gamma-\alpha
k)}$

The representation (18) of the stable density leads to an alternative infinite
sum representation

$\displaystyle\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)$
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-t)^{k}}{k!}\sin\pi\alpha
k\int_{0}^{\infty}\\{\rho_{\beta-\alpha\gamma}\star\rho_{1,u}\\}(1)\,u^{\alpha
k-1}\,du$ (115)

The observation of interest here is that
$\\{\rho_{\beta/\alpha-\gamma}\star\rho_{1,u}\\}(1)$ is a convolution of 2
gamma densities with different scale factors $0$ and $u$. More generally, the
gamma distribution has the well-known property that it is closed under
convolution of densities sharing a common scale parameter, i.e.
$\\{\rho_{\nu,\lambda}\star\rho_{\mu,\lambda}\\}(t)=\rho_{\nu+\mu,\lambda}(t)$
with scale parameter $\lambda$ throughout. The case of different scale
parameters is not as simple. In particular

$\displaystyle\\{\rho_{\nu}\star\rho_{\mu,\lambda}\\}(1)$
$\displaystyle=\int_{0}^{1}\rho_{\nu}(1-u)\rho_{\mu,\lambda}(u)du$ (116)
$\displaystyle=\frac{\lambda^{\mu}}{\Gamma(\nu)\Gamma(\mu)}\int_{0}^{1}(1-u)^{\nu-1}\,u^{\mu-1}e^{-\lambda
u}\,du$ (117)
$\displaystyle=\frac{\lambda^{\mu}}{\Gamma(\mu+\nu)}\int_{0}^{1}\beta_{\mu,\nu}(u)\,e^{-\lambda
u}\,du$ (118)

We recognise the integral in (118) as the Laplace transform of the beta
density ${\rm b}_{\mu,\nu}(u)$, also denoted by $M(\mu,\mu+\nu,-\lambda)$,
known as the confluent hypergeometric function (Abramowitz and Stegun [1,
13.2.1]). Hence, with $\nu=\beta-\alpha\gamma$ and $\mu=1$, (115) may be
written as

$\displaystyle\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)$
$\displaystyle=-\frac{1}{\pi}\sum_{k=0}^{\infty}\frac{(-t)^{k}\sin\pi\alpha
k}{k!\Gamma(\beta-\alpha\gamma+1)}\int_{0}^{\infty}u^{\alpha
k}\,M(1,\beta-\alpha\gamma+1,-u)\,du$ (119)

The integral may be looked upon as a Mellin transform of
$uM(1,\beta-\alpha\gamma+1,-u)$ defined by

$\displaystyle\int_{0}^{\infty}u^{\alpha
k-1}\,\\{uM(1,\beta-\alpha\gamma+1,-u)\\}\,du$

While this illustrates the link between Mittag-Leffler distributions and
special functions and Mellin transforms, (119) does not in our view provide
additional conceptual insight compared to the expressions for the convolution
$\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)$ given by Theorem
2.

##  10 Conclusion

We have used the convolution and mixing of gamma and stable densities to
construct densities of higher level distributions. The convolution between the
power density and the stable density has been the centrepiece of the
discussion. We applied the methodology to the construction of Mittag-Leffler
distributions, with a 4-parameter instance as the general case subsuming
published instances as special cases. The Laplace transforms of the Mittag-
Leffler densities are Mittag Leffler functions. We have shown that the
compositions of the latter with a power law Bernstein function can be
constructed from corresponding mixture densities.

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$\begin{array}[]{| c | c | c | }\cline{2-3}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\textrm{Probability Density}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\begin{array}[]{c}\textrm{Laplace Transform}\\\ \textrm{of density}\end{array}\hfil\lx<EMAIL_ADDRESS>\hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:MLconv1}: Mittag-Leffler density }\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize LT of ML density}\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize Parameters: $0<\alpha<1$ \, }\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 1}&\begin{aligned} p_{\alpha}(t)&=\\{\rho_{1-\alpha}\star f_{\alpha}(\cdot|t)\\}(1)\rule{0.0pt}{16.14578pt}\\\ &=\frac{1}{\alpha}f_{\alpha}(1|t)\,t^{-1}\equiv\frac{1}{\alpha}f_{\alpha}(t^{-1/\alpha})\,t^{-1/\alpha-1}\rule{0.0pt}{16.14578pt}\\\ &=\frac{1}{\pi\alpha t}\,{\rm Im}\int_{0}^{\infty}e^{-u}\,e^{-te^{-i\pi\alpha}u^{\alpha}}\,du\rule{0.0pt}{16.14578pt}\\\ &=\frac{1}{\pi\alpha}\sum_{k=0}^{\infty}\frac{(-t)^{k-1}}{k!}\sin(\pi\alpha k)\,\Gamma(\alpha k+1)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{aligned}&E_{\alpha}(-x)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:MLconv1Composition}: $\\{\alpha,\sigma\\}$ Mixing}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize Composition with $\lambda x^{\sigma}$ }\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize Parameters: $0<\alpha<1,0<\sigma<1;\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 2}&\lx@intercol\hfil\begin{array}[]{c}\displaystyle\int_{0}^{\infty}f_{\sigma}(t|\lambda u)\,dP_{\alpha}(u)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \begin{aligned} &=\phantom{-}\dfrac{1}{\pi}{\rm Im}\displaystyle\int_{0}^{\infty}e^{-tu}\,E_{\alpha}(-\lambda e^{-i\pi\sigma}u^{\sigma})\,du\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ &=-\dfrac{1}{\pi}\ \displaystyle\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\sigma k)\,\frac{\Gamma(\sigma k+1)}{\Gamma(\alpha k+1)}\,t^{-\sigma k-1}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{aligned}\end{array}\hfil\lx@intercol\vrule\lx@intercol&E_{\alpha}(-\lambda x^{\sigma})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:MLconv1Composition}: $\\{\alpha,\alpha\\}$ Mixing}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize Composition with $\lambda x^{\alpha}$ }\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize Parameters: $0<\alpha<1;\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 2a}&\lx@intercol\hfil\begin{array}[]{c}\displaystyle\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dP_{\alpha}(u)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ =\dfrac{\sin\pi\alpha}{\pi}\,\dfrac{\lambda\,t^{\alpha-1}}{\lambda^{2}+2\lambda t^{\alpha}\cos\pi\alpha+t^{2\alpha}}\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{array}\hfil\lx@intercol\vrule\lx@intercol&E_{\alpha}(-\lambda x^{\alpha})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\footnotesize\textrm{Notes:}\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$f_{\alpha}(t|u)$ is the stable density with Laplace transform $e^{-ux^{\alpha}}$ \, $(u>0)$ }\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$p_{\alpha}(t)$ is the density of the Mittag-Leffler distribution $P_{\alpha}(t)$ }\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$\rho_{\gamma}(t)$ is the unnormalised power density with Laplace transform $\widetilde{\rho}_{\gamma}(x)=x^{-\gamma}$ }\hfil\lx@intercol\\\ \end{array}\\\ $
Table 1: Theorem 3: Mittag-Leffler densities and their Laplace transforms

$\begin{array}[]{| c | c | c | }\cline{2-3}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\textrm{Probability Density}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\begin{array}[]{c}\textrm{Laplace Transform}\\\ \textrm{of density}\end{array}\hfil\lx<EMAIL_ADDRESS>\hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML3parconv1}: 3-parameter Mittag-Leffler density }\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize LT of 3-par ML density}\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize$0<\alpha<1$, $0<\alpha\gamma<\beta$ }\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 1}&\begin{aligned} p_{\alpha,\beta,\gamma}(t)&=\Gamma(\beta)\,\rho_{\gamma}(t)\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)\\\ &=\Gamma(\beta)\rho_{\gamma}(t)t^{\beta/\alpha-\gamma}\;\times\\\ &\qquad\displaystyle\int_{0}^{\infty}f_{\alpha}(1|tu)\rho_{\beta/\alpha-\gamma}(u-1)\mathbbm{1}_{[1,\infty)}du\\\ &=\frac{\Gamma(\beta)\rho_{\gamma}(t)}{\pi}{\rm Im}\int_{0}^{\infty}e^{-u}\,(e^{-i\pi}u)^{\alpha\gamma-\beta}\,e^{-t(e^{-i\pi}u)^{\alpha}}du\\\ &=\frac{\Gamma(\beta)}{\Gamma(\gamma)}\,\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{t^{\gamma+k-1}}{\Gamma(\beta-\alpha\gamma-\alpha k)}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{aligned}&\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-x)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML3parconv1Composition}: $\\{\alpha,\sigma\\}$ Mixing}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize Composition with $\lambda x^{\sigma}$ }\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize$0<\alpha<1,0<\sigma<1,0<\alpha\gamma<\beta;\,\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 2}&\lx@intercol\hfil\begin{array}[]{c}\displaystyle\int_{0}^{\infty}f_{\sigma}(t|\lambda u)\,dP_{\alpha,\beta,\gamma}(u)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \begin{aligned} &=\phantom{-}\dfrac{\Gamma(\beta)}{\pi}{\rm Im}\displaystyle\int_{0}^{\infty}e^{-tu}\,E^{\gamma}_{\alpha,\beta}(-\lambda e^{-i\pi\sigma}u^{\sigma})\,du\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ &=-\dfrac{1}{\pi}\ \displaystyle\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\sigma k)\,M_{\alpha,\beta,\gamma;k}\frac{\Gamma(\sigma k+1)}{t^{\sigma k+1}}\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{aligned}\end{array}\hfil\lx@intercol\vrule\lx@intercol&\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-\lambda x^{\sigma})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML3parconv1Composition}: $\\{\alpha,\alpha\\}$ Mixing (as above, $\sigma=\alpha$)}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize Composition with $\lambda x^{\alpha}$ }\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize$0<\alpha<1,0<\alpha\gamma<\beta;\,\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 2a}&\displaystyle\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dP_{\alpha,\beta,\gamma}(u)&\Gamma(\beta)E^{\gamma}_{\alpha,\beta}(-\lambda x^{\alpha})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\textrm{\footnotesize Notes: ($f_{\alpha}(t|u),\rho_{\gamma}(t)$ are as in Table~{}\ref{table:MLDensities})}\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$p_{\alpha,\beta,\gamma}(t)$ is the density of the 3-parameter Mittag-Leffler distribution $P_{\alpha,\beta,\gamma}(t)$ }\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$M_{\alpha,\beta,\gamma;k}=\dfrac{\Gamma(\beta)\Gamma(\gamma+k)}{\Gamma(\gamma)\,\Gamma(\beta+\alpha k)}$ is the $k^{\rm th}$ moment of<EMAIL_ADDRESS>\end{array}\\\ $
Table 2: Theorem 4: 3-parameter Mittag-Leffler densities and their Laplace
transforms

$\begin{array}[]{| c | c | c | }\cline{2-3}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\textrm{Probability Density}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\begin{array}[]{c}\textrm{Laplace Transform}\\\ \textrm{of density}\end{array}\hfil\lx<EMAIL_ADDRESS>\hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML4parconv1}: 4-parameter Mittag-Leffler density }\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize LT of 4-par ML density}\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize$0<\alpha<1$, $-\theta<\alpha\gamma<\beta$ }\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 1}&\begin{aligned} p_{\alpha,\beta,\gamma,\theta}(t)&=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(t)\\{\rho_{\beta-\alpha\gamma}\star f_{\alpha}(\cdot|t)\\}(1)\\\ &=\Gamma(\beta+\theta)\rho_{\gamma+\theta/\alpha}(t)t^{\beta/\alpha-\gamma}\;\times\\\ &\qquad\displaystyle\int_{0}^{\infty}f_{\alpha}(1|tu)\rho_{\beta/\alpha-\gamma}(u-1)\mathbbm{1}_{[1,\infty)}du\\\ &=\Gamma(\beta+\theta)\rho_{\gamma+\theta/\alpha}(t)\;\times\\\ &\qquad\frac{1}{\pi}{\rm Im}\int_{0}^{\infty}e^{-u}(e^{-i\pi}u)^{\alpha\gamma-\beta}e^{-t(e^{-i\pi}u)^{\alpha}}du\\\ &=\frac{\Gamma(\beta+\theta)}{\Gamma(\gamma+\theta/\alpha)}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{t^{\gamma+\theta/\alpha+k-1}}{\Gamma(\beta-\alpha\gamma-\alpha k)}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{aligned}&\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML4parconv1}: $\gamma=0,\beta>0,\theta>0$: $P_{\alpha,\beta,\theta}\equiv{\rm BML}(\alpha,\theta,\beta)$ }\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 1a}&p_{\alpha,\beta,\theta}(t)=\Gamma(\beta+\theta)\,\rho_{\theta/\alpha}(t)\\{\rho_{\beta}\star f_{\alpha}(\cdot|t)\\}(1)&\Gamma(\beta+\theta)E^{\theta/\alpha}_{\alpha,\beta+\theta}(-x)\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML2pardistribution}: $\beta-\alpha\gamma=1-\alpha$, $\theta>-\alpha\gamma$ }\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize$\gamma=(\beta-1)/\alpha+1$}\hfil\lx@intercol\vrule\lx@intercol\\\ \textrm{\scriptsize 2}&p_{\alpha,\beta,\gamma,\theta}(t)=\Gamma(\beta+\theta)\,\rho_{\gamma+\theta/\alpha}(t)\,p_{\alpha}(t)&\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML2pardistribution}: $\beta=\gamma=1$, $\theta>-\alpha$: $P_{\alpha,\theta}\equiv{\rm ML}(\alpha,\theta)$ }\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 2a}&\lx@intercol\hfil\begin{array}[]{c}p_{\alpha,\theta}(t)=\frac{\Gamma(1+\theta)}{\Gamma(1+\theta/\alpha)}\,t^{\theta/\alpha}\,p_{\alpha}(t)\rule{0.0pt}{16.14578pt}\\\ \end{array}\hfil\lx@intercol\vrule\lx@intercol&\Gamma(1+\theta)E^{1+\theta/\alpha}_{\alpha,1+\theta}(-x)\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML4parconv1Composition}: $\\{\alpha,\sigma\\}$ Mixing}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize Composition with $\lambda x^{\sigma}$ }\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize$0<\alpha<1,0<\sigma<1,-\theta<\alpha\gamma<\beta;\,\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 3}&\lx@intercol\hfil\begin{array}[]{c}\displaystyle\int_{0}^{\infty}f_{\sigma}(t|\lambda u)\,dP_{\alpha,\beta,\gamma,\theta}(u)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \begin{aligned} &=\dfrac{\Gamma(\beta+\theta)}{\pi}{\rm Im}\displaystyle\int_{0}^{\infty}e^{-tu}\,E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda e^{-i\pi\sigma}u^{\sigma})\,du\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ &=-\dfrac{1}{\pi}\ \displaystyle\sum_{k=0}^{\infty}(-\lambda)^{k}\sin(\pi\sigma k)\,M_{\alpha,\beta,\gamma,\theta;k}\frac{\Gamma(\sigma k+1)}{t^{\sigma k+1}}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{aligned}\end{array}\hfil\lx@intercol\vrule\lx@intercol&\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda x^{\sigma})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML4parconv1Composition}: $\\{\alpha,\alpha\\}$ Mixing (as above, with $\sigma=\alpha$)}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\textrm{\footnotesize Composition with $\lambda x^{\alpha}$ }\hfil\lx@intercol\vrule\lx@intercol\\\ &\lx@intercol\textrm{\footnotesize$0<\alpha<1,-\theta<\alpha\gamma<\beta;\,\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 3a}&\displaystyle\int_{0}^{\infty}f_{\alpha}(t|\lambda u)\,dP_{\alpha,\beta,\gamma,\theta}(u)&\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda x^{\alpha})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\footnotesize\textrm{Notes: {\footnotesize($f_{\alpha}(t|u),\rho_{\gamma}(t)$ are as in Table~{}\ref{table:MLDensities})} }\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$p_{\alpha,\beta,\gamma,\theta}(t)$ is the density of the 4-parameter Mittag-Leffler distribution $P_{\alpha,\beta,\gamma,\theta}(t)$ }\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$M^{\gamma}_{\alpha,\beta,\theta;k}=\dfrac{\Gamma(\beta+\theta)\Gamma(\gamma+\theta/\alpha+k)}{\Gamma(\gamma+\theta/\alpha)\,\Gamma(\beta+\theta+\alpha k)}$ is the $k^{\rm th}$ moment of<EMAIL_ADDRESS>\end{array}\\\ $
Table 3: Theorem 5: 4-parameter Mittag-Leffler densities and their Laplace
transforms.

$\begin{array}[]{| c | c | c || c |}\hline\cr\pagecolor{lightgray}\textrm{\footnotesize Distribution of $U$}&\pagecolor{lightgray}\textrm{\footnotesize Distribution of $V$}&\pagecolor{lightgray}\textrm{\footnotesize Distribution of $T=VU$}&\pagecolor{lightgray}\textrm{\footnotesize LT of density of $T$}\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize General Case of Theorem~{}\ref{thm:altML4pardistribution}: $0<\alpha<1$, $-\theta<\alpha\gamma<\beta$ }\hfil\lx@intercol\\\ \hline\cr{\rm Beta}\left(\tfrac{\theta}{\alpha}+\gamma,\tfrac{\beta}{\alpha}-\gamma\right)&{\rm ML}(\alpha,\beta+\theta)&{}^{\dagger}{\rm ML}(\alpha,\beta,\gamma,\theta)&\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize beta-Mittag-Leffler case \cite[cite]{[\@@bibref{Number}{Banderier}{}{}]}: $0<\alpha<1$, $\gamma=0,\beta>0,\theta>0$ }\hfil\lx@intercol\\\ \hline\cr{\rm Beta}\left(\tfrac{\theta}{\alpha},\tfrac{\beta}{\alpha}\right)&{\rm ML}(\alpha,\beta+\theta)&{\rm ML}(\alpha,\beta,0,\theta)&\Gamma(\beta+\theta)E^{\theta/\alpha}_{\alpha,\beta+\theta}(-x)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$0<\alpha<1$, $\beta-\alpha\gamma=1-\alpha\implies\gamma=(\beta-1)/\alpha+1,\theta>-\alpha\gamma$}\hfil\lx@intercol\\\ \hline\cr{\rm Beta}\left(\tfrac{\theta}{\alpha}+\gamma,\tfrac{1}{\alpha}-1\right)&{\rm ML}(\alpha,\beta+\theta)&{\rm ML}(\alpha,\beta,\gamma,\theta)&\Gamma(\beta+\theta)E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-x)\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\hfil\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize Mittag-Leffler Markov Chain \cite[cite]{[\@@bibref{Number}{Goldschmidt22, GoldschmidtHaas, HoJamesLau, RembartWinkel2018, RembartWinkel2023, Senizergues}{}{}]}: $0<\alpha<1$, $\beta=\gamma=1,\theta>-\alpha,n\geq 1$}\hfil\lx@intercol\\\ \hline\cr{\rm Beta}\left(\tfrac{\theta+n-1}{\alpha}+1,\tfrac{1}{\alpha}-1\right)&{\rm ML}(\alpha,\theta+n)&{\rm ML}(\alpha,\theta+n-1)&\lx@intercol\hfil\begin{array}[]{c}\Gamma(\theta+n)\,\times\rule{0.0pt}{16.14578pt}\\\ E^{1-1/\alpha+(\theta+n)/\alpha}_{\alpha,\theta+n}(-x)\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{array}\hfil\lx<EMAIL_ADDRESS>\hline\cr\lx@intercol\textrm{\normalsize${}^{\dagger}{\rm ML}(\alpha,\beta,\gamma,\theta)$ has density<EMAIL_ADDRESS>\lx@intercol\textrm{\footnotesize$\quad p_{\alpha,\beta,\gamma,\theta}(t)=\dfrac{\Gamma(1+\beta+\theta)\Gamma(1+\theta/\alpha)}{\Gamma(1+(\beta+\theta)/\alpha)\Gamma(1+\theta)}\displaystyle\int_{0}^{1}p_{\alpha,\beta+\theta}(t/u)\,{\rm beta}(u|\theta/\alpha+\gamma,\beta/\alpha-\gamma)\frac{du}{u}$<EMAIL_ADDRESS>\end{array}$
Table 4: Theorem 6: Distributions of products of beta and Mittag-Leffler
variables

$\begin{array}[]{| c | c | c | }\cline{2-3}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\textrm{Density}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\pagecolor{lightgray}\begin{array}[]{c}\textrm{Laplace Transform}\\\ \textrm{of density}\end{array}\hfil\lx<EMAIL_ADDRESS>\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:MLconv2}: $0<\alpha<1;\lambda>0$ }\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 1}&\lx@intercol\hfil\begin{array}[]{c}\dfrac{1}{\lambda\pi}\,{\rm Im}\left\\{\widetilde{\rho}_{1/\alpha-1}(e^{-i\pi\alpha}t^{\alpha})\,\widetilde{\rho}_{1,\lambda}(e^{-i\pi\alpha}t^{\alpha})\right\\}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ =\dfrac{\sin\pi\alpha}{\pi}\,\dfrac{\lambda\,t^{\alpha-1}}{\lambda^{2}+2\lambda t^{\alpha}\cos\pi\alpha+t^{2\alpha}}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{array}\hfil\lx@intercol\vrule\lx@intercol&E_{\alpha}(-\lambda x^{\alpha})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML3parconv2}: $0<\alpha<1,0<\alpha\gamma<\beta;\,\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 2}&\lx@intercol\hfil\begin{array}[]{c}\dfrac{1}{\lambda^{\gamma}\pi}\,{\rm Im}\left\\{\widetilde{\rho}_{\beta/\alpha-\gamma}(e^{-i\pi\alpha}t^{\alpha})\,\widetilde{\rho}_{\gamma,\lambda}(e^{-i\pi\alpha}t^{\alpha})\right\\}\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ =\dfrac{1}{\pi}\,{\rm Im}\dfrac{(e^{-i\pi}t)^{\alpha\gamma-\beta}}{(\lambda+e^{-i\pi\alpha}t^{\alpha})^{\gamma}}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{array}\hfil\lx@intercol\vrule\lx@intercol&x^{\beta-1}E^{\gamma}_{\alpha,\beta}(-\lambda x^{\alpha})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\hline\cr&\lx@intercol\textrm{\footnotesize Corollary~{}\ref{cor:ML4parconv2}: $0<\alpha<1,-\theta<\alpha\gamma<\beta;\,\lambda>0$}\hfil\lx@intercol\vrule\lx@intercol&\\\ \textrm{\scriptsize 3}&\lx@intercol\hfil\begin{array}[]{c}\dfrac{1}{\lambda^{\gamma+\theta/\alpha}\pi}\,{\rm Im}\left\\{\widetilde{\rho}_{\beta/\alpha-\gamma}(e^{-i\pi\alpha}t^{\alpha})\,\widetilde{\rho}_{\gamma+\theta/\alpha,\lambda}(e^{-i\pi\alpha}t^{\alpha})\right\\}\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ =\dfrac{1}{\pi}\,{\rm Im}\dfrac{(e^{-i\pi}t)^{\alpha\gamma-\beta}}{(\lambda+e^{-i\pi\alpha}t^{\alpha})^{\gamma+\theta/\alpha}}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \end{array}\hfil\lx@intercol\vrule\lx@intercol&x^{\beta+\theta-1}E^{\gamma+\theta/\alpha}_{\alpha,\beta+\theta}(-\lambda x^{\alpha})\rule{0.0pt}{16.14578pt}\rule[-15.0694pt]{0.0pt}{0.0pt}\\\ \hline\cr\lx@intercol\footnotesize\textrm{Notes:}\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$\rho_{\gamma,\lambda}(t)$ is the gamma density with Laplace transform $\widetilde{\rho}_{\gamma,\lambda}(x)=\lambda^{\gamma}/(\lambda+x)^{\gamma}$ $(\gamma,\lambda>0)$}\hfil\lx@intercol\\\ \lx@intercol\textrm{\footnotesize$\rho_{\gamma}(t)$ is the unnormalised power density with Laplace transform $\widetilde{\rho}_{\gamma}(x)=x^{-\gamma}$ }\hfil\lx@intercol\\\ \end{array}\\\ $
Table 5: Other densities and their Laplace transforms. Only Case 1 is a
probability density, coincident with Case 2a of Table 1.