Mellin convolutions of products and ratios

Usually, convolution refers to Laplace convolution in the literature, but Mellin convolutions can yield very ueful results. This aspect is illustrated in the coming sections. This study deals with Mellin convolutions of products and ratios. Functions belonging to the pathway family of functions are considered. Several types of integral representations, their equivalent representations in terms of G and H-functions, and their equivalent computable series representations are examined in this study.

Mathematics Subject Classification 2010: 26A33, 44A10, 33C60, 35J10.

1 Introduction

Laplace convolutions involving products of two functions are widely known in the literature (see [1, 2]), but Mellin convolutions of products and ratios are not widely used in the literature. Here, we examine Mellin convolutions of products and ratios involving some functions belonging to the pathway family of functions. The pathway family of functions was introduced by Mathai [3] for real-valued scalar functions with the arguments being rectangular matrices in the real domain. These results were extended to the complex domain in Mathai and Provost [4]. Here, we consider the real scalar variable case first, and then, some generalizations are also considered. Let x₁ > 0 and x₂ > 0 be real positive scalar variables associated with the functions f₁(x₁) and f₂(x₂), respectively, and with the joint function f₁(x₁)f₂(x₂) the product. Then, the Mellin transforms of f₁ and f₂, with Mellin parameter s, and denoted by M_f1(s) and M_f2(s), are the following:

$\begin{array}{ll}{M}_{f_1}(s)={\displaystyle {\int }_0^{\infty }}{x}_1^{s-1}{f}_1(x_1)\text{d}{x}_1\text{and }{M}_{f_2}(s)={\displaystyle {\int }_0^{\infty }}{x}_2^{s-1}{f}_2(x_2)\text{d}{x}_2& (1.1)\end{array}$

whenever the integrals are convergent. Let u = x₁x₂. Let g₁(u) be the function associated with u. Then, the Mellin transform of g₁(u) is the following:

$\begin{array}{ll}{M}_{g_1}(s)={\displaystyle {\int }_0^{\infty }}{u}^{s-1}{g}_1(u)\text{d}u={\displaystyle {\int }_0^{\infty }}{\displaystyle {\int }_0^{\infty }}{(x_1{x}_2)}^{s-1}{f}_1(x_1)f_2(x_2)\text{d}{x}_1\wedge \text{d}{x}_2=M_{f_1}(s)M_{f_2}(s)& (1.2)\end{array}$

since the joint function of x₁ and x₂ is assumed to be the product f₁(x₁)f₂(x₂). Consider the transformation

$u=x_1{x}_2,v=x_2\text{or }v=x_1\Rightarrow \text{d}{x}_1\wedge \text{d}{x}_2=\frac{1}{v}\text{d}u\wedge \text{d}v$

and the joint function of u and v is $\frac{1}{v}{f}_1(\frac{u}{v})f_2(v)$ or $\frac{1}{v}{f}_1(v)f_2(\frac{u}{v})$. Then, g₁(u) is the following:

$\begin{array}{ll}{g}_1(u)={\displaystyle {\int }_v}\frac{1}{v}{f}_1(\frac{u}{v})f_2(v)\text{d}v={\displaystyle {\int }_v}\frac{1}{v}{f}_1(v)f_2(\frac{u}{v})\text{d}v.& (1.3)\end{array}$

But, from Equation 1.2, g₁(u) is available as the inverse Mellin transform of M_f1(s)M_f2(s). That is,

$\begin{array}{ll}{g}_1(u)=\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}{M}_{f_1}(s)M_{f_2}(s)u^{-s}\text{d}s,i=\sqrt{(-1)}& (1.4)\end{array}$

whenever the inverse Mellin transform exists. From Equations 1.3, 1.4, we have

$\begin{array}{c}{g}_1(u)={\displaystyle {\int }_v}\frac{1}{v}{f}_1(\frac{u}{v})f_2(v)\text{d}v={\displaystyle {\int }_v}\frac{1}{v}{f}_1(v)f_2(\frac{u}{v})\text{d}v\\ =\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}{M}_{f_1}(s)M_{f_2}(s)u^{-s}\text{d}s& (1.5)\end{array}$

This Equation 1.5 is the Mellin convolution of a product property.

When x₁ > 0 and x₂ > 0 are real scalar random variables with the densities f₁(x₁) and f₂(x₂), respectively, and when x₁ and x₂ are statistically independently distributed, then

$\begin{array}{c}{M}_{f_1}(s)=E\left[x_1^{s-1}\right],M_{f_2}(s)=E\left[x_2^{s-1}\right],\\ E\left[{(x_1{x}_2)}^{s-1}\right]=E\left[x_1^{s-1}\right]E\left[x_2^{s-1}\right]=M_{f_1}(s)M_{f_2}(s)& (1.6)\end{array}$

due to statistical independence of x₁ and x₂, where E[·] denotes the expected value of [·]. The density of u = x₁x₂, denoted by g₁(u), is available as the first part of Equation 1.5 if we use transformation of variables and as the second part of Equation 1.5 if we use the result of arbitrary moments uniquely determining the corresponding density. Whenever f₁(x₁) and f₂(x₂) are statistical densities, we can also give a statistical interpretation of the Mellin convolution of a product as the unique density of the product of independently distributed real scalar positive random variables x₁ and x₂. Thus, the results in Equations 1.1–1.5 can be given statistical interpretations also whenever the functions f₁ and f₂ are statistical densities.

For the real scalar positive variables case, the pathway family consists of the generalized type-1 beta family given by

$\begin{array}{l}{p}_1(x)=\frac{\delta a^{\frac{\alpha }{\delta }}\Gamma (\frac{\alpha }{\delta }+\beta )}{\Gamma (\frac{\alpha }{\delta })\Gamma (\beta )}{x}^{\alpha -1}{(1-a{x}^{\delta })}^{\beta -1},0\le x\le 1,a>0,\\ \alpha >0,\beta >0,1-a{x}^{\delta }>0& (1.7)\end{array}$

and zero elsewhere, with the standard form given by

$p_{11}(x)=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}{x}^{\alpha -1}{(1-x)}^{\beta -1},0\le x\le 1,\alpha >0,\beta >0$

and zero elsewhere; the generalized type-2 beta model is given by

$\begin{array}{ll}{p}_2(x)=\frac{\delta a^{\frac{\alpha }{\delta }}\Gamma (\frac{\alpha }{\delta }+\beta )}{\Gamma (\frac{\alpha }{\delta })\Gamma (\beta )}{x}^{\alpha -1}{(1+a{x}^{\delta })}^{-(\frac{\alpha }{\delta }+\beta )},0\le x<\infty & (1.8)\end{array}$

for a > 0, α>0, β > 0, δ>0 and zero elsewhere, with the standard form given by

$p_{21}(x)=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}{x}^{\alpha -1}{(1+x)}^{-(\alpha +\beta )},0\le x<\infty ,\alpha >0,\beta >0$

and zero elsewhere; and the generalized gamma model is given by

$\begin{array}{ll}{p}_3(x)=\frac{\delta a^{\frac{\alpha }{\delta }}}{\Gamma (\frac{\alpha }{\delta })}{x}^{\alpha -1}{\text{e}}^{-a{x}^{\delta }},0\le x<\infty ,a>0,\delta >0,\alpha >0& (1.9)\end{array}$

and zero elsewhere with the standard form given by

$p_{31}(x)=\frac{a^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}{\text{e}}^{-ax},0\le x<\infty ,a>0,\alpha >0$

and zero elsewhere. In statistical densities, the parameters are usually real, and hence, the conditions are stated for the real parameters but the functions are available when the parameters are in the complex domain also. In that case, the same conditions hold on the real parts of the parameters. We will examine Mellin convolutions of products and ratios involving f₁(x₁) and f₂(x₂) belonging to the models (Equations 1.7–1.9) so that the results obtained will be readily applicable.

The concept of Mellin convolution of a ratio is coming from the following considerations. As before, let x₁ > 0 and x₂ > 0 be real scalar positive variables with their associated functions f₁(x₁) and f₂(x₂), respectively, and with the joint function f₁(x₁)f₂(x₂) the product. Let $u=\frac{x_2}{x_1}$ the ratio and let g₂(u) be the function corresponding to u. Then, proceeding as before we have

$\begin{array}{ll}E\left[u^{s-1}\right]=E\left[{(\frac{x_2}{x_1})}^{s-1}\right]=E\left[x_2^{s-1}\right]E\left[x_1^{-s+1}\right]=M_{f_2}(s)M_{f_1}(2-s).& (1.10)\end{array}$

Let $u=\frac{x_2}{x_1},v=x_2\Rightarrow \text{d}{x}_1\wedge \text{d}{x}_2=-\frac{v}{u^2}\text{d}u\wedge \text{d}v,x_2=v,x_1=\frac{v}{u}$. In addition, $u=\frac{x_2}{x_1},v=x_1\Rightarrow \text{d}{x}_1\wedge \text{d}{x}_2=v\text{d}u\wedge \text{d}v,x_1=v,x_2=uv$. Then, the Mellin conovlution of a ratio property is the following:

$\begin{array}{c}{g}_2(u)={\displaystyle {\int }_v}\frac{v}{u^2}{f}_1(\frac{v}{u})f_2(v)\text{d}v={\displaystyle {\int }_v}v{f}_1(v)f_2(uv)\text{d}v\\ =\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}{M}_{f_1}(2-s)M_{f_2}(s)u^{-s}\text{d}s.& (1.11)\end{array}$

Again, we will examine Mellin convolution of a ratio involving functions belonging to the pathway family (Equations 1.7–1.9) for the real scalar variable case first.

This study is organized as follows: Section 2 deals with Mellin convolution of a product involving functions in the standard form in Equations 1.7–1.9. One general form is illustrated at the end. Section 3 gives Mellin convolutions of a ratio involving the standard forms in Equations 1.7–1.9. Then, one illustration is given at the end for a general case. Section 4 examines some generalizations to the real matrix-variate case.

2 Mellin convolution of a product

Here, we consider only Mellin convolutions of products of two real-valued scalar functions of real scalar positive variables. The results can be extended to Mellin convolution of a product involving three or more real-valued scalar functions.

Problem 2.1. Type-2 beta vs. gamma

To start with, we consider

$\begin{array}{l}{f}_1(x_1)=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}{x}_1^{\alpha }{(1+x_1)}^{-(\alpha +1+\beta )}\\ x_1\ge 0,f_2(x_2)=\frac{a^{\rho +1}}{\Gamma (\rho +1)}{x}_2^{\rho }{\text{e}}^{-a{x}_2},x_2\ge 0,\end{array}$

for α > −1, ρ > −1, β > 0, a > 0 and let f₁ and f₂ be zero elsewhere. Let the joint function be f₁(x₁)f₂(x₂) and u = x₁x₂ with the corresponding function g₁(u). Then, for $c=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}\frac{a^{\rho +1}}{\Gamma (\rho +1)}$,

$\begin{array}{l}{g}_1(u)={\displaystyle {\int }_0^{\infty }}\frac{1}{v}{f}_1(\frac{u}{v})f_2(v)\text{d}v\\ =c{\displaystyle {\int }_0^{\infty }}\frac{1}{v}{(\frac{u}{v})}^{\alpha }{(1+\frac{u}{v})}^{-(\alpha +1+\beta )}{v}^{\rho }{\text{e}}^{-av}\text{d}v\\ =c{u}^{\alpha }{\displaystyle {\int }_0^{\infty }}{v}^{\rho +\beta }{(v+u)}^{-(\alpha +1+\beta )}{\text{e}}^{-av}\text{d}v,0\le u<\infty & (2.1)\end{array}$

$\begin{array}{l}{g}_1(u)={\displaystyle {\int }_0^{\infty }}\frac{1}{v}{f}_1(v)f_2(\frac{u}{v})\text{d}v\\ =c{u}^{\rho }{\displaystyle {\int }_0^{\infty }}{v}^{\alpha -\rho -1}{(1+v)}^{-(\alpha +1+\beta )}{\text{e}}^{-\frac{au}{v}}\text{d}v,0\le u<\infty .& (2.2)\end{array}$

Also, the Mellin transforms are the following:

$\begin{array}{l}{M}_{f_1}(s)=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}{\displaystyle {\int }_0^{\infty }}{x}_1^{\alpha +s-1}{(1+x_1)}^{-(\alpha +1+\beta )}\text{d}{x}_1\\ =\frac{\Gamma (\alpha +s)}{\Gamma (\alpha +1)}\frac{\Gamma (\beta -s+1)}{\Gamma (\beta )},\Re (\alpha +s)>0,\Re (\beta -s+1)>0& (2.3)\end{array}$

$\begin{array}{l}{M}_{f_2}(s)=\frac{a^{\rho +1}}{\Gamma (\rho +1)}{\displaystyle {\int }_0^{\infty }}{x}_2^{\rho +s-1}{\text{e}}^{-a{x}_2}\text{d}{x}_2\\ =\frac{\Gamma (\rho +s)}{\Gamma (\rho +1)}\frac{a}{a^s},\Re (\rho +s)>0& (2.4)\end{array}$

where ℜ(·) denotes the real part of (·). Then,

$\begin{array}{l}{g}_1(u)=\frac{1}{\Gamma (\alpha +1)\Gamma (\beta )}\frac{a}{\Gamma (\rho +1)}\\ \frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}\Gamma (\alpha +s)\Gamma (\rho +s)\Gamma (\beta +1-s){(au)}^{-s}\text{d}s& (2.5)\end{array}$

$\begin{array}{ll}=\frac{a}{\Gamma (\alpha +1)\Gamma (\beta )\Gamma (\rho +1)}{G}_{1,2}^{2,1}\left[au{|}_{\alpha .,\rho }^{-\beta }\right],0\le u<\infty & (2.6)\end{array}$

where the G(·) is the G-function. For the theory and applications of the G-function, see for example, Mathai [5]. The inverse Mellin transform in Equation 2.5 can be evaluated by using residue calculus. For α − ρ ≠ ±ν, ν = 0, 1, ... the poles of the integrand are simple. Then, the function is available as the sum of the residues at the poles of Γ(α+s) and Γ(ρ+s). These produce two confluent hypergeometric series, and the sum of these residues is the explicit value of the G-function for 0 ≤ u < ∞. g₁(u) above gives various representations of the density of u. Since the density is unique, we can compare the different representations of the unique density and obtain several results on the unique density or we can obtain several results on the G-function. Different types of results on the G-function are given as Theorem 2.1 below. Since the procedure is the same, we will not give the derivations in detail for the remaining theorems in Section 2; only the results will be listed as theorems. The proofs are parallel to the steps given above.

Theorem 2.1. For a > 0, u≥0, α > −1, ρ > 0, β > 0

$\begin{array}{l}{G}_{1,2}^{2,1}\left[au{|}_{\alpha ,\rho }^{-\beta }\right]=\Gamma (\alpha +1+\beta )a^{\rho }{u}^{\alpha }\\ {\displaystyle {\int }_0^{\infty }}{v}^{\rho +\beta }{(v+u)}^{-(\alpha +1+\beta )}{\text{e}}^{-av}\text{d}v\\ =\Gamma (\alpha +1+\beta )a^{\rho }{u}^{\rho }\\ {\displaystyle {\int }_0^{\infty }}{v}^{\alpha -\rho -1}{(1+v)}^{-(\alpha +1+\beta )}{\text{e}}^{-a\frac{u}{v}}\text{d}v\\ ={(au)}^{\alpha }\Gamma (\rho -\alpha )\Gamma (1+\alpha +\beta )\\ {}_1{{\displaystyle F}}_1(\beta +1+\alpha ;1+\alpha -\rho :au)\\ +{(au)}^{\rho }\Gamma (1+\beta +\rho )\Gamma (\alpha -\rho )\\ {}_1{{\displaystyle F}}_1(\beta +1+\rho ;1+\rho -\alpha ;au),\end{array}$

for 0 ≤ au < ∞, α − ρ ≠ ±ν, n = 0, 1, ....

Note that the integral can be evaluated at specific values of a and u such as u = 1; a = 1; a = 1, u = 1, giving rise to simpler integrals. The condition α−ρ≠±ν is needed only to write the G-function in terms of a confluent hypergeometric series. Each integral above multiplied by $\frac{1}{\Gamma (\alpha +1)\Gamma (\beta )}\frac{a}{\Gamma (\rho +1)}$ produces a statistical density g₁(u) for u also since our starting f₁ and f₂ are statistical densities. We have taken non-negative integrable functions with total integral unity for convenience. The results hold for other types of functions provided the Mellin transforms exist and the product of the Mellin transforms is invertible. A direct generalization of Problem 2.1 can be achieved by replacing 1+x₁ in f₁ by $1+a_1{x}_1^{{\delta }_1},a_1>0,{\delta }_1>0$ and replacing the exponent ax₂ in f₂ by $a{x}_2^{{\delta }_2},{\delta }_2>0$. Then, instead of the G-function, one will end up with a H-function and then one will obtain results on this H-function. For the theory and applications of the H-function, see for example, Mathai et al. [6]. Since x_j≥0 in Problem 2.1, we can obtain parallel results for δ_j < 0 also.

Problem 2.2. Type-2 beta vs. type-2 beta

Let

$f_j(x_j)=\frac{\Gamma ({\alpha }_j+1+{\beta }_j)}{\Gamma ({\alpha }_j+1)\Gamma ({\beta }_j)}{x}_j^{{\alpha }_j}{(1+x_j)}^{-({\alpha }_j+1+{\beta }_j)},$

for 0 ≤ x_j < ∞, α_j > −1, β_j > 0, j = 1, 2 and let f_j be zero elsewhere for j = 1, 2. Let the joint function of x₁ and x₂ be f₁(x₁)f₂(x₂) and u = x₁x₂ with the corresponding function g₁(u). Then, proceeding as in Problem 2.1, we have the following results:

Theorem 2.2. For α_j > −1, β_j > 0, j = 1, 2 and for $c=\left\{{\prod }_{j=1}^2\frac{\Gamma ({\alpha }_j+1+{\beta }_j)}{\Gamma ({\alpha }_j+1)\Gamma ({\beta }_j)}\right\}$,

$\begin{array}{l}{g}_1(u)=c{u}^{{\alpha }_1}{\displaystyle {\int }_0^{\infty }}{v}^{{\beta }_1+{\alpha }_2}{(u+v)}^{-({\alpha }_1+1+{\beta }_1)}{(1+v)}^{-({\alpha }_2+1+{\beta }_2)}\text{d}v,\\ 0\le u<\infty \\ =c{u}^{{\alpha }_2}{\displaystyle {\int }_0^{\infty }}{v}^{{\alpha }_1+{\beta }_2}{(1+v)}^{-({\alpha }_1+1+{\beta }_1)}{(v+u)}^{-({\alpha }_2+1+{\beta }_2)}\text{d}v,\\ 0\le u<\infty \\ =\left\{{\displaystyle \prod _{j=1}^2}\frac{1}{\Gamma ({\alpha }_j+1)\Gamma ({\beta }_j)}\right\}{G}_{2,2}^{2,2}\left[u{|}_{{\alpha }_1,{\alpha }_2}^{-{\beta }_1,-{\beta }_2}\right],0\le u<\infty \end{array}$

$\begin{array}{l}{G}_{2,2}^{2,2}\left[u{|}_{{\alpha }_1,{\alpha }_2}^{-{\beta }_1,-{\beta }_2}\right]\\ =\Gamma ({\alpha }_2-{\alpha }_1)\Gamma ({\beta }_1+1+{\alpha }_1)\Gamma ({\beta }_2+1+{\alpha }_1)u^{{\alpha }_1}\\ {\times }_2{F}_1({\beta }_1+1+{\alpha }_1,{\beta }_2+1+{\alpha }_1;1+{\alpha }_1-{\alpha }_2;u)\\ +\Gamma ({\alpha }_1-{\alpha }_2)\Gamma ({\beta }_1+1+{\alpha }_2)\Gamma ({\beta }_2+1+{\alpha }_2)u^{{\alpha }_2}\\ {\times }_2{F}_1({\beta }_1+1+{\alpha }_2,{\beta }_2+1+{\alpha }_2;1+{\alpha }_2-{\alpha }_1;u)\\ for0\le u<1,{\alpha }_1-{\alpha }_2\ne \pm \nu ,\nu =0,1,...\\ \Gamma ({\alpha }_1+1+{\beta }_1)\Gamma ({\alpha }_2+{\beta }_1+1)\Gamma ({\beta }_2-{\beta }_1)u^{-{\beta }_1-1}\\ {\times }_2{F}_1({\alpha }_1+{\beta }_1+1,{\alpha }_2+{\beta }_1+1;1+{\beta }_1-{\beta }_2;\frac{1}{u})\\ +\Gamma ({\alpha }_1+{\beta }_2+1)\Gamma ({\alpha }_2+{\beta }_2+1)\Gamma ({\beta }_1-{\beta }_2)u^{-{\beta }_2-1}\\ {\times }_2{F}_1({\alpha }_1+{\beta }_2+1,{\alpha }_2+{\beta }_2+1;1+{\beta }_2-{\beta }_1;\frac{1}{u})\\ for{\beta }_1-{\beta }_2\ne \pm \nu ,\nu =0,1,...,1\le u<\infty \end{array}$

A direct generalization of Problem 2.2 is to replace 1+x_j in f_j by $1+a_j{x}_j^{{\delta }_j},a_j>0,{\delta }_j>0,j=1,2$. Then, we will end up with a H-function, instead of the G-function above, and then parallel results will be available on this H-function. Since x_j≥0 in f_j here, one can allow δ_j to be negative also. Parallel results will be available for j = 1, 2.

Problem 2.3. Type-1 beta vs. type-1 beta

Let

$\begin{array}{l}{f}_j(x_j)=\frac{\Gamma ({\alpha }_j+1+{\beta }_j)}{\Gamma ({\alpha }_j+1)\Gamma ({\beta }_j)}{x}_j^{{\alpha }_j}{(1-x_j)}^{{\beta }_j-1},\\ 0\le x_j\le 1,{\alpha }_j>-1,{\beta }_j>0\end{array}$

and zero elsewhere for j = 1, 2. Let the joint function be f₁(x₁)f₂(x₂) and let u = x₁x₂ with the corresponding function denoted as g₁(u). Then, proceeding as in Problem 2.1, one has the following results, for ${\alpha }_j>-1,{\beta }_j>0,j=1,2,c={\prod }_{j=1}^2\frac{\Gamma ({\alpha }_j+1+{\beta }_j)}{\Gamma ({\alpha }_j+1)\Gamma ({\beta }_j)}$:

Theorem 2.3. For α_j > −1, β_j > 0, j = 1, 2 and for 0 ≤ u ≤ 1

$\begin{array}{l}{G}_{2,2}^{2,0}\left[u{|}_{{\alpha }_1,{\alpha }_2}^{{\alpha }_1+{\beta }_1,{\alpha }_2+{\beta }_2}\right]\\ =\frac{1}{\Gamma ({\beta }_1)\Gamma ({\beta }_2)}{u}^{{\alpha }_1}{\displaystyle {\int }_{v=u}^1}{v}^{{\alpha }_2-{\alpha }_1-{\beta }_1}{(v-u)}^{{\beta }_1-1}{(1-v)}^{{\beta }_2-1}\text{d}v,\\ 0\le u\le 1\\ =\frac{1}{\Gamma ({\beta }_1)\Gamma ({\beta }_2)}{u}^{{\alpha }_2}{\displaystyle {\int }_{v=u}^1}{v}^{{\alpha }_1-{\beta }_2-{\alpha }_2}{(v-u)}^{{\beta }_2-1}{(1-v)}^{{\beta }_1-1}\text{d}v,\\ 0\le u\le 1\\ =\frac{\Gamma ({\alpha }_2-{\alpha }_1)}{\Gamma ({\beta }_1)\Gamma ({\alpha }_2+{\beta }_2-{\alpha }_1)}{u}^{{\alpha }_1}\\ {\times }_2{F}_1(1-{\beta }_1,1+{\alpha }_1-{\alpha }_2-{\beta }_2;1+{\alpha }_1-{\alpha }_2;u)\\ +\frac{\Gamma ({\alpha }_1-{\alpha }_2)}{\Gamma ({\beta }_2)\Gamma ({\alpha }_1+{\beta }_1-{\alpha }_2)}{u}^{{\alpha }_2}\\ {\times }_2{F}_1(1-{\beta }_2,1+{\alpha }_2-{\alpha }_1-{\beta }_1;1+{\alpha }_2-{\alpha }_1;u),0\le u\le 1\\ for{\alpha }_1-{\alpha }_2\ne \pm \nu ,\nu =0,1,...,0\le u<1\end{array}$

This G-function multiplied by $\left\{{\prod }_{j=1}^2\frac{\Gamma ({\alpha }_j+1+{\beta }_j)}{\Gamma ({\alpha }_j+1)}\right\}$ gives a statistical density g₁(u) for u also. A direct generalization is available by replacing 1−x_j in f_j by $1-a_j{x}_j^{{\delta }_j}$ for a_j > 0, δ_j > 0, with the condition $1-a_j{x}_j^{{\delta }_j}>0,j=1,2$. Then, one will end up with a H-function, instead of the G-function above, and parallel results will be available for this H-function. It may be observed that a constant multiple of this H-function is a statistical density for u also.

Problem 2.4. Type-1 beta vs. gamma

Let

$\begin{array}{l}{f}_1(x_1)=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}{x}_1^{\alpha }{(1-x_1)}^{\beta -1},0\le x_1\le 1,\\ f_2(x_2)=\frac{a^{\gamma +1}}{\Gamma (\gamma +1)}{x}_2^{\gamma }{\text{e}}^{-a{x}_2},x_2\ge 0\end{array}$

for α > −1, β > 0, γ > −1, a > 0 and let f₁ and f₂ be zero elsewhere and let the joint function be f₁(x₁)f₂(x₂) and let u = x₁x₂ with the corresponding function denoted as g₁(u). Then, proceeding as in Problem 2.1, we have the following results:

Theorem 2.4. For α > −1, γ > −1, a > 0, β > 0

$\begin{array}{l}{G}_{1,2}^{2,0}\left[au{|}_{\alpha ,\gamma }^{\alpha +\beta }\right]=\frac{a^{\gamma }}{\Gamma (\beta )}{u}^{\alpha }\\ {\displaystyle {\int }_{v=u}^{\infty }{v}^{\gamma -\alpha -\beta }}(v-u{)}^{\beta -1}{\text{e}}^{-av}\text{d}v,0\le u<\infty \\ =\frac{a^{\gamma }}{\Gamma (\beta )}{u}^{\gamma }{\displaystyle {\int }_0^1{v}^{\alpha -\gamma -1}}(1-v{)}^{\beta -1}{\text{e}}^{-a\frac{u}{v}}\text{d}v,0\le u<\infty \\ =\frac{\Gamma (\gamma -\alpha )}{\Gamma (\beta )}(au{)}^{\alpha }{}_1{{\displaystyle F}}_1(1-\beta ;1+\alpha -\gamma :-au)\\ +\frac{\Gamma (\alpha -\gamma )}{\Gamma (\alpha +\beta -\gamma )}(au{)}^{\gamma }{}_1{{\displaystyle F}}_1(1+\gamma -\alpha -\beta ;\\ 1+\gamma -\alpha ;-au),0\le u<\infty \\ for\alpha -\gamma \ne \pm \nu ,\nu =0,1,\mathrm{...},0\le u<\infty \end{array}$

Note that this G-function multiplied by $\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)}\frac{a}{\Gamma (\gamma +1)}$ produces a statistical density g₁(u) for u also. A direct generalization is available by replacing 1−x₁ in f₁ by $1-a_1{x}_1^{{\delta }_1}$ with $1-a_1{x}_1^{{\delta }_1}>0,a_1>0,{\delta }_1>0$ and replacing the exponent ax₂ in f₂ by $a{x}_2^{{\delta }_2}$ with δ₂ > 0. Then, one can obtain parallel results on a H-function. Here, δ₂ can be negative also so that the integrals and the corresponding H-function will exist. If δ₁ is allowed to be negative, then it will create problems with the support there and the problem will be complicated.

Problem 2.5. Type-1 beta vs. type-2 beta

Let

$\begin{array}{l}{f}_1(x_1)=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}{x}_1^{\alpha }{(1-x_1)}^{\beta -1},\\ f_2(x_2)=\frac{\Gamma (\gamma +1+\delta )}{\Gamma (\gamma +1)\Gamma (\delta )}{x}_2^{\gamma }{(1+x_2)}^{-(\gamma +1+\delta )},\end{array}$

for 0 ≤ x₁ ≤ 1, x₂≥0, α > −1, γ > −1, δ>0, β > 0 and let f₁ and f₂ be zero elsewhere. Let the joint function be f₁(x₁)f₂(x₂) and let u = x₁x₂ with the corresponding function denoted as g₁(u). Then, proceeding as in Problem 2.1, we will have the following results:

Theorem 2.5. For α > −1, γ > −1, β > 0, δ > 0

$\begin{array}{l}{G}_{2,2}^{2,1}\left[u{|}_{\alpha ,\gamma }^{-\delta ,\alpha +\beta }\right]=\frac{\Gamma (\gamma +1+\delta )}{\Gamma (\beta )}{u}^{\alpha }\\ {\displaystyle {\int }_{v=u}^{\infty }{v}^{\gamma -\alpha -\beta }}(v-u{)}^{\beta -1}(1+v{)}^{-(\gamma +1+\delta )}\text{d}v,\\ 0\le u<\infty \\ =\frac{\Gamma (\gamma +1+\delta )}{\Gamma (\beta )}{u}^{\gamma }\\ {\displaystyle {\int }_{v=0}^1{v}^{\alpha +\delta }}(1-v{)}^{\beta -1}(v+u{)}^{-(\gamma +1+\delta )}\text{d}v,\\ 0\le u<\infty \\ =\{\begin{array}{l}\frac{\Gamma (\gamma -\alpha )\Gamma (1+\delta +\alpha )}{\Gamma (\beta )}{u}^{\alpha }{}_2{{\displaystyle F}}_1(1+\delta +\alpha ,1-\beta ;\hfill \\ 1+\alpha -\gamma ;-u)\hfill \\ +\frac{\Gamma (\alpha -\gamma )\Gamma (1+\delta +\gamma )}{\Gamma (\alpha +\beta -\gamma )}{u}^{\gamma }{}_2{{\displaystyle F}}_1(1+\delta \hfill \\ +\gamma ,1+\gamma -\alpha -\beta ;1+\gamma -\alpha ;-u)\hfill \\ for0\le u<1,\alpha -\gamma \ne \pm \nu ,\nu =0,1,\mathrm{...}\hfill \\ \frac{\Gamma (\alpha +1+\delta )\Gamma (\gamma +1+\delta )}{\Gamma (\alpha +\beta +1+\delta )}{u}^{-1-\delta }{}_2{{\displaystyle F}}_1(\alpha +1+\delta ,\hfill \\ \gamma +1+\delta ;\alpha +\beta +1+\delta ;-\frac{1}{u})\hfill \\ for1\le u<\infty .\hfill \end{array}\end{array}$

This G-function multiplied by $\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)}\frac{1}{\Gamma (\gamma +1)\Gamma (\delta )}$ produces a statistical density g₁(u) for u also. A direct generalization is available by replacing 1−x₁ in f₁ by $1-a_1{x}_1^{{\delta }_1}$ with $a_1>0,{\delta }_1>0,1-a_1{x}_1^{{\delta }_1}>0$ and replacing 1+x₂ in f₂ by $1+a_2{x}_2^{{\delta }_2}$ with a₂ > 0, δ₂ > 0. Then, one will obtain parallel results on a H-function. Also, note that in this Problem 2.5, if the type-2 beta density is replaced by a general density, then the first integral representation gives Erdélyi-Kober fractional integral of the second kind of order β and parameter α, see also Mathai and Haubold [7] and Fortin et al. [8, 9].

Problem 2.6. Gamma vs. gamma

Let

$f_j(x_j)=\frac{a_j^{{\alpha }_j+1}}{\Gamma ({\alpha }_j+1)}{x}_j^{{\alpha }_j}{\text{e}}^{-a_j{x}_j},x_j\ge 0,{\alpha }_j>-1,a_j>0,j=1,2$

and let the joint function be f₁(x₁)f₂(x₂) and u = x₁x₂ with the corresponding function denoted as g₁(u). Then, proceeding as in Problem 2.1, we will have the following results:

Theorem 2.6. For a_j > 0, α_j > −1, j = 1, 2

$\begin{array}{l}{G}_{0,2}^{2,0}\left[a_1{a}_{2}u{|}_{{\alpha }_1,{\alpha }_2}\right]=a_1^{{\alpha }_1}{a}_2^{{\alpha }_2}{u}^{{\alpha }_1}\\ {\displaystyle {\int }_{v=0}^{\infty }}{v}^{{\alpha }_2-{\alpha }_1-1}{\text{e}}^{-a_1\frac{u}{v}-a_{2}v}\text{d}v,0\le u<\infty \\ =a_1^{{\alpha }_1}{a}_2^{{\alpha }_2}{u}^{{\alpha }_2}\\ {\displaystyle {\int }_0^{\infty }}{v}^{{\alpha }_1-{\alpha }_2-1}{\text{e}}^{-a_{1}v-a_2\frac{u}{v}}\text{d}v,0\le u<\infty \\ ={(a_1{a}_{2}u)}^{{\alpha }_1}\Gamma ({\alpha }_2-{\alpha }_1)\\ {}_0{{\displaystyle F}}_1(;1+{\alpha }_1-{\alpha }_2;a_1{a}_{2}u)\\ +{(a_1{a}_{1}u)}^{{\alpha }_2}\Gamma ({\alpha }_1-{\alpha }_2)\\ {}_0{{\displaystyle F}}_1(;1+{\alpha }_2-{\alpha }_1;a_1{a}_{2}u),0\le u<\infty \\ for{\alpha }_1-{\alpha }_2\ne \pm \nu ,\nu =0,1,...\end{array}$

A direct generalization is available by replacing the exponent a_jx_j by $a_j{x}_j^{{\delta }_j},{\delta }_j>0,j=1,2$. Then, we will obtain parallel results on a H-function. Here, one can allow δ_j < 0, j = 1, 2 also, which will then produce parallel results on a H-function. Since we have started with statistical densities for f₁ and f₂, we may observe that a constant multiple of this G-function in Theorem 2.6 is a statistical density g₁(u) for u also. When δ₁ = 1 we have Krätzel integral and associated with it is the Krätzel transform, see Mathai and Haubold [10]. When δ₁ = δ₂ = 1, that is the case considered in Theorem 2.6, the integrand in the integral representations, normalized, gives inverse Gaussian density. The unconditional density in the Bayesian analysis when the prior density is gamma and when the scale parameter has a prior gamma distribution then the unconditional density has the structure in the general as well as in the particular case where δ₁ = δ₂ = 1. In this particular case, one has the Bessel integral given by the G-function representation in the simple poles cases also.

For the sake of illustration, one direct generalization of a Mellin convolution of a product will be derived here in detail. This will be listed as Problem 2.7.

Problem 2.7. Generalized type-2 beta vs. generalized type-2 beta

Let

$f_j(x_j)=\frac{{\delta }_j{a}_j^{\frac{{\alpha }_j+1}{{\delta }_j}}\Gamma (\frac{{\alpha }_j+1}{{\delta }_j}+{\beta }_j)}{\Gamma (\frac{{\alpha }_j+1}{{\delta }_j})\Gamma ({\beta }_j)}{x}_j^{{\alpha }_j}{(1+a_j{x}_j^{{\delta }_j})}^{-(\frac{{\alpha }_j+1}{{\delta }_j}+{\beta }_j)},$

for x_j≥0, a_j > 0, δ_j > 0, α_j > −1, j = 1, 2 and f_j is zero otherwise for j = 1, 2. Let the joint function be f₁(x₁)f₂(x₂) and let u = x₁x₂ and let the function corresponding to u be denoted as g₁(u). Then,

$\begin{array}{l}E\left[u^{s-1}\right]=E\left[x_1^{s-1}\right]E\left[x_2^{s-1}\right]\\ E\left[x_j^{s-1}\right]=\frac{{\delta }_j{a}_j^{\frac{{\alpha }_j+1}{{\delta }_j}}\Gamma (\frac{{\alpha }_j+1}{{\delta }_j}+{\beta }_j)}{\Gamma (\frac{{\alpha }_j+1}{{\delta }_j})\Gamma ({\beta }_j)}\\ {\displaystyle {\int }_0^{\infty }}{x}_j^{{\alpha }_j+s-1}{(1+a_j{x}_j^{{\delta }_j})}^{-(\frac{{\alpha }_j+1}{{\delta }_j}+{\beta }_j)}\text{d}{x}_j\\ =\frac{a_j^{\frac{1}{{\delta }_j}}}{a_j^{\frac{s}{{\delta }_j}}}\frac{\Gamma (\frac{{\alpha }_j+s}{{\delta }_j})}{\Gamma (\frac{{\alpha }_j+1}{{\delta }_j})}\frac{\Gamma ({\beta }_j-\frac{s-1}{{\delta }_j})}{\Gamma ({\beta }_j)}.& (2.7)\end{array}$

For convenience, let us consider the case δ₁ = δ₂ = δ. Then, for $c_1={\prod }_{j=1}^2\frac{a_j^{\frac{1}{\delta }}}{\Gamma (\frac{{\alpha }_j+1}{\delta })\Gamma ({\beta }_j)}$ and from Equation 2.7 by taking the inverse Mellin transform, we have g₁(u) as the following:

$\begin{array}{l}{g}_1(u)=c_1\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}\left[{\displaystyle \prod _{j=1}^2}\Gamma (\frac{{\alpha }_j+s}{\delta })\Gamma ({\beta }_j-\frac{s-1}{\delta })\right]{\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u\right]}^{-s}\text{d}s\\ =c_1{H}_{2,2}^{2,2}\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u{|}_{(\frac{{\alpha }_j}{\delta },\frac{1}{\delta }),j=1,2}^{(1-{\beta }_j-\frac{1}{\delta },\frac{1}{\delta }),j=1,2}\right],0\le u<\infty .& (2.8)\end{array}$

The inverse Mellin transform in Equation 2.8 will be evaluated by using residue calculus. Note that for α₁−α₂≠±ν, ν = 0, 1, ..., the poles of $\Gamma (\frac{{\alpha }_1+s}{\delta })\Gamma (\frac{{\alpha }_2+s}{\delta })$ are simple. The poles of $\Gamma (\frac{{\alpha }_j+s}{\delta })$ are at $\frac{{\alpha }_j+s}{\delta }=-\nu ,\nu =0,1,...\Rightarrow s=-{\alpha }_j-\delta \nu ,\nu =0,1,...$. The residue at s = −α₁−δν coming from $\Gamma (\frac{{\alpha }_1+s}{\delta })$ is $\underset{s\to -{\alpha }_1-\delta \nu }{lim}(s+{\alpha }_1+\delta \nu )\Gamma (\frac{{\alpha }_1+s}{\delta })=\delta \frac{{(-1)}^{\nu }}{\nu !}$. The rest of the steps are parallel to those in the derivations in Problem 2.1. For $0\le a_1{a}_2{u}^{\delta }<1$, we will end up with two Gauss' hypergeometric series, and the continuation part for $1\le a_1{a}_1{u}^{\delta }<\infty$ will give another two Gauss' hypergeometric series. If we consider the transformation of variables, then g₁(u) has the following forms where $c={\prod }_{j=1}^2\frac{\delta a_j^{\frac{{\alpha }_j+1}{\delta }}\Gamma (\frac{{\alpha }_j+1}{\delta }+{\beta }_j)}{\Gamma (\frac{{\alpha }_j+1}{\delta })\Gamma ({\beta }_j)}$:

$\begin{array}{l}{g}_1(u)=c{\displaystyle {\int }_v}\frac{1}{v}{(\frac{u}{v})}^{{\alpha }_1}{\left[1+a_1{(\frac{u}{v})}^{\delta }\right]}^{-(\frac{{\alpha }_1+1}{\delta }+{\beta }_1)}{v}^{{\alpha }_2}\\ {(1+a_{2}v)}^{-(\frac{{\alpha }_2+1}{\delta }+{\beta }_2)}\text{d}v\\ =c{u}^{{\alpha }_1}{\displaystyle {\int }_0^{\infty }}{v}^{{\alpha }_2-{\alpha }_1-1}{\left[1+\frac{a_1{u}^{\delta }}{v^{\delta }}\right]}^{-(\frac{{\alpha }_1+1}{\delta }+{\beta }_1)}\\ {(1+a_{2}v)}^{-(\frac{{\alpha }_2+1}{\delta }+{\beta }_2)}\text{d}v.& (2.9)\end{array}$

The second form is available by interchanging (α₁, β₁) and (α₂, β₂) in Equation 2.9. Note that

$g_1(u)={\delta }^2\left\{{\displaystyle \prod _{j=1}^2}{a}_j^{\frac{{\alpha }_j}{\delta }}\Gamma (\frac{{\alpha }_j+1}{\delta }+{\beta }_j)\right\}\times \text{(the H-function)}$

where the H-function has the following explicit forms:

$\begin{array}{l}H=H_{2,2}^{2,2}\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u{|}_{(\frac{{\alpha }_j}{\delta },\frac{1}{\delta }),j=1,2}^{(1-{\beta }_j-\frac{1}{\delta },\frac{1}{\delta }),j=1,2}\right]\\ =\{\begin{array}{l}\delta \Gamma (\frac{{\alpha }_2-{\alpha }_1}{\delta })\Gamma ({\beta }_1+\frac{{\alpha }_1+1}{\delta })\Gamma ({\beta }_2+\frac{{\alpha }_1+1}{\delta }){\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u\right]}^{{\alpha }_1}\\ {\times }_2{F}_1({\beta }_1+\frac{{\alpha }_1+1}{\delta },{\beta }_2+\frac{{\alpha }_1+1}{\delta };1+\frac{{\alpha }_1-{\alpha }_2}{\delta };a_1{a}_1{u}^{\delta })\\ +\delta \Gamma (\frac{{\alpha }_1-{\alpha }_2}{\delta })\Gamma ({\beta }_1+\frac{{\alpha }_2+1}{\delta })\Gamma ({\beta }_2+\frac{{\alpha }_2+1}{\delta }){\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u\right]}^{{\alpha }_2}\\ {\times }_2{F}_1({\beta }_1+\frac{{\alpha }_2+1}{\delta },{\beta }_2+\frac{{\alpha }_2+1}{\delta };1+\frac{{\alpha }_2-{\alpha }_1}{\delta };a_1{a}_2{u}^{\delta }),\\ \text{for }0\le a_1{a}_2{u}^{\delta }<1\text{ and for }{\alpha }_1-{\alpha }_2\ne \pm \nu ,\nu =0,1,...\\ \delta \Gamma ({\beta }_1+\frac{{\alpha }_1+1}{\delta })\Gamma ({\beta }_1+\frac{{\alpha }_2+1}{\delta })\Gamma ({\beta }_2-{\beta }_1){\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u\right]}^{-({\beta }_1\delta +1)}\\ {\times }_2{F}_1({\beta }_1+\frac{{\alpha }_1+1}{\delta },{\beta }_1+\frac{{\alpha }_2+1}{\delta };1+{\beta }_1-{\beta }_2;\frac{1}{a_1{a}_2{u}^{\delta }})\\ +\delta \Gamma ({\beta }_2+\frac{{\alpha }_1+1}{\delta })\Gamma ({\beta }_2+\frac{{\alpha }_2+1}{\delta })\Gamma ({\beta }_1-{\beta }_2)\\ {\left[{(a_1{a}_2)}^{\frac{1}{\delta }}u\right]}^{-({\beta }_2\delta +1)}\\ {\times }_2{F}_1({\beta }_2+\frac{{\alpha }_1+1}{\delta },{\beta }_2+\frac{{\alpha }_2+1}{\delta };1+{\beta }_2-{\beta }_1;\frac{1}{a_1{a}_2{u}^{\delta }})\\ \text{for }{a}_1{a}_2{u}^{\delta }\ge 1\text{ and for }{\beta }_1-{\beta }_2\ne \pm \nu ,\nu =0,1,...\end{array}\end{array}$

This completes the calculations and the representations of g₁(u) in two different integral representations and in terms of a H-function with explicit computable series form representations.

3 Mellin convolution of a ratio

Here, we examine a ratio. Let x₁ > 0 and x₂ > 0 be real scalar positive variables associated with the functions f₁(x₁) and f₂(x₂), respectively. Let the joint function be f₁(x₁)f₂(x₂). Let $u=\frac{x_2}{x_1}$ the ratio. Let the Mellin transforms of f₁ and f₂ be M_f1(s) and M_f2(s), respectively with the Mellin parameter s. Let the function associated with the ratio u be denoted by g₂(u). Then, the Mellin transform of g₂(u) is available from the joint function as M_f2(s)M_f1(2−s) which can easily be seen if x₁ and x₂ are statistically independently distributed random variables with the densities f₁(x₁) and f₂(x₂), respectively. Then,

$\begin{array}{ll}E\left[u^{s-1}\right]=E\left[{(\frac{x_2}{x_1})}^{s-1}\right]=E\left[x_2^{s-1}\right]E\left[x_1^{-s+1}\right]=M_{f_2}(s)M_{f_1}(2-s)& (3.1)\end{array}$

whenever the Mellin transforms exist. Then, from the inverse Mellin transform, g₂(u) is available as the following:

$\begin{array}{ll}{g}_2(u)=\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}{M}_{f_2}(s)M_{f_1}(2-s)u^{-s}\text{d}s,i=\sqrt{(-1)}& (3.2)\end{array}$

whenever the integral is convergent. We can also reach g₂(u) through transformation of variables. Let x₂ = v and $u=\frac{x_2}{x_1}$, then $x_1=\frac{v}{u}$ and $\text{d}{x}_1\wedge \text{d}{x}_2=-\frac{v}{u^2}\text{d}u\wedge \text{d}v$. If one takes x₁ = v, then x₂ = uv and dx₁∧dx₂ = vdu∧dv. Then, we have two different integral representations for g₂(u), namely,

$\begin{array}{ll}{g}_2(u)={\displaystyle {\int }_v}\frac{v}{u^2}{f}_1(\frac{v}{u})f_2(v)\text{d}v& (3.3)\end{array}$

$\begin{array}{ll}={\displaystyle {\int }_v}v{f}_1(v)f_2(uv)\text{d}v.& (3.4)\end{array}$

Then, Equations 3.2–3.4 give three different representations for the same function g₂(u). We will consider various functions belonging to the pathway family of functions for convenience.

Problem 3.1. Gamma vs. gamma

Let

$f_j(x_j)=\frac{a_j^{{\alpha }_j+1}}{\Gamma ({\alpha }_j+1)}{x}_j^{{\alpha }_j}{\text{e}}^{-a_j{x}_j},x_j\ge 0,{\alpha }_j>-1,a_j>0,j=1,2$

and let the joint function be f₁(x₁)f₂(x₂) and let $c={\prod }_{j=1}^2\frac{a_j^{{\alpha }_j+1}}{\Gamma ({\alpha }_j+1)}$. Then, from Equation 3.3

$\begin{array}{r}{g}_2(u)=c\int \frac{v}{u^2}{(\frac{v}{u})}^{{\alpha }_1}{\text{e}}^{-a_1\frac{v}{u}}{v}^{{\alpha }_2}{\text{e}}^{-a_{2}v}\text{d}v\\ =c\Gamma ({\alpha }_1+{\alpha }_2+2)u^{{\alpha }_2}{(a_1+a_{2}u)}^{-({\alpha }_1+{\alpha }_2+2)}& (3.5)\end{array}$

and from Equation 3.4

$\begin{array}{r}{g}_2(u)=c{\displaystyle {\int }_v}v{v}^{{\alpha }_1}{\text{e}}^{-a_{1}v}{(uv)}^{{\alpha }_2}{\text{e}}^{-a_{2}uv}\text{d}v\\ =c{u}^{{\alpha }_2}\Gamma ({\alpha }_1+{\alpha }_2+2){(a_1+a_{2}u)}^{-({\alpha }_1+{\alpha }_2+2)},& (3.6)\end{array}$

for 0 ≤ u < ∞, which is the same as the one in Equation 3.5. Also,

$\begin{array}{r}{M}_{f_2}(s)=\frac{a_2^{{\alpha }_2+1}}{\Gamma ({\alpha }_2+1)}{\displaystyle {\int }_0^{\infty }}{x}_2^{{\alpha }_2+s-1}{\text{e}}^{-a_2{x}_2}\text{d}{x}_2\\ =\frac{a_2}{a_2^s}\frac{\Gamma ({\alpha }_2+s)}{\Gamma ({\alpha }_2+1)},\Re ({\alpha }_2+s)>0& (3.7)\end{array}$

and

$\begin{array}{r}{M}_{f_1}(2-s)=\frac{a_1^{{\alpha }_1+1}}{\Gamma ({\alpha }_1+1)}{\displaystyle {\int }_0^{\infty }}{x}_1^{{\alpha }_1-s+1}{\text{e}}^{-a_1{x}_1}\text{d}{x}_1\\ =\frac{a_1^s}{\Gamma ({\alpha }_1+1)a_1}\Gamma (2+{\alpha }_1-s),\Re (2+{\alpha }_1-s)>0.& (3.8)\end{array}$

Then,

$\begin{array}{l}{g}_2(u)=\frac{a_2}{a_1}\frac{1}{\Gamma ({\alpha }_1+1)\Gamma ({\alpha }_2+1)}\frac{1}{2\pi i}\\ {\displaystyle {\int }_{c-i\infty }^{c+i\infty }}\Gamma ({\alpha }_2+s)\Gamma (2+{\alpha }_1-s){(\frac{a_{2}u}{a_1})}^{-s}\text{d}s\\ =\frac{a_2}{a_1\Gamma ({\alpha }_1+1)\Gamma ({\alpha }_2+1)}{G}_{1,1}^{1,1}\left[\frac{a_{2}u}{a_1}{|}_{{\alpha }_2}^{-1-{\alpha }_1}\right],0\le u<\infty .& (3.9)\end{array}$

This G-function can be evaluated as the sum of the residues at the poles of Γ(α₂+s) for $0\le \frac{a_{2}u}{a_1}<1$ and as the sum of the residues at the poles of Γ(2+α₁−s) for $1\le \frac{a_{2}u}{a_1}<\infty$. Thus, we have

$\begin{array}{ll}\begin{array}{r}\hfill g_2(u)=\frac{\Gamma ({\alpha }_1+{\alpha }_2+2)}{\Gamma ({\alpha }_1+1)\Gamma ({\alpha }_2+1)}\\ \hfill \times \frac{1}{u}\{\begin{array}{l}{(\frac{a_{2}u}{a_1})}^{{\alpha }_2+1}{}_1{{\displaystyle F}}_0(2+{\alpha }_1+{\alpha }_2; ;-\frac{a_{2}u}{a_1}),0\le \frac{a_{2}u}{a_1}<1\hfill \\ {(\frac{a_1}{a_{2}u})}^{{\alpha }_1+1}{}_1{{\displaystyle F}}_0(2+{\alpha }_1+{\alpha }_2; ;-\frac{a_1}{a_{2}u}),1\le \frac{a_{2}u}{a_1}<\infty .\hfill \end{array}\end{array}& (3.10)\end{array}$

Then, from Equations 3.5, 3.6, 3.10, we have the following theorem:

Theorem 3.1. For $a_j>0,{\alpha }_j>-1,c=\left\{{\prod }_{j=1}^2\frac{a_j^{{\alpha }_j+1}}{\Gamma ({\alpha }_1+1)}\right\}$,

$\begin{array}{l}{g}_2(u)=c\Gamma ({\alpha }_1+{\alpha }_2+2)u^{{\alpha }_2}(a_1+a_{2}u{)}^{-({\alpha }_1+{\alpha }_2+2)},0\le u<\infty \\ =\frac{\Gamma ({\alpha }_1+{\alpha }_2+2)}{\Gamma ({\alpha }_1+1)\Gamma ({\alpha }_2+1)}\\ \times \frac{1}{u}\{\begin{array}{l}{(\frac{a_{2}u}{a_1})}^{{\alpha }_2+1}{}_1{{\displaystyle F}}_0(2+{\alpha }_1+{\alpha }_2; ;-\frac{a_{2}u}{a_1}),0\le \frac{a_{2}u}{a_1}<1\hfill \\ {(\frac{a_1}{a_{2}u})}^{{\alpha }_1+1}{}_1{{\displaystyle F}}_0(2+{\alpha }_1+{\alpha }_2; ;-\frac{a_1}{a_{2}u}),1\le \frac{a_{2}u}{a_1}<\infty ,\hfill \end{array}\end{array}$

where g₂(u) is a statistical density for u also.

Generalization is available by replacing a_jx_j in f_j(x_j) by $a_j{x}_j^{{\delta }_j},{\delta }_j>0,j=1,2.$ In this case, one will end up with a H-function instead of the G-function above. Then, various integral and series representations of that H-function will be available. Here, we will be able to consider δ_j < 0, j = 1, 2 also giving rise to parallel results. Since the derivations of the results in Problem 3.1 are given in detail, the remaining results will be stated without the detailed derivations. The derivations will be parallel to those provided in Problem 3.1.

Problem 3.2. Gamma vs. type-2 beta

Let

$\begin{array}{l}{f}_1(x_1)=\frac{a_1^{{\alpha }_1+1}}{\Gamma ({\alpha }_1+1)}{x}_1^{{\alpha }_1}{\text{e}}^{-a_1{x}_1},\\ f_2(x_2)=\frac{\Gamma ({\alpha }_2+1+{\beta }_2)}{\Gamma ({\alpha }_2+1)\Gamma ({\beta }_2)}{x}_2^{{\alpha }_2}{(1+x_2)}^{-({\alpha }_2+1+{\beta }_2)}\end{array}$

for α_j > −1, 0 ≤ x_j < ∞, j = 1, 2, a₁ > 0, β₂ > 0 and let the joint function be f₁(x₁)f₂(x₂). Let $u=\frac{x_2}{x_1}$ with the corresponding function denoted as g₂(u). Then, following through the derivations parallel to those in Problem 3.1, we have the following results:

Theorem 3.2. For α_j > −1, j = 1, 2, β₂ > 0, a₁ > 0

$\begin{array}{l}{G}_{2,1}^{1,2}\left[\frac{u}{a_1}{|}_{{\alpha }_2}^{-{\beta }_2,-1-{\alpha }_1}\right]\\ =a_1^{{\alpha }_1+2}\Gamma ({\alpha }_2+{\beta }_2+1)u^{-{\alpha }_1-2}\\ {\displaystyle {\int }_0^{\infty }}{v}^{{\alpha }_1+{\alpha }_2+1}{(1+v)}^{-(1+{\alpha }_2+{\beta }_2)}{\text{e}}^{-a_1\frac{v}{u}}\text{d}v,0\le u<\infty \\ =a_1^{{\alpha }_1+2}\Gamma ({\alpha }_2+{\beta }_2+1)u^{{\alpha }_2}\\ {\displaystyle {\int }_0^{\infty }}{v}^{{\alpha }_1+{\alpha }_2+1}{(1+uv)}^{-(1+{\alpha }_2+{\beta }_2)}{\text{e}}^{-a_{1}v}\text{d}v,0\le u<\infty \\ =\Gamma ({\alpha }_2+{\beta }_2+1)\Gamma (1+{\alpha }_1-{\beta }_2){(\frac{a_1}{u})}^{1+{\beta }_2}\\ {}_1{{\displaystyle F}}_1(1+{\alpha }_2+{\beta }_2;{\beta }_2-{\alpha }_1;\frac{a_1}{u})\\ +\Gamma (2+{\alpha }_1+{\alpha }_2)\Gamma ({\beta }_2-{\alpha }_1-1){(\frac{a_1}{u})}^{2+{\alpha }_1}\\ {}_1{{\displaystyle F}}_1(2+{\alpha }_1+{\alpha }_2;2+{\alpha }_1-{\beta }_2;\frac{a_1}{u}),0\le u<\infty \\ for{\beta }_2-{\alpha }_1-1\ne \pm \nu ,\nu =0,1,...\end{array}$

Generalizations are possible by replacing the exponent a₁x₁ in f₁ by $a_1{x}_1^{{\delta }_1},{\delta }_1>0$ and replacing 1+x₂ in f₂ by $1+a_2{x}_2^{{\delta }_2},a_2>0,{\delta }_2>0$. Then, one will end up with a H-function, and results will be available for this H-function. In this case, δ_j < 0, j = 1, 2 will also work and parallel results will be available.

Problem 3.3. Type-2 beta vs. gamma

Let

$\begin{array}{l}{f}_1(x_1)=\frac{\Gamma (\gamma +1+\delta )}{\Gamma (\gamma +1)\Gamma (\delta )}{x}_1^{\gamma }{(1+x_1)}^{-(\gamma +1+\delta )},\\ f_2(x_2)=\frac{a^{\alpha +1}}{\Gamma (\alpha +1)}{x}_2^{\alpha }{\text{e}}^{-a{x}_2}\end{array}$

for γ > −1, α > −1, a > 0, δ>0, 0 ≤ x_j < ∞, j = 1, 2 and let the joint function be f₁(x₁)f₂(x₂). Let $u=\frac{x_2}{x_1}$. Then, proceeding as in the derivations in Problem 3.1, we will have the following results:

Theorem 3.3. For α > −1, γ > −1, δ>0, a > 0

$\begin{array}{l}{G}_{1,2}^{2,1}\left[au{|}_{\alpha ,\delta -1}^{-1-\gamma }\right]=\Gamma (\gamma +1+\delta )a^{\alpha }{u}^{\delta -1}\\ {\displaystyle {\int }_0^{\infty }}{v}^{\alpha +\gamma +1}{(u+v)}^{-(\gamma +1+\delta )}{\text{e}}^{-av}\text{d}v,0\le u<\infty \\ =\Gamma (\gamma +1+\delta )a^{\alpha }{u}^{\alpha }\\ {\displaystyle {\int }_0^{\infty }}{v}^{\alpha +\gamma +1}{(1+v)}^{-(\gamma +1+\delta )}{\text{e}}^{-auv}\text{d}v,0\le u<\infty \\ =\Gamma (\delta -1-\alpha )\Gamma (2+\gamma +\alpha ){(au)}^{\alpha }\\ {}_1{{\displaystyle F}}_1(2+\gamma +\alpha ;2+\alpha -\delta ;au)\\ +\Gamma (1+\gamma +\delta )\Gamma (1+\alpha -\delta ){(au)}^{\delta -1}\\ {}_1{{\displaystyle F}}_1(1+\gamma +\delta ;\delta -\alpha ;au),0\le u<\infty \\ for\delta -1-\alpha \ne \pm \nu ,\nu =0,1,...\end{array}$

Generalizations are possible by replacing 1+x₁ in f₁ by $1+b_1{x}_1^{{\delta }_1},b_1>0,{\delta }_1>0$ and by replacing the exponent in f₂ by $a{x}_2^{{\delta }_2},{\delta }_2>0$. This will produce a H-function and several results on this H-function. In this case, δ_j < 0, j = 1, 2 will also work, resulting in parallel results.

Problem 3.4. Gamma vs. type-1 beta

Let

$f_1(x_1)=\frac{a^{\gamma +1}}{\Gamma (\gamma +1)}{x}_1^{\gamma }{\text{e}}^{-a{x}_1},f_2(x_2)=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}{x}_2^{\alpha }{(1-x)}^{\beta -1}$

for a > 0, β > 0, γ > −1, α > −1, x₁≥0, 0 ≤ x₂ ≤ 1, and f₁ and f₂ are zero otherwise. Let the joint function be f₁(x₁)f₂(x₂) and let $u=\frac{x_2}{x_1}$. Then, proceeding as in the derivations for Problem 3.1, we have the following results:

Theorem 3.4. For a > 0, β > 0, α > −1, γ > −1

$\begin{array}{l}{G}_{2,1}^{1,1}\left[\frac{u}{a}{|}_{\alpha }^{-1-\gamma ,\alpha +\beta }\right]=\frac{a^{\gamma +2}}{\Gamma (\beta )}{u}^{-\gamma -2}\\ {\displaystyle {\int }_0^1}{v}^{\alpha +\gamma +1}{(1-v)}^{\beta -1}{\text{e}}^{-a\frac{v}{u}}\text{d}v,0<u<\infty \\ =\frac{a^{\gamma +2}}{\Gamma (\beta )}{u}^{\alpha }\\ {\displaystyle {\int }_0^{\frac{1}{u}}}{v}^{\alpha +\gamma +1}{(1-uv)}^{\beta -1}{\text{e}}^{-av}\text{d}v,0\le u<\infty \\ =\frac{\Gamma (2+\alpha +\gamma )}{\Gamma (2+\alpha +\gamma +\beta )}{(\frac{u}{a})}^{-\gamma -2}\\ {}_1{{\displaystyle F}}_1(2+\alpha +\gamma ;2+\alpha +\gamma +\beta ;-\frac{a}{u}),\end{array}$

for 0 ≤ u < ∞. Generalization can be achieved by replacing the exponent ax₁ in f₁ by $a{x}_1^{{\delta }_1},{\delta }_1>0$ and replacing 1−x₂ in f₂ by $1-a_2{x}_2^{{\delta }_2},a_2>0,{\delta }_2>0,1-a_2{x}_2^{{\delta }_2}>0$. Then, one will end up with a H-function and several results on this H-function.

Problem 3.5. Type-1 beta vs. gamma

Let

$f_1(x_1)=\frac{\Gamma (\alpha +1+\beta )}{\Gamma (\alpha +1)\Gamma (\beta )}{x}_1^{\alpha }{(1-x_1)}^{\beta -1},f_2(x_2)=\frac{a^{\gamma +1}}{\Gamma (\gamma +1)}{x}_2^{\gamma }{\text{e}}^{-a{x}_2}$

for α > −1, γ > −1, a > 0, β > 0, x₂≥0, 0 ≤ x₁ ≤ 1, and f₁ and f₂ are zero otherwise. Let the joint function be f₁(x₁)f₂(x₂) and let $u=\frac{x_2}{x_1}$. Then, proceeding as in the derivations in Problem 3.1, we have the following results:

Theorem 3.5. For α > −1, γ > −1, a > 0, β > 0

$\begin{array}{l}{G}_{1,2}^{1,1}\left[au{|}_{\gamma ,-1-\alpha -\beta }^{-1-\alpha }\right]=\frac{a^{\gamma }}{\Gamma (\beta )}{u}^{-\beta -\alpha -1}\\ {\displaystyle {\int }_{v=0}^u}{v}^{\alpha +\gamma +1}{(u-v)}^{\beta -1}{\text{e}}^{-av}\text{d}v,0<u<\infty \\ =\frac{a^{\gamma }}{\Gamma (\beta )}{u}^{\gamma }{\displaystyle {\int }_0^1}{v}^{\alpha +\gamma +1}{(1-v)}^{\beta -1}{\text{e}}^{-auv}\text{d}v,\\ 0<u<\infty \\ =a^{\gamma }\frac{\Gamma (2+\alpha +\gamma )}{\Gamma (2+\alpha +\gamma +\beta )}{u}^{\gamma }\\ {}_1{{\displaystyle F}}_1(2+\alpha +\gamma ;2+\alpha +\gamma +\beta ;-au),0\le u<\infty .\end{array}$

A direct generalization is achieved by replacing 1−x₁ in f₁ by $1-a_1{x}_1^{{\delta }_1},a_1>0,{\delta }_1>0,1-a_1{x}_1^{{\delta }_1}>0$ and by replacing the exponent ax₂ in f₂ by $a{x}_2^{{\delta }_2},{\delta }_2>0$. Then, one will end up with a H-function, and several results will be available on this H-function. It may be observed that if in f₁, α is replaced by α−1, and if f₂ is a general function, then the first integral representation in Theorem 3.5 is also Erdélyi-Kober fractional integral of the first kind of order β and parameter α, see also Mathai and Haubold [7].

Problem 3.6. Type-1 beta vs. type-1 beta

Let

$\begin{array}{l}{f}_j(x_j)=\frac{\Gamma ({\alpha }_j+1+{\beta }_j)}{\Gamma ({\alpha }_j+1)\Gamma ({\beta }_j)}{x}_j^{{\alpha }_j}{(1-x_j)}^{{\beta }_j-1},\\ 0\le x_j\le 1,{\alpha }_j>-1,{\beta }_j>0,j=1,2.\end{array}$

Let the joint function be f₁(x₁)f₂(x₂) and let $u=\frac{x_2}{x_1}$. Then, proceeding as in the derivation of Problem 3.1, we have the following results:

Theorem 3.6. For α_j > −1, β_j > 0, j = 1, 2

$\begin{array}{l}G=G_{2,2}^{1,1}\left[u{|}_{{\alpha }_2,-1-{\alpha }_1-{\beta }_1}^{-1-{\alpha }_1,{\alpha }_2+{\beta }_2}\right]\\ =\{\begin{array}{l}\frac{1}{\Gamma ({\beta }_1)\Gamma ({\beta }_2)}{u}^{{\alpha }_2}{\displaystyle {\int }_0^{\frac{1}{u}}{v}^{{\alpha }_1+{\alpha }_2+1}}{(1-v)}^{{\beta }_1-1}{(1-uv)}^{{\beta }_2-1}\text{d}v,\hfill \\ 0\le u<1\hfill \\ \frac{1}{\Gamma ({\beta }_1)\Gamma ({\beta }_2)}{u}^{-{\alpha }_1-{\beta }_1-1}{\displaystyle {\int }_0^1{v}^{{\alpha }_1+{\alpha }_2+1}}{(u-v)}^{{\beta }_1-1}{(1-v)}^{{\beta }_2-1}\text{d}v,\hfill \\ 1\le u<\infty \hfill \end{array}\\ G=\{\begin{array}{l}\frac{\Gamma (2+{\alpha }_1+{\alpha }_2)}{\Gamma (2+{\alpha }_1+{\alpha }_2+{\beta }_1)\Gamma ({\beta }_2)}{u}^{{\alpha }_2}{}_2{{\displaystyle F}}_1(1-{\beta }_2,2+{\alpha }_1+{\alpha }_2;\hfill \\ 2+{\alpha }_1+{\alpha }_2+{\beta }_1;u),0\le u<1\hfill \\ \frac{\Gamma (2+{\alpha }_1+{\alpha }_2)}{\Gamma (2+{\alpha }_1+{\alpha }_2+{\beta }_2)\Gamma ({\beta }_1)}{u}^{-2-{\alpha }_1}{}_2{{\displaystyle F}}_1(1-{\beta }_1,2+{\alpha }_1+{\alpha }_2;\hfill \\ 2+{\alpha }_1+{\alpha }_2+{\beta }_2;\frac{1}{u}),\hfill \\ \text{for }1\le u<\infty .\hfill \end{array}\end{array}$

A direct generalization will be available by replacing 1−x_j by $1-a_j{x}_j^{{\delta }_j},a_j>0,{\delta }_j>0,1-a_j{x}_j^{{\delta }_j}>0,j=1,2.$ Then, we will end up with a H-function, and several results will be available on this H-function. One general case of the ratio $u=\frac{x_2}{x_1}$ will be discussed here for the sake of illustration. This will be listed here as Problem 3.7.

Problem 3.7. Generalized type-2 beta vs. generalized gamma

Let

$\begin{array}{l}{f}_1(x_1)=\delta \frac{a^{\frac{\alpha +1}{\delta }}\Gamma (\frac{\alpha +1}{\delta }+\beta )}{\Gamma (\frac{\alpha +1}{\delta })\Gamma (\beta )}{x}_1^{\alpha }{(1+a{x}_1^{\delta })}^{-(\frac{\alpha +1}{\delta }+\beta )},\\ f_2(x_2)=\frac{\rho b^{\frac{\gamma }{\rho }}}{\Gamma (\frac{\gamma +1}{\rho })}{x}_2^{\rho }{\text{e}}^{-b{x}_2^{\rho }}\end{array}$

for α > −1, γ > −1, δ>0, ρ>0, a > 0, b > 0, β > 0, and f₁ and f₂ are zero elsewhere. Let $u=\frac{x_2}{x_1}$ with the associated function g₂(u). The Mellin transforms of f₁ and f₂ are the following:

$M_{f_2}(s)=\frac{b^{\frac{1}{\rho }}}{\Gamma (\frac{\gamma +1}{\rho })}\Gamma (\frac{\gamma +s}{\rho }){(b^{\frac{1}{\rho }})}^{-s},\Re (\frac{\gamma +s}{\rho })>0$

and

$M_{f_1}(2-s)=\frac{1}{a^{\frac{1}{\delta }}\Gamma (\frac{\alpha +1}{\delta })\Gamma (\beta )}\Gamma (\frac{2+\alpha }{\delta }-\frac{s}{\delta })\Gamma (\beta -\frac{1}{\delta }+\frac{s}{\delta }){(a^{\frac{1}{\delta }})}^s$

for ℜ(2+α−s)>0, ℜ(βδ−1+s)>0. Then

$\begin{array}{l}{M}_{f_1}(2-s)M_{f_2}(s)=\frac{b^{\frac{1}{\rho }}}{a^{\frac{1}{\delta }}\Gamma (\frac{\alpha +1}{\delta })\Gamma (\frac{\gamma +1}{\rho })\Gamma (\beta )}\\ \times \Gamma (\frac{\gamma }{\rho }+\frac{s}{\rho })\Gamma (\beta -\frac{1}{\delta }+\frac{s}{\delta })\Gamma (\frac{2+\alpha }{\delta }-\frac{s}{\delta }){(\frac{b^{\frac{1}{\rho }}}{a^{\frac{1}{\delta }}})}^{-s}.\end{array}$

Let

$c=\frac{\delta \rho a^{\frac{\alpha +1}{\delta }}{b}^{\frac{\gamma +1}{\rho }}\Gamma (\frac{\alpha +1}{\delta }+\beta )}{\Gamma (\frac{\alpha +1}{\delta })\Gamma (\beta )\Gamma (\frac{\gamma +1}{\rho })},c_1=\frac{b^{\frac{1}{\rho }}}{a^{\frac{1}{\delta }}\Gamma (\frac{\alpha +1}{\delta })\Gamma (\beta )\Gamma (\frac{\gamma +1}{\rho })}.$

Then, from the inverse Mellin transform we have

$\begin{array}{l}{g}_2(u)=\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}{M}_{f_1}(2-s)M_{f_2}(s)u^{-s}\text{d}s\\ =c_1\frac{1}{2\pi i}{\displaystyle {\int }_{c-i\infty }^{c+i\infty }}\Gamma (\frac{\gamma }{\rho }+\frac{s}{\rho })\Gamma (\beta -\frac{1}{\delta }+\frac{s}{\delta })\\ \Gamma (\frac{2+\alpha }{\delta }-\frac{s}{\delta }){\left[\frac{b^{\frac{1}{\rho }}u}{a^{\frac{1}{\delta }}}\right]}^{-s}\text{d}s\\ =c_1{H}_{1,2}^{2,1}\left[\frac{b^{\frac{1}{\rho }}u}{a^{\frac{1}{\delta }}}{|}_{(\frac{\gamma }{\rho },\frac{1}{\rho }),(\beta -\frac{1}{\delta },\frac{1}{\delta })}^{(1-\frac{2+\alpha }{\delta },\frac{1}{\delta })}\right],0\le u<\infty .\end{array}$

From the integral representations of g₂(u), we have the following forms and for δ = ρ and for γ+ρν≠βδ+δλ, v = 0, 1, ..., λ = 0, 1, ... the poles of the integrand are simple, and then, we have the following series representations for the H-function for δ = ρ. We will state this as a theorem.

Theorem 3.7. For the conditions, c, c₁ stated above, we have

$\begin{array}{l}{g}_2(u)=c{u}^{-\alpha -2}{\displaystyle {\int }_0^{\infty }}{v}^{\alpha +\gamma +1}{\left[1+a{(\frac{v}{u})}^{\delta }\right]}^{-(\frac{\alpha +1}{\delta }+\beta )}{\text{e}}^{-b{v}^{\rho }}\text{d}v\\ =c{u}^{-\gamma }{\displaystyle {\int }_0^{\infty }}{v}^{\alpha +\gamma +1}{\left[1+a{v}^{\delta }\right]}^{-(\frac{\alpha +1}{\delta }+\beta )}{\text{e}}^{-b{(uv)}^{\rho }}\text{d}v\\ =c_1\text{H-function above}\end{array}$

where for ρ = δ,

$\begin{array}{l}H=H_{1,2}^{2,1}\left[{(\frac{b}{a})}^{\frac{1}{\delta }}u{|}_{(\frac{\gamma }{\delta },\frac{1}{\delta }),(\beta -\frac{1}{\delta },\frac{1}{\delta })}^{(1-\frac{2+\alpha }{\delta },\frac{1}{\delta })}\right]\\ =\delta \{\begin{array}{l}\Gamma (\beta -\frac{\gamma +1}{\delta })\Gamma (\frac{2+\alpha +\gamma }{\delta }){[{(\frac{b}{a})}^{\frac{1}{\delta }}u]}^{\gamma }{}_1{{\displaystyle F}}_1(\frac{2+\alpha +\gamma }{\delta };\hfill \\ 1+\frac{\gamma +1}{\delta }-\beta ;\frac{b{u}^{\delta }}{a})\hfill \\ +\Gamma (\frac{\gamma +1}{\delta }-\beta )\Gamma (\frac{\alpha +1}{\delta }+\beta ){\left[{(\frac{b}{a})}^{\frac{1}{\delta }}u\right]}^{\beta \delta -1}{}_1{{\displaystyle F}}_1(\frac{\alpha +1}{\delta }+\beta ;\hfill \\ 1+\beta -\frac{\alpha +1}{\delta };\frac{b}{a}{u}^{\delta }),\hfill \\ \text{for }0\le u<\infty ,\gamma -\delta \beta +1\ne \pm \nu ,\nu =0,1,\mathrm{...}\hfill \end{array}\end{array}$

4 Generalization to matrix-variate cases

Here, all the matrices appearing are p×p real symmetric positive definite when in the real domain or p×p Hermitian positive definite when in the complex domain. Let X₁>O and X₂>O be p×p real symmetric positive definite matrices with the associated functions f₁(X₁) and f₂(X₂), respectively, and with the joint function f₁(X₁)f₂(X₂), capital letters representing matrices and lower case letters denoting scalar variables where f_j(X_j) is a real-valued scalar function of X_j, j = 1, 2. When f_j(X_j)≥0 for all X_j, in the domain of X_j, with $\underset{X_j}{\int }{f}_j(X_j)\text{d}{X}_j=1$, then f_j(X_j) is a statistical density also. Since we are dealing with symmetric matrices, we will be defining symmetric product corresponding to the scalar case x₁x₂ as $X_2^{\frac{1}{2}}{X}_1{X}_2^{\frac{1}{2}}$ where $X_2^{\frac{1}{2}}$ is the symmetric positive definite square root of X₂, and symmetric ratio corresponding to $\frac{x_2}{x_1}$ as $X_2^{\frac{1}{2}}{X}_1^{-1}{X}_2^{\frac{1}{2}}$ where $X_1^{-1}$ is the regular inverse of X₁. We can also define symmetric product corresponding to x₁x₂ as $X_1^{\frac{1}{2}}{X}_2{X}_1^{\frac{1}{2}}$ and symmetric ratio corresponding to $\frac{x_2}{x_1}$ as $X_1^{-\frac{1}{2}}{X}_2{X}_1^{-\frac{1}{2}}$ also.

4.1 Symmetric product in the real matrix-variate case

Let $U=X_2^{\frac{1}{2}}{X}_1{X}_2^{\frac{1}{2}}$ with V = X₂ so that $X_1=V^{-\frac{1}{2}}U{V}^{-\frac{1}{2}}$. We can show that

$\begin{array}{ll}\text{d}{X}_1\wedge \text{d}{X}_2=|V{|}^{-\frac{p+1}{2}}\text{d}U\wedge \text{d}V& (4.1)\end{array}$

see, for example, Mathai [11], where |(·)| denotes the determinant of (·). Let X_j = (x_jik) with the (i, k)-th element x_jik for i = 1, ..., p, k = 1, ..., p. Since $X_j=X_j^{\prime }$, a prime denoting the transpose, there are only p(p+1)/2 distinct elements in X_j and only p(p+1)/2 differentials dx_jik, i ≤ k or i≥k. Then, dX_j = ∧_{i ≤ k}dx_jik. Note that the wedge product in Equation 4.1 remains the same if we take V = X₁ instead of X₂. For illustrative purposes, we will discuss one problem involving M-convolution of a product. For M-convolutions and M-transforms, see Mathai [11].

Problem 4.1. Real p×p matrix-variate gamma vs. real p×p matrix-variate gamma

Let

$f_j(X_j)=\frac{|B_j{|}^{{\alpha }_j}}{{\Gamma }_p({\alpha }_j)}|X_j{|}^{{\alpha }_j-\frac{p+1}{2}}{\text{e}}^{-\text{tr}(B_j{X}_j)},X_j>O,B_j>O,j=1,2$

where $X_j=X_j^{\prime }>O$ is a real matrix of a p(p+1)/2 distinct (functionally independent) real scalar variables x_jik as elements, $B_j=B_j^{\prime }>O$ is a p×p constant positive definite matrix, tr(·) means the trace of (·), and Γ_p(α_j) is a real matrix-variate gamma defined as

$\begin{array}{ll}{\Gamma }_p(\alpha )={\pi }^{\frac{p(p-1)}{4}}\Gamma (\alpha )\Gamma (\alpha -\frac{1}{2})...\Gamma (\alpha -\frac{p-1}{2}),\Re (\alpha )>\frac{p-1}{2}& (4.2)\end{array}$

$\begin{array}{ll}={\displaystyle {\int }_{Y>O}}|Y{|}^{\alpha -\frac{p+1}{2}}{\text{e}}^{-\text{tr}(Y)}\text{d}Y,Y>O,\Re (\alpha )>\frac{p-1}{2}& (4.3)\end{array}$

and thus, the real matrix-variate gamma Γ_p(α) is associated with the real matrix-variate gamma integral in Equation 4.3. Hence, Γ_p(α) is called as the real matrix-variate gamma. Let g₂(U) be the real-valued scalar function associated with the matrix $U=X_2^{\frac{1}{2}}{X}_1{X}_2^{\frac{1}{2}}$. Then, from the joint function f₁(X₁)f₂(X₂), we have for $c={\prod }_{j=1}^2\frac{|B_j{|}^{{\alpha }_j}}{{\Gamma }_p({\alpha }_j)}$,

$\begin{array}{l}{g}_2(U)=c{\displaystyle {\int }_{V>O}}\left|V{|}^{-\frac{p+1}{2}}\right|V^{-\frac{1}{2}}U{V}^{-\frac{1}{2}}{|}^{{\alpha }_1-\frac{p+1}{2}}|V{|}^{{\alpha }_2-\frac{p+1}{2}}\\ {\text{e}}^{-\text{tr}(B_1{V}^{-\frac{1}{2}}U{V}^{-\frac{1}{2}})-\text{tr}(B_{2}V)}\text{d}V\\ =c\left|U{|}^{{\alpha }_1-\frac{p+1}{2}}{\displaystyle {\int }_{V>O}}\right|V{|}^{{\alpha }_2-{\alpha }_1-\frac{p+1}{2}}{\text{e}}^{-\text{tr}(B_1{V}^{-\frac{1}{2}}U{V}^{-\frac{1}{2}})-\text{tr}(B_{2}V)}\text{d}V& (4.4)\end{array}$

$\begin{array}{ll}=c\left|U{|}^{{\alpha }_2-\frac{p+1}{2}}{\displaystyle {\int }_{V>O}}\right|V{|}^{{\alpha }_1-{\alpha }_2-\frac{p+1}{2}}{\text{e}}^{-\text{tr}(B_{1}V)-\text{tr}(B_2{V}^{-\frac{1}{2}}U{V}^{-\frac{1}{2}})}\text{d}V.& (4.5)\end{array}$

Note that Equations 4.4, 4.5 can be taken as real matrix-variate generalization of Bessel integrals, Krätzel integral, Bayesian structures, etc. The integrand, normalized can also be taken as a real matrix-variate inverse Gaussian density when f_j(X_j), j = 1, 2 are densities. In this case, g₂(U) is the unique density of the symmetric product U also. All the pairs of functions that we considered in Section 2 can be generalized to the corresponding matrix-variate cases except Problem 2.7. In Problem 2.7, generalization to the matrix-variate case is possible for some specific values of δ₁ and δ₂.

5 Concluding remarks

In Sections 2 and 3, elementary functions belonging to the pathway family are considered so that the results in Sections 2 and 3 will be readily applicable to various practical problems. Immediate generalizations of all pairs in Sections 2 and 3 are also useful in situations in real scalar variable cases. All problems in Sections 2 and 3, except Problems 2.7 and 3.7, can be generalized to the corresponding matrix-variate cases in the real and complex domains. In all those cases, when f_j(X_j), j = 1, 2 are statistical densities, then g₁(U) and g₂(U) will be unique densities corresponding to symmetric product and symmetric, ratios respectively. There are not many results in the literature on Mellin convolutions of products and ratios involving more than two real scalar variables. Let u = x₁x₂...x_k with the joint function f₁(x₁)...f_k(x_k). Then, the integral representations and series representations of g₁(u) can be obtained in many different ways which will produce various results. When it comes to the ratio, then ratios can be defined in many different ways when more than two real scalar variables are involved. Then, the corresponding g₂(u) can take different forms and shapes.

“Convolution” in mathematical literature often means Laplace convolution. Mellin convolutions of products and ratios used to appear very rarely in mathematical literature. In the area of special functions, when the functions are defined through Mellin-Barnes representations, Mellin transforms and Mellin convolutions appear automatically. The class of hypergeometric functions appears in very many practical problems as models and as series solutions of differential equations. Hypergeometric function _pF_q(·) has a Mellin-Barnes representation. Mittag-Leffler function is considered as the queen function in the area of fractional calculus. These functions also have Mellin-Barnes representations. Wright's function is another prominent function in the literature having Mellin-Barnes representation. Generalized scalar variable special functions such as G and H-functions have Mellin-Barnes representations. In all these cases, Mellin convolutions of products and ratios appear. Products and ratios of random variables appear in many real-life situations. In industrial production processes, the money value of the output has the structure of the product of two real scalar positive random variables, namely, quantity of output and price per unit. In a series of papers, starting from 2009, Mathai has shown that all types fractional integrals introduced by various authors from time to time are nothing but Mellin convolutions of products and ratios of real scalar variables. In the overview paper Mathai [12], it is shown that the whole area of fractional calculus, including fractional integrals and fractional derivatives etc., can be studied by using Mellin convolutions of products and ratios. It is shown that the approach through Mellin convolutions enable us to go to the Mellin transforms thereby to the corresponding inverse Mellin transforms which provide explicit evaluations of fractional integrals. It is also shown that when the arbitrary functions in fractional integrals belong to some general classes, then the inverse Mellin transform will end up in G and H-functions. Then, one could make use of the G-function differential equation to construct differential equations for fractional integrals. It is also shown in Mathai [12] that the approach through statistical distribution theory or Mellin convolutions enables us to extend fractional integrals to functions of matrix arguments and to the complex domains.

Explicit computable series representations for various categories of G-functions, including the general G-function, are given in Mathai [5] book. The same techniques work in obtaining explicit computable series forms for the H-function also. With the help of these computable representations, computer packages are also produced. Programs are available in MAPLE and MATHEMATICA for analytical and numerical computations of G and H-functions Hence, numerical computations and graphs are not attempted in the present study.

In the case of Mellin convolutions of products involving two functions, it is shown that when these two functions belong to generalized gamma densities, then one can obtain Krätzel integrals and Krätzel transforms, reaction-rate probability integrals in nuclear reaction-rate theory, and inverse Gaussian density in stochastic processes. The large contributions of Mathai and Haubold in astrophysics area started with the evaluation of a reaction-rate probability integral by using the density of product of real scalar positive random variables, which is nothing but the Mellin convolution of a product involving generalized gamma densities. A summary of the study in this area of astrophysics is available in the monograph Mathai and Haubold [13], and an updated version of this book is scheduled to appear in early 2025 from Springer.

In the matrix-variate case, Mathai [11] defined M-convolutions and M-transforms, corresponding to Mellin convolutions and Mellin transforms in the real scalar case. Jacobians of matrix transformations and functions of matrix arguments in the real and complex domains are given there, along with the necessary conditions for the existence of various items. Detailed existence conditions for G-functions are given in Mathai [5]. These M-convolutions are shown to be densities of symmetric product and symmetric ratios of matrices when the basic functions involved are matrix-variate statistical densities in the real or complex domain. Distributions of symmetric products and symmetric ratios of real positive definite or Hermitian positive definite matrix-variate random variables appear in a large number of practical situations such as in neutrino problems [8], matrix-variate fractional calculus [12], the analysis of multiple and multi-look data in radar, and sonar [14]. Matrix-variate statistical distributions including singular distributions in the real and complex domains and their applications in various areas are given in Mathai et al. [15]. In this book, detailed existence conditions of the various results in the real and complex domains are also given.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

HH: Conceptualization, Writing – review & editing. AM: Conceptualization, Writing – original draft.

Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

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The author(s) declare that no Gen AI was used in the creation of this manuscript.

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