The Meijer G and Fox H Functions

Abstract

In this chapter besides briefly introducing the Meijer G and Fox H functions through their usual definitions, the authors open the way to the establishment of several cases where there are finite representations for these functions which have not been previously identified and that as such are not recognized by any of the available software. These cases are related to the distribution of three extended products of independent Beta random variables, which are treated in Chap. 3.

Appendix: Notation and Expressions for the Probability Density and Cumulative Distribution Functions of the Gamma, GIG, and EGIG Distributions

This appendix is used to establish the notation used concerning the p.d.f. and c.d.f. of the GIG (Generalized Integer Gamma) and the EGIG (Exponentiated Generalized Integer Gamma) distributions.

We say that the r.v. (random variable) X has a Gamma distribution with shape parameter r (> 0) and rate parameter λ (> 0), and we will denote this fact by X ∼ Γ(r, λ), if the p.d.f. of X is

$$\displaystyle \begin{aligned} f^{}_X(x)=\frac{\lambda^r}{\varGamma(r)}\,e^{-\lambda x}\,x^{r-1}~~~~(x>0)\,. \end{aligned}$$

Let X _j ∼ Γ(r _j, λ _j) (j = 1, …, p) be a set of p independent r.v.’s and consider the r.v.

$$\displaystyle \begin{aligned} W=\sum^p_{j=1}X_j\,. \end{aligned}$$

In case all the $r_j\in \mathbb {N}$, the distribution of W is what we call a GIG distribution (Coelho, 1998, 1999). If all the λ _j are different, W has a GIG distribution of depth p, with shape parameters r _j and rate parameters λ _j, with p.d.f.

$$\displaystyle \begin{aligned} f^{}_W(w)=f^{\,\,GIG}\Big(w\,\big|\,\{r_j\}_{j=1:p};\{\lambda_j\}_{j=1:p};p\Big)=K\sum^p_{j=1} P_j(w)\,e^{-\lambda_j w}\,,~~~~(w>0) {} \end{aligned} $$

(2.18)

and c.d.f.

$$\displaystyle \begin{aligned} F^{}_W(w)=F^{GIG}\Big(w\,\big|\,\{r_j\}_{j=1:p};\{\lambda_j\}_{j=1:p};p\Big)=1-K\sum^p_{j=1}P^*_j(w)\,e^{-\lambda_jw}\,,~~~~(w>0) {} \end{aligned} $$

(2.19)

where

$$\displaystyle \begin{aligned} K=\prod^p_{j=1}\lambda_j^{r_j}\,,~~~~~~P_j(w)=\sum^{r_j}_{k=1} c_{j,k}\,w^{k-1} \end{aligned}$$

and

$$\displaystyle \begin{aligned} P^*_j(w)=\sum^{r_j}_{k=1}c_{j,k}(k-1)!\sum^{k-1}_{i=0}\frac{w^i}{i!\,\lambda_j^{k-i}}\,, \end{aligned}$$

with

(2.20)

and

$$\displaystyle \begin{aligned} c_{j,r_j-k}=\frac{1}{k}\sum_{i=1}^k\frac{(r_j-k+i-1)!}{(r_j-k-1)!}\,R(i,j,p)c_{j,r_j-(k-i)}\,,~~~~(k=1,\dots,r_j-1;\,j=1,\dots,p) {} \end{aligned} $$

(2.21)

where

(2.22)

In case some of the λ _j assume the same value as other λ _j’s, the distribution of W still is a GIG distribution, but in this case with a reduced depth. In this more general case, let {λ _ℓ;ℓ = 1, …, g(≤ p)} be the set of different λ _j’s and let {r _ℓ;ℓ = 1, …, g(≤ p)} be the set of the corresponding shape parameters, with r _ℓ being the sum of all r _j (j ∈{1, …, p}) which correspond to the λ _j assuming the value λ _ℓ. In this case W will have a GIG distribution of depth g, with shape parameters r _ℓ and rate parameters λ _ℓ (ℓ = 1, …, g).

Let us consider the r.v. Z = e ^−W. Then the r.v. Z has what Arnold et al. (2013) call an Exponentiated Generalized Integer Gamma (EGIG) distribution of depth p, with p.d.f.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \displaystyle f^{}_Z(z) & = & \displaystyle f^{\,EGIG}\Big(z~\big|\,\{r_j\}_{j=1:p};\{\lambda_j\}_{j=1:p};p\Big)\\ & = & \displaystyle f^{\,\,GIG}\Big(-\log\,z~\big|\,\{r_j\}_{j=1:p};\{\lambda_j\}_{j=1:p};p\Big)\,\frac{1}{z}~~~~~~~~~~(0<z<1) \end{array} {} \end{aligned} $$

(2.23)

and c.d.f.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \displaystyle F^{}_Z(z) & = & \displaystyle F^{EGIG}\Big(z~\big|\,\{r_j\}_{j=1:p};\{\lambda_j\}_{j=1:p};p\Big)\\ & = & \displaystyle 1-F^{GIG}\Big(-\log\,z~\big|\,\{r_j\}_{j=1:p};\{\lambda_j\}_{j=1:p};p\Big)~~~~~~~~~~(0<z<1)\,. \end{array} {} \end{aligned} $$

(2.24)