(2.1) and (2.2)] made use of the extended beta function [B.sub.p]([alpha], [beta]) in (1) to extend the

Gauss hypergeometric function and confluent hypergeometric function as follows:

PEREZ SINUSIA, A simplification of Laplace's method: applications to the gamma function and the

Gauss hypergeometric function, J.

Two special cases of (10) are the confluent hypergeometric function and the

Gauss hypergeometric function denoted by [PHI] and F, respectively.

It has been shown by van der Merwe and Roux [2] that the above density can be obtained as a limiting case of a density involving the

Gauss hypergeometric function of matrix argument.

Let [alpha] > 0, [beta], [eta] [member of] M then the Saigo fractional integral [I.sup.[alpha],[beta],[eta].sub.0,t] (in terms of the

Gauss hypergeometric function) of order a for a real-valued continuous function f (t) is defined by ([12], see also [13, p.

In terms of the

Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

Applying the transformation (see [6, [section]2.8]) in terms of the

Gauss hypergeometric functionTo do this, we consider the

Gauss hypergeometric function F (a, b; c; z), which is defined by (see [9])

The results such as the cumulative distribution function, the mean and the variance of the beta type 3 random variable involve

Gauss hypergeometric function which is easily computable using the software Mathematica.

Note that (42) applies also for N = 0, that is, for [epsilon] = 0, for which the Heun function degenerates to the

Gauss hypergeometric function u = [sub.2][F.sub.1]([alpha], [beta]; [gamma]; z) provided q = 0.

Based on the beta function, the

Gauss hypergeometric function, denoted by F(a, b; c; z), and the confluent hypergeometric function, denoted by [PHI](b; c; z), for Re(c) > Re(b) > 0, are defined as (see Luke [1]),

where F(a,b;c;x) denotes the

Gauss hypergeometric function, and the parameters involved are complex numbers.