problems and historical background

Legendre’s Duplication Formula
In view of the definition (2) it is not difficult to show that
(13)
which follows also from Legendre’s duplication formula for the Gamma function, viz [Erdelyi et al. (1953, p. 5, Equation (15))]
(14)
Gauss’s Multiplication Theorem
For every positive integer m, we have
(15)
which reduces to (13) when m = 2. Starting from (15) with , it can be proved that [Erdelyi et al. (1953, p. 4, Equation (11))]
(16)

which is known in the literature as Gauss’s multiplication theorem for the Gamma function.

The Gaussian Hypergeometric Series

In terms of the Pochhammer symbol defined by (2), we can rewrite the definition 1.1(1) in the form:

(17)
The infinite series in (17) obviously reduces to the elementary geometric series
(18)
in its special cases when
(19) (i) a = c and b=1; (ii) a = 1 and b = c.
Hence it is called the hypergeometric series or, more precisely, Gauss’s hypergeometric series after the famous German mathematician Carl Friedrich Gauss (1777-1855) who in the year 1812 introduced this scries into analysis and gave the F-notation for it.
By d’Alembert’s ratio test, it is easily seen that the hypergeometric series in (17) converges absolutely within the unit circle, that is, when |z| < 1, provided that the denominator parameter c is neither zero nor a negative integer. Notice, however, that if either or both of the • numerator parameters a and b in (17) is zero or a negative integer, the hypergeometric series terminates in view of (12), and the question of convergence does not enter the discussion.
Further tests readily show that the hypergeometric series in (17), when |z| = 1 (that is, on the unit circle), is

1. 
   absolutely convergent if Re(c - a - b)> 0;
   
2. 
   conditionally convergent
   
3. 
   divergent if
   

As a matter of fact, in Case (i) we are led to the well-known Gauss’s summation theorem: