problems and historical background

1.2 GAUSSIAN HYPERGEOMETRIC SERIES AND ITS GENERALIZATSIONS
With a view to introducing formally the Gaussian hypergeometric series and its generalizations, we recall here some definitions and identities involving Pochhammer’s symbol , and the Gamma function defined by
Introduction and Definitions

1. 
   
   

Pochhammer’s Symbol and the Factorial Function
Throughout this work we shall find it convenient to employ the Pochhammer symbol defined by

2. 
   
   

Since (1)„ = n!, may be looked upon as a generalization of the elementary factorial; hence the symbol is also referred to as the factorial function.
In terms of Gamma functions, we have

(3)

which can easily be verified. Furthermore, the binomial coefficient may now be expressed as
(4)
or, equivalently, as
(5)
It follows from (4) and (5) that
(6)
which, for , yields
(7)
Sec. 1.2]
Gaussian Hypergeometric Series and Its Generalizations
Equations (3) and (7) suggest the definition:
(8)
Equation (3) also yields
(9)
which, in conjunction with (8), gives
(10)
For , we have
(11)
which may alternatively be written in the form:
(12)