content_image/295164.jpg content_image/295166.jpg content_image/295163.jpg content_image/295162.jpg
known hypergeometric function is the Gauss hypergeometric function, $_{2}$F$_{1}$ [ a , b ; c ; z ], of which the Chebyshev, Gegenbauer, Jacobi and Legendre polynomials are special cases. The Gauss hypergeometric function has the integral representation 

<img src='content_image/295164.jpg'>

At z = 1, the Gauss hypergeometric function reduces to 

<img src='content_image/295166.jpg'>

It is interesting to note that there are recurrence relations relating $_{2}$F$_{1}$ [ a , b ; c ; z ] and the contiguous functions $_{2}$F$_{1}$ [ a ± 1 , b ± 1 ; c ± 1 ; z ] (See Ref. [7]). In Ref. [119] it is pointed out that these are closely related to the generalized recurrence relations discussed in Chapter 2. 

## B.3 Polylogarithms 

In Chapter 2 it was pointed out that loop integrals that include an external momentum and a mass often lead to generalized hypergeometric functions whose indices of are d - dependent. Higher order terms in the epsilon expansion of these often involve Polylogarithm functions [76]. The simplest Polylogarithm is the Dilogarithm, which is defined as 

<img src='content_image/295163.jpg'>

which has the same branch cut as the generalized hypergeometric function (B.12). The numerical value at z = 1 is given by 

<img src='content_image/295162.jpg'>

