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 At z = 1, the Gauss hypergeometric function reduces to 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. 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 which has the same branch cut as the generalized hypergeometric function (B.12). The numerical value at z = 1 is given by 111 Li$_{2}$ (1) = π 2 6 . (B.19) Li$_{2}$ ( z ) = − ∫ z 0 ds log (1 − s ) s , (B.18) $_{2}$F$_{1}$ [ a , b ; c ; z ] = Γ ( c ) Γ ( a ) Γ ( c − b ) ∫ 1 0 dt t b − $^{1}$(1 − t ) c − b − $^{1}$(1 − tz ) − a , Re ( c ) > Re ( b ) > 0 . (B.16) B.3 Polylogarithms $_{2}$F$_{1}$ [ a , b ; c ; 1] = Γ ( c ) Γ ( c − a − b ) Γ ( c − a ) Γ ( c − b ) , Re ( c − a − b ) > 0 . (B.17)