Gauss hypergeometric function : reduction , ε - expansion for integer / half - integer parameters and Feynman diagrams

The Gauss hypergeometric functions 2 F 1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hyperge-ometric functions. The ε-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order ε-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and ε−expansion of q–loop off-shell propagator diagrams with one massive line and q massless lines and q–loop bubble with two-massive lines and q − 1 massless lines are considered.