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

The Gauss hypergeometric functions ₂F₁ 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 hypergeometric 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 ofq–loop off-shell propagator diagrams with one massive line andq massless lines and q–loop bubble with two-massive lines andq−1 massless lines are considered.