All order epsilon-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters

It is proved that the Laurent expansion of the following Gauss hypergeometric functions, are an arbitrary integer nonnegative numbers, a, b, c are an arbitrary numbers and ε is an arbitrary small parameters, are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with polynomial coefficients. An efficient algorithm for the calculation of the higher-order coefficients of Laurent expansion is constructed. Some particular cases of Gauss hypergeometric functions are also discussed. Contents 1. Introduction 1 2. All-order ε-expansion 3 2.1 Non-zero values of the ε-dependent part 3 2.1.1 Integer values of of ε-independent parameters 4 2.1.2 Half-integer values of of ε-independent parameters 8 2.2 Zero-values of the ε-dependent part of upper parameters 10 3. Some particular cases 11 3.1 The generalized log-sine functions and their generalization 11 3.2 Special cases: all-order ε-expansion in terms of Nielsen polylogarithms 13 4. Conclusions 14 A. The Laurent expansion of Gauss hypergeometric functions with half-integer values of parameters around z = 1 15