Towards all-order Laurent expansion of generalised hypergeometric functions about rational values of parameters

Abstract

We prove the following theorems: (1) the Laurent expansions in ε of the Gauss hypergeometric functions F 1 2 ( I 1 + a ε , I 2 + b ε ; I 3 + p q + c ε ; z ) , F 1 2 ( I 1 + p q + a ε , I 2 + p q + b ε ; I 3 + p q + c ε ; z ) and F 1 2 ( I 1 + p q + a ε , I 2 + b ε ; I 3 + p q + c ε ; z ) , where I 1 , I 2 , I 3 , p , q are arbitrary integers, a , b , c are arbitrary numbers and ε is an infinitesimal parameter, are expressible in terms of multiple polylogarithms of q-roots of unity with coefficients that are ratios of polynomials; (2) the Laurent expansion of the Gauss hypergeometric function F 1 2 ( I 1 + p q + a ε , I 2 + b ε ; I 3 + c ε ; z ) is expressible in terms of multiple polylogarithms of q-roots of unity times powers of logarithm with coefficients that are ratios of polynomials; (3) the multiple inverse rational sums ∑ j = 1 ∞ Γ ( j ) Γ ( 1 + j − p q ) z j j c S a 1 ( j − 1 ) × ⋯ × S a p ( j − 1 ) and the multiple rational sums ∑ j = 1 ∞ Γ ( j + p q ) Γ ( 1 + j ) z j j c S a 1 ( j − 1 ) × ⋯ × S a p ( j − 1 ) , where S a ( j ) = ∑ k = 1 j 1 k a is a harmonic series and c is an arbitrary integer, are expressible in terms of multiple polylogarithms; (4) the generalised hypergeometric functions F p − 1 p ( A → + a → ε ; B → + b → ε , p q + B p − 1 ; z ) and F p − 1 p ( A → + a → ε , p q + A p ; B → + b → ε ; z ) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials with complex coefficients.