Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters
نویسندگان
چکیده
We prove the following theorems: 1) The Laurent expansions in ε of the Gauss hypergeometric functions 2F1(I1 + aε, I2 + bε; I3 + p q + cε; z), 2F1(I1 + p q + aε, I2 + p q + bε; I3+ p q + cε; z) and 2F1(I1+ p q +aε, I2+ bε; I3 + p q + cε; z), where I1, I2, I3, 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 2F1(I1 + p q + aε, I2 + bε; I3 + cε; 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 jcSa1(j − 1) · · · Sap(j − 1) and the multiple rational sums ∑∞ j=1 Γ “
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