Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

doi:10.1088/1126-6708/2007/10/048

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Mikhail Yu. Kalmykov et al JHEP10(2007)048 doi:10.1088/1126-6708/2007/10/048

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Mikhail Yu. Kalmykov^1,2, Bennie F.L. Ward¹ and Scott A. Yost¹

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We continue the study of the construction of analytical coefficients of the ε-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results:
Theorem A: The multiple (inverse) binomial sums
$\sum_{j=1}^\infty\frac{1}{\binom{2j}{j}^k}\frac{z^j}{j^c} S_{a_1}(j-1) \cdots S_{a_p}(j-1) ,$
where k = ±1, S_a(j) is a harmonic series, S_a(j) = ∑^j_{k = 1} 1/k^a, and c is any integer number are expressible in terms of Remiddi-Vermaseren functions;
Theorem B: The hypergeometric functions
${}_pF_{p-1}\bigg(\vec{A} \!+\! \vec{a}\ep; \vec{B} \!+\! \vec{b} \ep, \frac{1}{2} \!+\! B_{p-1}; z\bigg) , \qquad {}_pF_{p-1}\bigg(\vec{A} \!+\! \vec{a}\ep, \frac{1}{2} \!+\! A_{p} ; \vec{B} \!+\! \vec{b} \ep ;z\bigg) ,$
are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials.

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Journal of High Energy Physics

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

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