Mellin-Barnes integrals of Gamma functions for the sums of coth(pi n)/n^(4m+1), and a hypergeometric evaluation for m = 1
Year: 2026
Type: preprint
Abstract: For every integer m≥1 we express S4m+1 = ∑n≥1 coth(πn)/n4m+1, divided by π4m, as a rational linear combination of π and of m absolutely convergent Mellin–Barnes integrals of dimensions 1,3,…,2m−1, each divided by a power of Γ(1/4). The integrands are quotients of products of Gamma functions with the parameters 1/4, 1/2, 1, and the integrals of even dimension from the same family reduce to these. For k≥2, the integral of dimension k−1, divided by 4Γ(1/4)2k, is the moment ∫0∞ [χ(1−χ)]k dr/r of the distribution function χ of the probability density proportional to (1+r4)−1/2 on (0,∞). At its fixed point, Ramanujan's formula for ζ(2n+1) evaluates ∑n≥1 coth(πn)/ns for s ≡ 3 (mod 4) and is void for s ≡ 1 (mod 4). For m = 1 the integral is a value of Meijer's G-function, and we evaluate it by a hypergeometric series: ∑n≥1 (coth(πn))/(n5) = (5π5)/1512+(64 π6)/(81 Γ(1/4)8) 6F5(3/4,1,1,1,1,1;7/4,3/2,3/2,3/2,3/2;1). The same series gives the torsional rigidity of a bar of square cross-section and the principal value of ∫−11 P−1/6(x)4 dx/x for the Ferrers function P−1/6. A conjectural second evaluation, by values of Meijer's G-function at −1, is also stated. MSC 2020: Primary 33C60; Secondary 33C20, 11F67, 11M36, 33E05, 74B05. The record contains the paper, its LaTeX source, Python programs with recorded outputs that reproduce the numerical statements, and a Lean archive formally verifying the m = 1 one-fold integral formula.
Author: Na, Sungsoo
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225800803
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