Multiple Clausen values and deformed Apéry-like series

2026/07/16 by Zhi-Wei Sun, Yajun Zhou
#math.NT #hep-th #math.CO

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Abstract

With generalized central binomial coefficients \binom2xx:=(Γ(2x+1))/([Γ(x+1)]²) defined through Euler's gamma function, we represent deformed Apéry-like series \mathscr A_s,n:=∑_k=1^∞. (∂ⁿ)/(∂ xⁿ)\frac1x^s\binom2xx|_x=k by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level 3. For example, exploiting provable algebraic relations among MCVs, we show that \mathscr A_1,5=-\frac9[495L(χ_-3,6)-30π²L(χ_-3,4)-2π⁴L(χ_-3,2)]4and\mathscr A_4,4=\frac352ζ_5,315+\frac752537π⁸10206000,where L(χ_-3,s):=∑_n=0^∞[(3n+1)^-s-(3n+2)^-s] and ζ_5,3:=∑_m>n>0m^-5n^-3.

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