number theory

Multiple Clausen values and deformed Apéry-like series

arXiv:2607.14646

summary

The paper expresses deformed Apéry‑like series, defined via derivatives of generalized central binomial coefficients, in terms of multiple Clausen values, a subclass of cyclotomic multiple zeta values at level 3, and provides explicit evaluations.

Abstract

With generalized central binomial coefficients defined through Euler's gamma function, we represent deformed Apéry-like series \[ \mathscr A_{s,n}:=\sum_{k=1}^\infty\left.\!\frac{\partial^n}{\partial x^n}\frac{1}{x^s\binom{2x}{x}}\right|_{x=k} \] by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level . For example, exploiting provable algebraic relations among MCVs, we show that \[\mathscr A_{1,5}=-\frac{9[495L(χ_{-3},6)-30π^{2}L(χ_{-3},4)-2π^{4}L(χ_{-3},2)]}{4}\]and\[\mathscr A_{4,4}=\frac{352ζ_{5,3}}{15}+\frac{752537π^{8}}{10206000},\]where and .

29 pages, 9 tables. Maple worksheet and Mathematica notebook available as ancillary files

Topics & keywords

#multiple clausen values#apery-like series#cyclotomic multiple zeta values#central binomial coefficients#special functionsmultiple clausen valuescyclotomic multiple zeta valuesApéry-like seriescentral binomial coefficientsgamma functionL-functionszeta_{5,3}
Multiple Clausen values and deformed Apéry-like series · wovepaper