Multiple Clausen values and deformed Apéry-like series

Abstract

With generalized central binomial coefficients $\left(\genfrac{}{}{0ex}{}{2x}{x}\right):=\frac{\Gamma (2x+1)}{[\Gamma (x+1){]}^2}$ 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 $3$. 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 $L({\chi }_{-3},s):={\sum }_{n=0}^{\infty }\left[(3n+1{)}^{-s}-(3n+2{)}^{-s}\right]$ and ${\zeta }_{5,3}:={\sum }_{m>n>0}{m}^{-5}{n}^{-3}$.

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