Recurrence equation and integral representation of Apéry sums

Abstract

AbstractVarious methods are used to investigate sums involving a reciprocal central binomial coefficient and a power term. In the first part, new functions are introduced for calculation of sums with a negative exponent in the power term. A recurrence equation for the functions provides an integral representation of the sums using polylogarithm functions. Thus polylogarithms and, in particular, zeta values can be expressed via these functions, too. In the second part, a straightforward recurrence formula is derived for sums having a positive exponent in the power term. Finally, two interesting cases of double sums are presented.