Simultaneous Generation of Koecher and Almkvist-Granville's Apéry-Like Formulae

Abstract

We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the family {\small $(\zeta(2r+4s+3))_{r,s\ge 0}$}: it unifies two identities, proved by Koecher in 1980 and Almkvist and Granville in 1999, for the generating functions of {\small $(\zeta(2r+3))_{r\ge 0}$} and {\small $(\zeta(4s+3))_{s\ge 0}$}, respectively. As a consequence, we obtain that, for any integer {\small $j \ge 0$}, there exists at least {\small $[j/2]+1$ } Apéry-like formulae for {\small $\zeta(2j+3)$}.