## Simultaneous generation of Koecher and Almkvist-Granville’s Apéry-like formulae.(English)Zbl 1127.11057

Summary: We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the family $$(\zeta(2r+4s+3))_{r,s\geq 0}$$: it unifies two identities, proved by M. Koecher [Letter to the Editor. Math. Intell. 2, 62–64 (1980)] in 1980 and G. Almkvist and A. Granville [Exp. Math. 8, No. 2, 197–203 (1999; Zbl 0976.11035)] in 1999, for the generating functions of {$$(\zeta(2r+3))_{r\geq 0}$$ and $$(\zeta(4s+3))_{s\geq 0}$$, respectively. As a consequence, we obtain that, for any integer $$j \geq 0$$, there exists at least $$[j/2]+1$$ Apéry-like formulae for $$\zeta(2j+3)$$.}

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 05A15 Exact enumeration problems, generating functions 11J72 Irrationality; linear independence over a field 11Y60 Evaluation of number-theoretic constants

Zbl 0976.11035
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### References:

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