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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

Citations:

Zbl 0976.11035
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References:

[1] Almkvist G., Exp. Math. 8 (2) pp 197– (1999)
[2] Apéry R., Astérisque 61 pp 11– (1979)
[3] DOI: 10.1007/s002220100168 · Zbl 1058.11051
[4] Borwein J., Exp. Math. 6 (3) pp 181– (1997) · Zbl 0887.11037
[5] Cohen H., Sém. de Théorie des Nombres de Grenoble. (1978)
[6] Cohen H., Bull. Soc. Math. France 109 pp 269– (1981) · Zbl 0478.10018
[7] Graham R. L., Concrete Mathematics: A Foundation for Computer Science (1994)
[8] DOI: 10.1007/BF03023373 · Zbl 1219.11123
[9] DOI: 10.1090/S0002-9939-96-03042-0 · Zbl 0843.15005
[10] DOI: 10.1016/0022-314X(81)90020-2 · Zbl 0468.10006
[11] DOI: 10.1016/S0764-4442(00)01624-4 · Zbl 0973.11072
[12] DOI: 10.1007/BF03028234 · Zbl 0409.10028
[13] van der Poorten, A. ”Some Wonderful Formulas... An Introduction to Polylogarithms.”. Queen’s Papers in Pure and Applied Mathematics, (Proc. 1979 Queen’s Number Theory Conference, 1980). Vol. 54, pp.269–286. [van der Poorten 80]
[14] Zudilin W., J. Théor. Nombres Bordeaux (2004)
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