A third-order Apery-like recursion for $ζ(5)$

Details A third-order Apery-like recursion for $ζ(5)$ A third-order Apery-like recursion for $ζ(5)$

2002-06-18

arxiv.org/abs/math/0206178v2

Mat. Zametki 72:5 (2002), 796--800 (Russian); English transl., Math. Notes 72:5 (2002), 733--737

Mathematics

Number Theory

5 pages, AmSTeX; to appear in Mat. Zametki [Math. Notes] 72 (2002)

Scientific paper

10.1023/A:1021473409544

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, a similar second-order difference equation for $\zeta(4)$ has been discovered. The note contains a possible generalization of the above results for the number $\zeta(5)$.

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