Zeta Functions
September 18 - 22, 2006, Moscow, Russia
Laboratoire J.-V. Poncelet
Laboratoire J.-V. Poncelet
Wadim Zudilin
MGU, Moscow, Russia
Magic of Apery's numbers
The integers $ a_n=\sum_{k=0}^n{\binom nk}^2{\binom{n+k}k}^2, n=0,1,2,... $ appear as denominators of rational approximations to $\zeta(3)=\sum_{k=1}^\infty k^{-3}$ in Apery's proof of the irrationality of $\zeta(3)$. The sequence satisfies a polynomial recursion, namely, $ (n+1)^3a_{n+1}-(2n+1)(17n^2+17n+5)a_n+n^3a_{n-1}=0$, where $n=0,1,2,...$, and admits several curious congruence properties. We survey some recent results and open problems related to Apery's numbers $a_n$, $n=0,1,2,...$, and their generalizations.
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