Zeta FunctionsSeptember 18  22, 2006, Moscow, RussiaLaboratoire J.V. Poncelet 

Wadim Zudilin MGU, Moscow, Russia Magic of Apery's numbersThe 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_{n1}=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. Go to the Laboratoire Poncelet home page. 

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