# Zeta Functions

## September 18 - 22, 2006, Moscow, Russia

### Laboratoire J.-V. Poncelet

General

Announcement

Participants

Practical details

Program

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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