Hypergeometric series over finite fields and Apéry numbers
Let \(A,B\) be multiplicative characters on \(\mathbb{F}_ p\) with values in \(\mathbb{Q}_ p\). Denote \({A\choose B}^*= {{B(-1)} \over {p-1}} J(A,b^{- 1})\) where \(J\) is the Jacobi sum. Let \(\omega\) be the Teichmüller character. Then, for any \(0\leq i,k\leq p-2\) we have \({\omega^ k \choose \omega^ i}^* \equiv {k \choose i} \pmod {p}\) (0 if \(k<i\)). The hypergeometric series over the finite field \(\mathbb{F}_ p\) is defined by \[ _{n+1} F_ n^* \left({{A_ 0,A_ 2,\ldots,A_ n} \atop {B_ 1,\ldots,B_ n}}\mid x \right) =\sum_ \chi {{A_ 0\chi} \choose \chi}^* {{A_ 1\chi} \choose {B_ 1\chi}}^* \cdots {{A_ n\chi} \choose {B_ n\chi}}^* \chi(x). \] Using the congruence the author identifies the socalled Apéry sums \(a_ n^{(m,\ell)}= \sum_{k=0}^ n {{n+k} \choose k}^ m {n\choose k}^ \ell\) with certain hypergeometric series over \(\mathbb{F}_ p\). Using Greene's evaluations at special points the author obtains a number of congruences for the numbers \(a_ n^{(m,\ell)}\) several of which have been proved earlier by other means. It is perhaps interesting to compare this approach with the paper of P. T. Young who uses \(p\)-adic hypergeometric functions.
scientific article; zbMATH DE number 169012