Fleck's congruence, associated magic squares and a zeta identity

Abstract

Let the \emph{Fleck numbers}, $C_n(t,q)$, be defined such that \[ C_n(t,q)=\sum_{k\equiv q (mod n)}(-1)^k\binom{t}{k}. \] For prime $p$, Fleck obtained the result $C_p(t,q)\equiv 0 (mod p^{{\left \lfloor (t-1)/(p-1)\right \rfloor}}} )$, where $\lfloor.\rfloor$ denotes the usual floor function. This congruence was extended 64 years later by Weisman, in 1977, to include the case $n=p^\alpha$. In this paper we show that the Fleck numbers occur naturally when one considers a symmetric $n\times n$ matrix, $M$, and its inverse under matrix multiplication. More specifically, we take $M$ to be a symmetrically constructed $n\times n$ associated magic square of odd order, and then consider the reduced coefficients of the linear expansions of the entries of $M^t$ with $t\in \mathbb{Z}$. We also show that for any odd integer, $n=2m+1$, $n\geq 3$, there exist geometric polynomials in $m$ that are linked to the Fleck numbers via matrix algebra and $p$-adic interaction. These polynomials generate numbers that obey a reciprocal type of congruence to the one discovered by Fleck. As a by-product of our investigations we observe a new identity between values of the Zeta functions at even integers. Namely \[ \zeta{(2j)}=(-1)^{j+1}\left (\frac{j\pi^{2j}}{(2j+1)!}+\sum_{k=1}^{j-1}\frac{(-1)^k\pi^{2j-2k}}{(2j-2k+1)!}\zeta{(2k)}\right ). \] We conclude with examples of combinatorial congruences, Vandermonde type determinants and Number Walls that further highlight the symmetric relations that exist between the Fleck numbers and the geometric polynomials.