Functiones et Approximatio Commentarii Mathematici

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

Matthew C. Lettington

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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.

Article information

Funct. Approx. Comment. Math., Volume 45, Number 2 (2011), 165-205.

First available in Project Euclid: 12 December 2011

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Zentralblatt MATH identifier

Primary: 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx] 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A19: Combinatorial identities, bijective combinatorics 11C20: Matrices, determinants [See also 15B36] 11E95: $p$-adic theory 11S05: Polynomials

combinatorial identities combinatorial functions matrices determinants p-adic theory and binomial coefficients


Lettington, Matthew C. Fleck's congruence, associated magic squares and a zeta identity. Funct. Approx. Comment. Math. 45 (2011), no. 2, 165--205. doi:10.7169/facm/1323705813.

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