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

Lettington, Matthew C. ORCID: https://orcid.org/0000-0001-9327-143X 2011. Fleck's congruence, associated magic squares and a zeta identity. Functiones et Approximatio, Commentarii Mathematici 45 (2) , pp. 165-205. 10.7169/facm/1323705813

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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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	combinatorial identities; combinatorial functions; matrices; determinants; p-adic theory and binomial coefficients
Additional Information:	Zentralblatt MATH identifier: 05989419 Mathematical Reviews number (MathSciNet): MR2895154
Publisher:	Project Euclid
Last Modified:	02 Oct 2023 19:50
URI:	https://orca.cardiff.ac.uk/id/eprint/15644

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