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Fleck’s congruence, associated magic squares and a zeta identity. (English) Zbl 1246.11043

Summary: Let the Fleck numbers, \(C_n(t,q)\), be defined such that \[ C_n(t,q)=\sum_{k\equiv q \pmod n}(-1)^k\binom{t}{k}. \] For prime \(p\), Fleck (1913) obtained the result \(C_p(t,q)\equiv 0 \pmod {p^{\left \lfloor (t-1)/(p-1)\right \rfloor}}\), where \(\lfloor\cdot\rfloor\) denotes the usual floor function. This congruence was extended 64 years later by C. S. Weisman [ Mich. Math. J. 24, 141–151 (1977; Zbl 0377.10003)], 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.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
05B15 Orthogonal arrays, Latin squares, Room squares

Citations:

Zbl 0377.10003

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