On a link between Dirichlet kernels and central multinomial coefficients

Abstract

The central coefficients of powers of certain polynomials with arbitrary degree in x form an important family of integer sequences. Although various recursive equations addressing these coefficients do exist, no explicit analytic representation has yet been proposed. In this article, we present an explicit form of the integer sequences of central multinomial coefficients of polynomials of even degree in terms of finite sums over Dirichlet kernels, hence linking these sequences to discrete nth-degree Fourier series expansions. The approach utilizes the diagonalization of circulant Boolean matrices, and is generalizable to all multinomial coefficients of certain polynomials with even degree, thus forming the base for a new family of combinatorial identities.