On sums of binomial coefficients, wavelets, complex anallysis and operator theory

Abstract

In this short note we describe and demonstrate an interesting (analytical) tool for evaluating certain (discrete) sums of binomial coefficients. Its essence consists in complex analysis observation that spectral functions of Toeplitz operators acting on the true-poly-analytic Bergman spaces over the upper half-plane with a vertical symbol can be written as a sum of spectral functions of Toeplitz operators acting on weighted Bergman spaces. This connection is expressed using Laguerre functions and demonstrated on a few examples to get a closed form of certain sums. Our note contibutes to a demonstration how "continuous" mathematics may serve as a useful tool for purposes of "discrete" mathematics.