Binomial prolate spheroidal functions, Pascal matrices, and arithmetic of elliptic curves.

Abstract

The [Formula: see text] symmetric Pascal matrix [Formula: see text] is a generalized discrete time and band-limiting operator for the binomial transform and its eigenvectors are generalized discrete prolate spheroidal wave functions which we call binomial prolates. Their generating functions are also generalized prolate spheroidal functions in the sense that they are simultaneously eigenfunctions of a third-order differential operator and an integral operator over the line [Formula: see text]. For even, positive integers <i>N</i>, we obtain an explicit formula for the generating function of an eigenvector of the symmetric Pascal matrix with eigenvalue 1. When [Formula: see text] for an odd prime <i>p</i>, we show that the generating function is equivalent modulo <i>p</i> to [Formula: see text], where [Formula: see text] is the number of points on the Legendre elliptic curve [Formula: see text] over the finite field [Formula: see text]. Furthermore when [Formula: see text], our generating function is the square of a period of [Formula: see text] modulo [Formula: see text] in the open <i>p</i>-adic unit disk.