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

Abstract

The ( N + 1 ) × ( N + 1 ) symmetric Pascal matrix T N 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 { z ∈ C :   Re ( z ) = 1 / 2 } . For even, positive integers N , we obtain an explicit formula for the generating function of an eigenvector of the symmetric Pascal matrix with eigenvalue 1. When N = p − 1 for an odd prime p , we show that the generating function is equivalent modulo p to ( # E z ( F p ) − 1 ) 2 , where # E z ( F p ) is the number of points on the Legendre elliptic curve y 2 = x ( x − 1 ) ( x − z ) over the finite field F p . Furthermore when N = p n − 1 , our generating function is the square of a period of E z modulo p n in the open p -adic unit disk.