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

• 成果类型: Article
• 署名作者: Casper, W. Riley
• 署名单位: California State University System; California State University Fullerton
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.2529171123
• 发表日期: 2026-06-23
• 页码: e2529171123
• 关键词: prolate spheroidal wave functions; zeta functions; Pascal matrices; elliptic curves; level-spacing distributions; differential-equations; fourier-analysis; wave-functions; COEFFICIENT MATRIX; uncertainty

摘要

The (N+1)& times;(N+1) symmetric Pascal matrix TN 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 is an element of 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 (#Ez(Fp)-1)2, where #Ez(Fp) is the number of points on the Legendre elliptic curve y2=x(x-1)(x-z) over the finite field Fp. Furthermore when N=pn-1, our generating function is the square of a period of Ez modulo pn in the open p-adic unit disk.