Quadrature formulas for Bessel polynomials

TL;DR

This paper develops quadrature formulas for Bessel polynomials, extending previous work on classical orthogonal polynomials, and employs advanced mathematical tools including the Riesz–Shohat theorem and elliptic curves.

Contribution

It extends the theory of rational quadrature formulas to Bessel polynomials within the Askey-scheme, using novel proof techniques.

Findings

Established existence and non-existence results for rational quadrature formulas for Bessel polynomials.

Connected the problem to number theory via rational points on elliptic curves.

Applied the Riesz–Shohat theorem and Newton polygons in proofs.

Abstract

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring's problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz--Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.