An explicit formula for the linearization coefficients of Bessel polynomials

TL;DR

This paper derives a single sum explicit formula for the linearization coefficients of Bessel polynomials, generalizing previous formulas and confirming positivity in special cases, while also extending an integral formula by Boros and Moll.

Contribution

It provides a new explicit single sum formula for Bessel polynomial linearization coefficients, unifying and extending prior results.

Findings

Formula reduces to known results in special cases

Confirms positivity of coefficients in certain cases

Generalizes an existing integral formula

Abstract

We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat's formulas in their proof of the positivity results about these coefficients (Constructive Approximation, 27(2008), 15-32). As a bonus we also obtain a generalization of an integral formula of Boros and Moll (J. Comput. Appl. Math. 106 (1999), 361-368).