An explicit formula for the linearization coefficients of Bessel polynomials II

TL;DR

This paper presents a single sum formula for the linearization coefficients of Bessel polynomials, unifying previous formulas and simplifying hypergeometric function sums, with implications for positivity proofs.

Contribution

It introduces a new explicit single sum formula for Bessel polynomial linearization coefficients, connecting and simplifying existing results.

Findings

Unified formula reduces to known special cases.

Simplifies hypergeometric function sums from $_3F_2$ to $_2F_1$.

Supports positivity results of coefficients.

Abstract

In this paper, a single sum formula for the linearization coefficients of the Bessel polynomials is given. In three special cases this formula reduces indeed to either Atia and Zeng's formula (Ramanujan Journal, Doi 10.1007/s11139-011-9348-4) or Berg and Vignat's formulas in their proof of the positivity results about these coefficients (Constructive Approximation, {\bf 27} (2008), 15-32). As a bonus, a formula reducing a sum of hypergeometric functions $_3F_2$ to $_2F_1$ is obtained.