On a nonorthogonal polynomial sequence associated with Bessel operator

TL;DR

This paper introduces a polynomial sequence linked to the Bessel operator, detailing its properties, explicit form, and connections to Euler numbers and integral transforms, despite its nonorthogonality.

Contribution

It constructs a new polynomial sequence associated with the Bessel operator and explores its properties, including explicit expression and measure association under certain conditions.

Findings

Explicit expression of the polynomial sequence

Connection with Euler numbers and Kontorovich-Lebedev transform

Existence of a positive-definite measure under constraints

Abstract

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them, its explicit expression, the connection with the Euler numbers, its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality, it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed.