Central factorials under the Kontorovich-Lebedev transform of polynomials

TL;DR

This paper explores how modifications of the Kontorovich-Lebedev transform create automorphisms of polynomial spaces, leading to new identities between central factorials and Euler polynomials.

Contribution

It introduces a modified KL-transform that maps monomials to central factorial polynomials and characterizes polynomial sequences with the canonical KL-transform.

Findings

Modified KL-transform acts as an automorphism on polynomial space

Characterization of polynomial sequences with canonical KL-transform

New identities linking central factorials and Euler polynomials

Abstract

We show that slight modifications of the Kontorovich-Lebedev transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL-transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found.