On the Carlitz-Mehler formula for Hermite polynomials

TL;DR

This paper provides a direct proof of a further generalization of Carlitz's extension of Mehler's formula for Hermite polynomials, building on previous work by Carlitz and Srivastava et al.

Contribution

It introduces a new direct proof for an extended form of Carlitz's formula, advancing the understanding of Hermite polynomial identities.

Findings

A new direct proof of the generalized Carlitz-Mehler formula is established.

The proof simplifies previous approaches to Hermite polynomial identities.

The generalization broadens the applicability of Mehler's formula in mathematical analysis.

Abstract

Carlitz proved a few generalizations of Mehler's formula. Later, Srivastava et al. gave a new proof for some extensions of Carlitz's formula. Here, a direct proof of the further generalization is given.