Mehler's formulas for the univariate complex Hermite polynomials and applications

TL;DR

This paper derives two broad Mehler's formulas for univariate complex Hermite polynomials, providing simpler proofs and applications, extending classical formulas to the complex domain.

Contribution

It introduces two new, more general Mehler's formulas for complex Hermite polynomials with simplified proofs and explores their applications.

Findings

Derived two broad Mehler's formulas for complex Hermite polynomials

Provided simpler, direct proofs for these formulas

Applied formulas to derive new identities and results

Abstract

We give two widest Mehler's formulas for the univariate complex Hermite polynomials $H_{m,n}^\nu$, by performing double summations involving the products $u^m H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^\nu (w,\overline{w})}$ and $u^m v^n H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^{\nu'} (w,\overline{w})}$. They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level $m$. The second Mehler's formula generalizes the one appearing as a particular case of the so-called Kibble-Slepian formula. The proofs, we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.