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

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.