New relations for two-dimensional Hermite polynomials

TL;DR

This paper introduces new formulas and generating functions for two-dimensional Hermite polynomials, explores their asymptotic behavior, and discusses applications in quantum physics such as squeezed states and harmonic oscillators.

Contribution

It provides effective reduction formulas, novel generating functions, and asymptotic formulas for two-dimensional Hermite polynomials, with applications in quantum physics.

Findings

Derived effective reduction formulas to classical polynomials

Established new one-parameter generating functions

Obtained asymptotic formulas for large indices

Abstract

The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. New one-parameter generating functions for these polynomials are derived. Asymptotical formulas for large values of indices are found. The applications to the squeezed one-mode states and to the time-dependent quantum harmonic oscillator are considered.