New relations for two-dimensional Hermite polynomials

Abstract

The effective formulas reducing the two-dimensional Hermite polynomials to the classical one-dimensional orthogonal polynomials by Jacobi, Gegenbauer, Legendre, Laguerre, and Hermite are given. New one-parameter generating functions for the ‘‘diagonal’’ multidimensional Hermite polynomials are derived. The factorial moments and cumulants of the distribution functions related to the Hermite polynomials of two variables with equal indices are expressed in terms of the Legendre and Chebyshev polynomials. Asymptotical formulas for the two-dimensional polynomials with large values of indices and zero arguments are found. The applications to the squeezed one-mode states and to the time-dependent quantum harmonic oscillator are considered.