Holomorphic Hermite polynomials in two variables

On Pseudo Hermite Matrix Polynomials of Two Variables

Abstract. The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix p...

Erratum: On a Hermite interpolation by polynomials of two variables

TWO CURIOUS q-ANALOGUES OF HERMITE POLYNOMIALS

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider two new q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci polynomials and the recent works on the combinatorics of the Matrix Ansatz of the PASEP.

New relations for two-dimensional Hermite polynomials

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.

A note on Hermite polynomials of several variables

The present paper deals with an extension of certain results obtained by Burchnall for Her-mite polynomials to similar results for Hermite polynomials of several variables.