About sum rules for Gould-Hopper polynomials

TL;DR

This paper demonstrates how identities involving Gould-Hopper polynomials can be derived from the invariance properties of Gaussian distributions, providing new stochastic representations for Gaussian vectors and matrices.

Contribution

It introduces a novel approach linking sum rules for Gould-Hopper polynomials to the invariance of Gaussian distributions, offering new stochastic representations.

Findings

Identities for Gould-Hopper polynomials can be derived from Gaussian invariance.

A stochastic representation for inner products of Gaussian vectors and matrices is established.

The approach connects polynomial identities with probabilistic invariance principles.

Abstract

We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful stochastic representation for the inner product of two non-centered Gaussian vectors and two non-centered Gaussian matrices. [1] J. Daboul, S. S. Mizrahi, O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed state, J. Phys. A: Math. Gen. 38 (2005) 427-448 [3] P. Graczyk, A. Nowak, A composition formula for squares of Hermite polynomials and its generalizations, C. R. Acad. Sci. Paris, Ser 1 338 (2004)