# Two-dimensional $(p,q)$-heat polynomials of Gould--Hopper type

## TL;DR

This paper introduces a new class of two-variable holomorphic polynomials extending Gould--Hopper polynomials, exploring their properties, identities, differential equations, and connections to hypergeometric functions.

## Contribution

It presents the first comprehensive study of two-dimensional $(p,q)$-heat polynomials of Gould--Hopper type, including operational, generating, and recurrence relations.

## Key findings

- Derived new generating functions and recurrence relations.
- Established multiplication, addition, and Nielsen type formulas.
- Connected these polynomials to hypergeometric functions and differential equations.

## Abstract

We introduce a new class of holomorphic polynomials extending the classical Gould--Hopper to two complex variables. The considered polynomials include the $1$-D and $2$-D holomorphic and polyanalytic It\^o--Hermite polynomials as particular cases. We emphasize studying their operational representation, various generating functions, and recurrence relations. We also establish some special identities including multiplication and addition formulas of Runge type, as well as the Nielson type formulas. Higher-order partial differential equations are analyzed and the connection to Gould-Hopper polynomials and hypergeometric functions are investigated.