# Operational-umbral approach to bivariate degenerate Hermite polynomials and their partial orthogonality

## TL;DR

This paper uses umbral calculus to analyze bivariate degenerate Hermite polynomials, revealing their differential equations, operational identities, and partial orthogonality, especially for negative orders, advancing their theoretical understanding.

## Contribution

It introduces an operational-umbral framework for bivariate degenerate Hermite polynomials, deriving new identities and properties, including partial orthogonality and differential equations.

## Key findings

- Derived new operational identities for the polynomials
- Established partial orthogonality conditions
- Extended analysis to negative order polynomials

## Abstract

The operational calculus associated with special polynomials has proven to be a powerful tool for analyzing and simplifying their properties. This article examines the bivariate degenerate Hermite polynomials with a focus on their differential equations, monomiality properties, operational identities, and partial orthogonality conditions. These polynomials are redeveloped within the framework of umbral formalism, which is further extended to derive certain new results. The study concludes with key observations and insights that highlight the significance of this approach in advancing the understanding of degenerate Hermite polynomials of negative order.