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

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.