New mixed recurrence relations of two-variable orthogonal polynomials via differential operators

TL;DR

This paper introduces new recurrence relations for two-variable orthogonal polynomials, such as Jacobi and Legendre polynomials, using novel differential operators, enhancing understanding of their structural properties.

Contribution

The paper presents novel recurrence relations for two-variable orthogonal polynomials derived via two specific differential operators, expanding theoretical tools for these polynomials.

Findings

Derived new recurrence relations for Jacobi, Bateman's, and Legendre polynomials.

Established special cases of the main recurrence relations.

Enhanced the theoretical framework for two-variable orthogonal polynomials.

Abstract

In this paper, we derive new recurrence relations for two-variable orthogonal polynomials for example Jacobi polynomial, Bateman's polynomial and Legendre polynomial via two different differential operators $\Xi =\left(\frac{\partial }{\partial z} +\sqrt{w} \frac{\partial }{\partial w} \right)$ and $\Delta =\left(\frac{1}{w} \frac{\partial }{\partial z} +\frac{1}{z} \frac{\partial }{\partial w} \right)$. We also derive some special cases of our main results.