# Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra

## TL;DR

This paper derives new convolution identities for $q=-1$ analog orthogonal polynomials, utilizing Lie superalgebra representations, and constructs a bilinear generating function for Big -1 Jacobi polynomials.

## Contribution

It introduces novel convolution identities for $q=-1$ orthogonal polynomials using Lie superalgebra $rak{osp}(1|2)$ representations, expanding the understanding of their algebraic structure.

## Key findings

- Convolution identities for specialized Chihara, dual -1 Hahn, and Big -1 Jacobi polynomials.
- Convolution identities for Big -1 Jacobi and Bannai-Ito polynomials using Racah coefficients.
- A bilinear generating function for Big -1 Jacobi polynomials.

## Abstract

New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the Lie superalgebra $\mathfrak{osp}(1|2)$ in the positive discrete series representation. Using the Clebsch-Gordan coefficients, a convolution identity for the Specialized Chihara, the dual -1 Hahn and the Big -1 Jacobi polynomials is found. Using the Racah coefficients, a convolution identity for the Big -1 Jacobi and the Bannai-Ito polynomials is found. Finally, these results are applied to construct a bilinear generating function for the Big -1 Jacobi polynomials.