Jacobi polynomials and some connection formulas in terms of the action of linear differential operators

Abstract

In this paper, we introduce the concept of the $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials, where $\mathbb{J}_{\alpha, \beta}$ is the raising operator $\mathbb{J}_{\alpha, \beta}:=(x^2-1)\frac{d}{dx}+ \big((\alpha+\beta)x-\alpha+\beta\big)\mathbb{I}$, with $\alpha$ and $\beta$ nonzero complex numbers and $\mathbb{I}$ representing the identity operator. Then, we show that the Jacobi polynomials $P^{(\alpha, \beta)}_n(x),\ n\geq0$, where $\alpha, \beta\in\mathbb{C}\backslash\{0,-1,-2,\ldots\}$, $(\alpha+\beta\neq-m, \ m\geq0)$, are the only $\mathbb{J}_{\alpha, \beta}$-classical orthogonal polynomials. As an application, we give some new connection formulas satisfied by the polynomials solution of our problem.