Orthogonal polynomials which are eigenfunctions of a partial differential operator

Abstract

We study orthogonal polynomials of $d = d_1+d_2$ variables with respect to a wrapped product weight function ${\bm W}({\bm x},{\bm y}) = W_1({\bm x}/\rho({\bm y})) W_2(\bm y)$ for $(\bm x, \bm y) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$, where $\rho$ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For $d =2$, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in $d= d_1+d_2$ variables that have not been studied before if either $d_1>1$ or $d_2>1$, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when $d = 3$. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.