Hypergeometric orthogonal polynomials of Jacobi type

Joseph Bernstein, Dmitry Gourevitch, Siddhartha Sahi

Published 2024-01-26Version 1

Motivated by the theory of hypergeometric orthogonal polynomials, we consider quasi-orthogonal polynomial families, i.e. those that are orthogonal with respect to a non-degenerate bilinear form defined by a linear functional, and in which the ratio of successive coefficients is given by a rational function $f(u,s)$ which is polynomial in $u$. We call this a family of Jacobi type. Our main result is that there are precisely five families of Jacobi type. These are the classical families of Jacobi, Laguerre and Bessel polynomials, and two more one parameter families $E_n^{(c)},F_n^{(c)}$. The last two families can be expressed through Lommel polynomials, and they are orthogonal with respect to a positive measure on $\mathbb{R}$ for $c>0$ and $c>-1$ respectively. Each of the five families can be obtained as a suitable specialization of some hypergeometric series.