A Generalization of Laguerre Polynomials

Abstract

The authors study orthogonal polynomials on $[0, + \infty )$ with respect to an inner product involving derivatives that cannot be derived from a weight function. These polynomials can be written as a ${}_3 F_3 $ hypergeometric series and they satisfy a second-order differential equation and a five term recurrence relation. At most one zero of each polynomial is located outside $(0, + \infty )$, the interior of the interval of orthogonality. As a special case Koornwinder’s Laguerre polynomials $\{ L_n^{\alpha ,M} (x)\} _{n = 0}^{ + \infty } $ are included.