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arXiv · 0807.3939

An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

David Gomez-Ullate·Niky Kamran·Robert Milson
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Abstract

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the the compact interval $[-1,1]$ or the half-line $[0,\infty)$, respectively, and they are a basis of the corresponding $L^2$ Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second order operator has a complete set of polynomial eigenfunctions $\{p_i\}_{i=1}^\infty$, then it must be either the $X_1$-Jacobi or the $X_1$-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the $X_1$ polynomial sequences.

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David Gomez-Ullate, Niky Kamran, Robert Milson. 2008-07-24. An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. https://doi.org/10.1016/j.jmaa.2009.05.052

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