An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
By D. Gómez‐Ullate, Niky Kamran, Robert Milson. 2008
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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 compact interval [ − 1 , 1 ] or the half-line [ 0 , ∞ ) , 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 ∞ , 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.
Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/e71b91736b8d825f9bb60cf3e1e24e4d8d78147a
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In Applied Mathematics
In Atomic and Molecular Physics, and Optics
In Statistical and Nonlinear Physics
Citations in the corpus
- Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials (2014, background): “The last few years have seen a great deal of activity in the area of exceptional and exceptional discrete orthogonal polynomials (see, for instance, [5, 8, 9, 15], [16] (where the adjective exceptional for this topic was introduced), [17, 18, 19, 20, 21, 26, 29, 30, 32, 36, 37, 39] and the reference”