Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters alpha =beta =-1 |
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Authors: | Andrea Bruder Lance L. Littlejohn |
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Affiliation: | 1. Department of Mathematics and Computer Science, Tutt Science Center, Colorado College, 14 E. Cache la Poudre St., Colorado Springs, CO, 80903, USA 2. Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX, 76798-7328, USA
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Abstract: | In this paper, we consider the second-order differential expression $$begin{aligned} ell [y](x)=(1-x^{2})(-(y^{prime }(x))^{prime }+k(1-x^{2})^{-1} y(x))quad (xin (-1,1)). end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $alpha =beta =-1$ in contrast to the classical case when $alpha ,beta >-1$ . For fixed $kge 0$ and appropriate values of the spectral parameter $lambda ,$ the equation $ell [y]=lambda y$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions ${P_{n}^{(-1,-1)} }_{n=0}^{infty }.$ These Jacobi polynomial solutions of degree $ge 2$ form a complete orthogonal set in the Hilbert space $L^{2}((-1,1);(1-x^{2})^{-1})$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $ge 0$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $T$ , generated by $ell [cdot ],$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $T$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman. |
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