Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture |
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Authors: | K. H. Kwon L. L. Littlejohn G. J. Yoon |
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Affiliation: | Department of Mathematics, KAIST, Taejon, 305-701, Koreaf1;Department of Mathematics and Statistics, Utah State University, Logan, Utah, 84322-3900, U.S.A., f2;Department of Mathematics, KAIST, Taejon, 305-701, Korea, f3 |
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Abstract: | Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Qn(x)}∞n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Qn(x)}∞n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: LN[y](x)=∑Ni=1 ℓi(x) y(i)(x)=λny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Qn(x)}∞n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions. |
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Keywords: | differential equations Bochner– Krall orthogonal polynomials Magnus' conjecture |
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