Publication: When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?
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2010-01
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Elsevier
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Abstract
Given {Pn}n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e.,
Qn(x) = Pn(x) + a1Pn-1(x) + ... + akPn-k(x), ak<>0, n>k.
Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 is shown.
Qn(x) = Pn(x) + a1Pn-1(x) + ... + akPn-k(x), ak<>0, n>k.
Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 is shown.
Description
7 pages, no figures.-- MSC2000 codes: 33C45; 42C05.-- Issue title: "Special Functions, Information Theory, and Mathematical Physics" (Special issue dedicated to Professor Jesús Sánchez Dehesa on the occasion of his 60th birthday).
Keywords
Orthogonal polynomials, Recurrence relations, Linear functionals, Chebyshev polynomials, Difference equations
Bibliographic citation
Journal of Computational and Applied Mathematics, 2010, vol. 233, n. 6, p. 1446-1452