Christoffel-Darboux-type formulae and a recurrence for biorthogonal polynomials |
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Authors: | A. Iserles S. P. Nørsett |
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Affiliation: | 1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England 2. Institutt for Matematiske Fag, Norges Tekniske H?gskole, Trondheim, Norway
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Abstract: | It is proved that biorthogonal polynomials obey two different kinds of Christoffel-Darboux-type formulae, one linking polynomials with a different parameter and one combining polynomials with different degrees. This is used to produce a mixed recurrence relation, which is valid for all biorthogonal polynomials. This recurrence relation establishes several results on interlacing property of zeros of successive biorthogonal polynomials and leads to a new result on the interlace of zeros of orthogonal polynomials (of equal degrees) with respect to two distributionsdψ(x) andx p dψ(x), 0<p≤1, with support in either [0, 1] or [1, ∞). |
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