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Nonlinear recurrence relations and induced orthogonal polynomials
Journal article   Peer reviewed

Nonlinear recurrence relations and induced orthogonal polynomials

J. Wimp and H. Kiesel
Computers & mathematics with applications (1987), v 28(1), pp 325-332
1994
DOI: https://doi.org/10.1016/0898-1221(94)00120-0

Abstract

Jacobi polynomials Mass points Nonlinear difference equations Nonlinear recurrences Orthogonal polynomials
We consider a class of polynomials Q n ( x) defined by Q n(x) = (x + b n) P n−1 (x) + d nP n (x), n = 0, 1, 2,…,d o ≠ 1 , where the P n ( x) are polynomials orthogonal with respect to some real linear functional. We ask what conditions the sequences b n , d n must satisfy so that the polynomials Q n ( x) also form an orthogonal set. It turns out that the quantities b n , d n must satisfy second order nonlinear recurrences, but that these recurrences reduce to first order recurrences. The values b 0, b 1, d 0, d 1 are arbitrary. We determine the weight function for the polynomials and discuss a number of special cases. We show that a specialization of our results leads to some polynomials discussed by Koornwinder.

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Mathematics, Applied
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