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Jacobi polynomials from compatibility conditions

Authors: Yang Chen and Mourad Ismail
Journal: Proc. Amer. Math. Soc. 133 (2005), 465-472
MSC (2000): Primary 33C45; Secondary 42C05
Published electronically: August 30, 2004
MathSciNet review: 2093069
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Abstract: We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable $z$(spectral parameter) and the other a recurrence relation in $n$ (the lattice variable). For the Jacobi weight

\begin{displaymath}w(x)=(1-x)^{\alpha}(1+x)^{\beta},\qquad x\in[-1,1],\end{displaymath}

we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.

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Additional Information

Yang Chen
Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, United Kingdom

Mourad Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Received by editor(s): February 21, 2003
Received by editor(s) in revised form: October 2, 2003
Published electronically: August 30, 2004
Additional Notes: This research was supported by NSF grant DMS 99-70865 and by EPSRC grant GR/S14108.
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

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