Jacobi polynomials from compatibility conditions
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- by Yang Chen and Mourad Ismail PDF
- Proc. Amer. Math. Soc. 133 (2005), 465-472 Request permission
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 \[ w(x)=(1-x)^{\alpha }(1+x)^{\beta },\qquad x\in [-1,1],\] we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.References
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Additional Information
- Yang Chen
- Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, United Kingdom
- Email: y.chen@imperial.ac.uk
- Mourad Ismail
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- MR Author ID: 91855
- Email: ismail@math.ucf.edu
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 465-472
- MSC (2000): Primary 33C45; Secondary 42C05
- DOI: https://doi.org/10.1090/S0002-9939-04-07566-5
- MathSciNet review: 2093069