Jacobi polynomials from compatibility conditions

Chen, Yang; Ismail, Mourad

doi:10.1090/S0002-9939-04-07566-5

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.

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