Jacobi polynomials from compatibility conditions

By Y. Ismail. 2003

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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) α (1 + x) β , x ∈ [-1, 1], we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7a6b44b666a1ea83f57450cb2c4ae62e227fdc19