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q-orthogonal polynomials

In: A Comprehensive Treatment of q-Calculus

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  • Thomas Ernst

    (Uppsala University, Department of Mathematics)

Abstract

This chapter and the next one have many things in common. The generating function technique by Rainville is used to prove recurrences for q-Laguerre polynomials. We prove product expansions and bilinear generating functions for q-Laguerre polynomials by using operator formulas. Many formulas for q-Laguerre polynomials are special cases of q-Jacobi polynomial formulas. A certain Rodriguez operator turns up, a generalization of the Rodriguez formula. We will prove orthogonality for both the above-mentioned polynomials by using q-integration by parts, a method equivalent to recurrences. Many of the operator formulas for q-Jacobi polynomial formulas are formal, because of the limited convergence region of the q-shifted factorial. The q-Legendre polynomials are defined by the Rodrigues formula to enable an easy orthogonality relation. q-Legendre polynomials have been given before, but these do not have the same orthogonality range in the limit as ordinary Legendre polynomials. We also find q-difference equations for these polynomials.

Suggested Citation

  • Thomas Ernst, 2012. "q-orthogonal polynomials," Springer Books, in: A Comprehensive Treatment of q-Calculus, edition 127, chapter 0, pages 309-358, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-0431-8_9
    DOI: 10.1007/978-3-0348-0431-8_9
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