# A Comprehensive Treatment of q-Calculusq-orthogonal polynomials

A Comprehensive Treatment of q-Calculus: q-orthogonal polynomials [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Comprehensive Treatment of q-Calculusq-orthogonal polynomials

50 pages      /lp/springer-journals/a-comprehensive-treatment-of-q-calculus-q-orthogonal-polynomials-h0RDXqJsT0
Publisher
Springer Basel
ISBN
978-3-0348-0430-1
Pages
309 –358
DOI
10.1007/978-3-0348-0431-8_9
Publisher site
See Chapter on Publisher Site

### 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.]

Published: Jun 18, 2012

Keywords: Rodriguez Formula; Orthogonal Range; Generating Function Technique; Operator Formulas; Feldheim