Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0835.33011
Al-Salam, Waleed A.; Ismail, Mourad E.H.
A $q$-beta integral on the unit circle and some biorthogonal rational functions. (English)
[J] Proc. Am. Math. Soc. 121, No.2, 553-561 (1994). ISSN 0002-9939; ISSN 1088-6826/e

The authors start with a pair of polynomial sets, due to {\it P. I. Pastro} [J. Math. Anal. Appl. 112, 517-540 (1985; Zbl 0582.33010)], biorthogonal on the unit circle with respect to a complex weight function. Using generating functions, they turn the biorthogonality relation into a certain $q$-beta integral, which in turn leads to a pair of sets of rational functions biorthogonal on the unit circle with respect to another complex weight function. The asymptotics of these biorthogonal pairs are exhibited. Remarkably, in both cases the weight function can be read off from this asymptotics, in a way reminiscent of the Szegö theory for orthogonal polynomials. To be more precise, if $\{P_n (z)\}$ and $\{Q_n (z)\}$ is one of these pairs and $K(z)$ is the weight function of the corresponding biorthogonality measure, then (under suitable conditions) $P_n (z) \overline {(Q_n 1/z}) \sim 1/K(z)$ as $n$ tends to $\infty$.
[Ch.Krattenthaler (Wien)]

MSC 2000:: *33D45 Basic hypergeometric functions and integrals in several variables
33D05 q-gamma functions, q-beta functions and integrals
42A65 Completeness of sets of functions

Keywords: biorthogonal sets of polynomials; biorthogonal sets of functions; asymptotics of biorthogonal rational functions; $q$-beta integral

Citations: Zbl 0582.33010

Cited in: Zbl 0932.33002

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]