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The Associated Ultraspherical Polynomials and their q-Analogues

Published online by Cambridge University Press:  20 November 2018

Joaquin Bustoz  and
Mourad E. H. Ismail
Joaquin Bustoz
Affiliation:
Arizona State University, Tempe, Arizona
Mourad E. H. Ismail
Affiliation:
Arizona State University, Tempe, Arizona
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A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that

1.1

A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence

1.2

with

1.3

Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by

1.4

with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 718 - 736
Copyright
Copyright © Canadian Mathematical Society 1982

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