The Associated Ultraspherical Polynomials and their q-Analogues

1982/06/01 by Joaquin Bustoz, J. Bustoz, Mourad E. H. Ismail · 5 citations
Mathematics · Computer Science · #Mathematical functions and polynomials #Algebraic and Geometric Analysis #Matrix Theory and Algorithms

paper · pdf · doi:10.4153/cjm-1982-049-6

Abstract

A sequence of polynomials P n ( x ) is orthogonal if P n ( x ) is of precise degree n and there is a finite positive measure d μ such that 1.1 A necessary and sufficient condition for orthogonality [ 9 ] is that P n ( x ) satisfies a three term recurrence 1.2 with 1.3 Given a sequence of orthogonal polynomials P n ( x ) satisfying (1.2), the associated polynomials P n ( γ ) ( x ), γ > 0, are defined by 1.4 with P ( γ ) -1( x ) = 0, P 0 ( γ ) ( x ) = 1, when A n + γ and B n + γ are well-defined.

Cited by

• Explicit Formulas for the Associated Jacobi Polynomials and Some Applications
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients
• Bilateral q-ultraspherical functions
• Solutions to the associated q-Askey-Wilson polynomial recurrence\n relation

Related

• Orthogonality of quasi-orthogonal polynomials
• Alternative Orthogonal Polynomials
• A characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind
• New proofs of theorems on q-orthogonal functions
• Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials