The orthogonality of some systems of polynomials

Geronimus, J.

doi:10.1215/s0012-7094-47-01441-5

Cited by 14 publications

(7 citation statements)

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“…A d-Geronimus problem type consists to characterize all d-OPSs {P n } n 0 which take the form (2.1) for fixed conditions on {γ n (k)} n 0, 0 k n and {x n } n 0 . This appellation is justified by the fact that Geronimus [26] was the first to pose this problem for d = 1 and γ n (k) = a n−k b k where {a n } n 0 and {b n } n 0 are two arbitrary complex sequences such that b n = 0, n 0. He gave necessary and sufficient conditions on the sequences {a n }, {b n } and {x n } for which {P n } n 0 is an OPS, but to identify such PS {P n } n 0 more conditions, on the function f for instance, are required.…”

Section: A D-geronimus Problem Typementioning

confidence: 99%

Some generalized hypergeometric d-orthogonal polynomial sets

Cheikh

Ouni²

2008

Journal of Mathematical Analysis and Applications

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In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d = 1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.

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Section: A D-geronimus Problem Typementioning

confidence: 99%

Some generalized hypergeometric d-orthogonal polynomial sets

Cheikh

Ouni²

2008

Journal of Mathematical Analysis and Applications

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“…Note that the study of orthogonal polynomials P n (x) having an expansion (1.1) with respect to the Newtonian basis ϕ n (x) was initiated in the pioneering paper of Geronimus [6]. For orthogonal polynomials of several variables, the Newtonian interpolation scheme plays an important role as well [9,10].…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases

Vinet

Zhedanov²

2016

SIGMA

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Abstract. We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions (LP n (x) = λ n P n (x)) of some abstract operator L which is 2-diagonal in the Newtonian basis ϕ n (x): Lϕ n (x) = λ n ϕ n (x) + τ n (x)ϕ n−1 (x) with some coefficients λ n , τ n . We find the necessary and sufficient conditions for the polynomials P n (x) to be orthogonal. For the special cases where the sets λ n correspond to the classical grids, we find the complete solution to these conditions and observe that it leads to the most general Askey-Wilson polynomials and their special and degenerate classes.

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“…When this property is related to the explicit expression of P n in certain basis, the corresponding characterization problem will be called d-Geronimus type problem. This is justified by the fact that Geronimus [28] was the first to raise a characterization problem for d = involving polynomials of the form…”

Section: D-geronimus Type Problemmentioning

confidence: 99%