Interlacing theorems for the zeros of some orthogonal polynomials from different sequences

We study the interlacing properties of the zeros of orthogonal polynomials p n and r m , m = n or n − 1 where { p n } n = 1 ∞ and { r m } m = 1 ∞ are different sequences of orthogonal polynomials. The results obtained extend a conjecture by Askey, that the zeros of Jacobi polynomials p n = P n ( α ,...

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Bibliographic Details
Published in:Applied numerical mathematics 2009-08, Vol.59 (8), p.2015-2022
Main Authors: Jordaan, Kerstin, Toókos, Ferenc
Format: Article
Language:English
Subjects:
Exact sciences and technology
Fourier analysis
Hahn polynomials
Interlacing of zeros
Krawtchouk polynomials
Mathematical analysis
Mathematics
Meixner polynomials
Meixner–Pollaczek polynomials
Numerical analysis
Numerical analysis. Scientific computation
Orthogonal polynomials
Sciences and techniques of general use
Separation of zeros
Special functions
Zeros
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Summary:We study the interlacing properties of the zeros of orthogonal polynomials p n and r m , m = n or n − 1 where { p n } n = 1 ∞ and { r m } m = 1 ∞ are different sequences of orthogonal polynomials. The results obtained extend a conjecture by Askey, that the zeros of Jacobi polynomials p n = P n ( α , β ) and r n = P n ( γ , β ) interlace when α < γ ⩽ α + 2 , showing that the conjecture is true not only for Jacobi polynomials but also holds for Meixner, Meixner–Pollaczek, Krawtchouk and Hahn polynomials with continuously shifted parameters. Numerical examples are given to illustrate cases where the zeros do not separate each other.
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2009.04.002