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Characterizations of ultraspherical polynomials and their $q$-analogues

Author: Stefan Kahler
Journal: Proc. Amer. Math. Soc. 144 (2016), 87-101
MSC (2010): Primary 33C45, 33D45; Secondary 42C05, 42C10
Published electronically: September 4, 2015
MathSciNet review: 3415579
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Abstract: We investigate symmetric, suitably normalized orthogonal polynomial sequences $(P_n(x))_{n\in \mathbb {N}_0}$ and characterize the class of ultraspherical polynomials in terms of certain constancy properties of the Fourier coefficients which belong to $(P_{2n-1}^\prime (x))_{n\in \mathbb {N}}$. Similar characterizations are obtained for the discrete, resp. continuous, $q$-ultraspherical polynomials after replacing the derivative $\frac {\mathrm {d}}{\mathrm {d}x}$ by the $q$-difference operator $D_{q^{-1}}$, resp. Askey–Wilson operator $\mathcal {D}_q$. Our results extend earlier work of Lasser–Obermaier and Ismail–Obermaier where the whole sequences $(P_n^\prime (x))_{n\in \mathbb {N}}$, $(D_{q^{-1}}P_n(x))_{n\in \mathbb {N}}$ and $(\mathcal {D}_q P_n(x))_{n\in \mathbb {N}}$ had to be taken into account; we shall see that the characterizing properties concerning the even indices turn out to be redundant. We also characterize a large subclass of the continuous $q$-ultraspherical polynomials via the averaging operator $\mathcal {A}_q$, and we show that this characterization cannot be extended to the whole class.

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Additional Information

Stefan Kahler
Affiliation: Department of Mathematics, Chair for Mathematical Modelling, Chair for Mathematical Modeling of Biological Systems, Technische Universität München, Boltzmannstr. 3, 85747 Garcing b. München, Germany

Keywords: Ultraspherical polynomials, discrete $q$-ultraspherical polynomials, continuous $q$-ultraspherical polynomials, $q$-difference operator, Askey–Wilson operator, averaging operator, expansions of orthogonal polynomials, Fourier coefficients
Received by editor(s): October 15, 2013
Received by editor(s) in revised form: August 20, 2014
Published electronically: September 4, 2015
Additional Notes: The author gratefully acknowledges support from the graduate program TopMath of the ENB (Elite Network of Bavaria) and the TopMath Graduate Center of TUM Graduate School at Technische Universität München. During large parts of his research the author was partially supported by a scholarship from the Max Weber-Programm within the ENB and by a scholarship from the Studienstiftung des deutschen Volkes.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society