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Symmetry, Integrability and Geometry: Methods and Applications, 2023, Volume 19, 103, 18 pp.
DOI: https://doi.org/10.3842/SIGMA.2023.103
(Mi sigma1998)
 

Expansions and Characterizations of Sieved Random Walk Polynomials

Stefan Kahlerabc

a Fachgruppe Mathematik, RWTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Germany
b Department of Mathematics, Chair for Mathematical Modelling, Chair for Mathematical Modeling of Biological Systems, Technical University of Munich, Boltzmannstr. 3, 85747 Garching b. München, Germany
c Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
References:
Abstract: We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$ $n\in\mathbb{N}$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every $k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$ arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and $c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansions in the Chebyshev basis $\{T_n(x)\colon n\in\mathbb{N}_0\}$. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of the Askey–Wilson operator $\mathcal{D}_q$. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative and obtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity. However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for $k$-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator $\mathrm{A}_k$.
Keywords: random walk polynomials, sieved polynomials, Askey–Wilson operator, averaging operator, polynomial expansions, Fourier coefficients.
Funding agency Grant number
Elite Network of Bavaria
The research was begun when the author worked at Technical University of Munich, and 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 Technical University of Munich.
Received: July 3, 2023; in final form December 1, 2023; Published online December 22, 2023
Document Type: Article
MSC: 42C05, 33C47, 42C10
Language: English
Citation: Stefan Kahler, “Expansions and Characterizations of Sieved Random Walk Polynomials”, SIGMA, 19 (2023), 103, 18 pp.
Citation in format AMSBIB
\Bibitem{Kah23}
\by Stefan~Kahler
\paper Expansions and Characterizations of Sieved Random Walk Polynomials
\jour SIGMA
\yr 2023
\vol 19
\papernumber 103
\totalpages 18
\mathnet{http://mi.mathnet.ru/sigma1998}
\crossref{https://doi.org/10.3842/SIGMA.2023.103}
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