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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
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.
Received: July 3, 2023; in final form December 1, 2023; Published online December 22, 2023
Citation:
Stefan Kahler, “Expansions and Characterizations of Sieved Random Walk Polynomials”, SIGMA, 19 (2023), 103, 18 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1998 https://www.mathnet.ru/eng/sigma/v19/p103
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Abstract page: | 32 | Full-text PDF : | 7 | References: | 18 |
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