Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 103, 18 pages      arXiv:2306.16411      https://doi.org/10.3842/SIGMA.2023.103

Expansions and Characterizations of Sieved Random Walk Polynomials

Stefan Kahler ^abc
a) Fachgruppe Mathematik, RWTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Germany
^b) Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
^c) 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

Received July 03, 2023, in final form December 01, 2023; Published online December 22, 2023

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 $\mathrm{A}_k$.

Key words: random walk polynomials; sieved polynomials; Askey-Wilson operator; averaging operator; polynomial expansions; Fourier coefficients.

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