Computation of Al-Salam Carlitz and Askey-Wilson moments using Motzkin paths
Summary: In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where \(d=0\). Their values have already been computed by \textit{J. S. Kim} and \textit{D. Stanton} in [J. Math. Anal. Appl. 421, No. 1, 501--520 (2015; Zbl 1302.33017)], however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with \(d=0\) only, providing the first fully combinatorial proof for that case.
scientific article; zbMATH DE number 7368130
scientific article