\(q\)-analogue of the Krawtchouk and Meixner orthogonal polynomials
The classical orthogonal polynomials and their discrete analogues arise in many different ways. One place where some of these arise in the group \(\text{SU} (2)\). The authors summarize some known results about Krawtchouk polynomials which arise naturally in this setting, and then give some similar results for one of the basic hypergeometric orthogonal polynomial extensions of Krawtchouk polynomials. Similar results are stated for a \(q\)-version of Meixner polynomials. The authors say there are two types of \(q\)-Krawtchouk polynomials, and that they have similar results for the other type. Many of the identities they state, and similar ones for all of the classical type hypergeometric and basic hypergeometric orthogonal polynomials were given by \textit{R. Koekoek} and \textit{R. Swarttouw} in Report 94-05, ``The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue'', T.U. Delft. They include four sets of polynomials which are called \(q\)-Krawtchouk, with an adjective or two to distinguish them. The authors of the paper under review follow the usual pattern in physics and do not use the notation of hypergeometric and basic hypergeometric functions. This is an error, for the notation used makes it hard to see exactly what the functions considered are. Hypergeometric notation is just a canonical way to write some important series, with all of the shifted factorials being raising ones, rather than some raising and others falling as in the present paper.
scientific article; zbMATH DE number 770337
scientific article