symmetry in generating functions对称性在生成函数.pdf

Symmetry 2010, 2, 346-365; doi:10.3390/sym2010345 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Review Symmetry in Generating Functions Shigeru Watanabe The University of Aizu, Ikki-machi Tsuruga, Aizu-Wakamatsu City, Fukushima 965-8580, Japan; E-Mail: sigeru-w@u-aizu.ac.jp. Received: 23 February 2010 / Accepted: 16 March 2010 / Published: 19 March 2010 Abstract: Generating functions play important roles in theory of orthogonal polynomials. In particular, it is important to consider generating functions that have symmetry. This paper is a survey on generating functions that define unitary operators. First, classical generating functions that define unitary operators are discussed. Next, group theoretical approach to generating functions that have unitarity are discussed. Keywords: Generating function; Orthogonal polynomial; Zonal spherical function Classification: MSC 43A90 22E30 33C45 33C50 46E20 1. Introduction This paper deals with generating functions that define unitary operators. Problems of this kind were discussed first by Bargmann [ 1]. He constructed a unitary operator given by an integral operator whose kernel is a generating function of the Hermite polynomials. He also gave a similar construction for the Laguerre polynomials without proof, and noticed as follows ([ 1], p.203). “It is worth noting that a similar interpretation may be given to other classical generating functions.” We turned our interest to the Gegenbauer polynomials which give the zonal spherical functions on the pair SOn, SOn−, and in [2] we showed that a similar construc