[工程科技]Special functions and q-commuting variables.pdfVIP

Special functions and q-commuting variables Tom H. Koornwinder Department of Mathematics University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands email thk@wins.uva.nl 6 9 Abstract: This paper is mostly a survey, with a few new results. The first part deals with func- 9 tional equations for q-exponentials, q-binomials and q-logarithms in q-commuting variables and 1 more generally under q-Heisenberg relations. The second part discusses translation invariance of g Jackson integrals, q-Fourier transforms and the braided line. u A Last modified: August 26, 1996 6 2 Note: Report No. 1, Institut Mittag-Leffler, Djursholm, Sweden, 1995/96; q-alg/9608008; 2 to appear in “Special Functions, q-Series and Related Topics”, The Fields Institute Communications v Series. 8 0 0 8 1. Introduction 0 6 Identities for special functions often involve several variables, even if the special function itself 9 depends on only one variable. In general these variables are real or complex, so they commute with / each other. The theory of quantum groups has been quite successful in producing identities for g l q-special funcions, in particular addition formulas, see e.g. the survey in Koelink [21, Section 1] and a - further references given there. Although quantum groups themselves abound of non-commuting q variables satisfying certain relations, one usually does not find back a similar type of variables in : v the