OPDAM FUNCTIONS: PRODUCT FORMULA AND CONVOLUTION STRUCTURE IN DIMENSION

OPDAM FUNCTIONS: PRODUCT FORMULA AND CONVOLUTION STRUCTURE IN DIMENSION 1 J.-PH. ANKER, F. AYADI AND M. SIFI Abstract. Let G(?,?)? (x) be the eigenfunctions of the Dunkl-Cherednik oper- ator T (?,?) on R, with ? ≥ ? ≥ ? 12 . In this paper we express the product G(?,?)? (x)G (?,?) ? (y) as an integral in terms of G (?,?) ? (z) with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called ratio- nal limit, we recover the product formula for the Dunkl kernels proved in [13]. We then define and study a convolution structure associated to G(?,?)? . 1. Introduction The Opdam hypergeometric functions G(?,?)? on R are normalized eigenfunctions { T (?,?)G(?,?)? (x) = i?G (?,?) ? (x) G(?,?)? (0) = 1 of the differential-difference operator T (?,?)f(x) = f ?(x) + ( (2?+ 1) cothx+ (2? + 1) tanhx )f(x)? f(?x) 2 ? ?f(?x).

• jacobi-dunkl functions


• kunze-stein phenomenon


• associated convolution


• sinh


• jacobi functions


• cosh z


• called opdam- cherednik transform