PROBABILITY MEASURES ON ℂ ARISING FROM THE JACOBI–SZEGÖ PARAMETERS FOR CONTINUOUS DUAL HAHN POLYNOMIALS

Abstract

In this paper, we shall construct probability measures on ℂ, which can be expressed by the Mellin convolution product of the modified Bessel functions. Our measures on ℂ are related to the Jacobi–Szegö parameters for the continuous dual Hahn polynomials Sn(x2, a, b, c) under the special choice of parameters a, b, c. This paper contains new materials which go further than our previous results in Ref.7. The most interesting thing is that the Mellin convolution of two modified Bessel functions can again be expressed in terms of the modified Bessel functions, by choosing parameters a, b, c appropriately. The origin of our research in this direction goes back to the Bargmann–Fock representation of the classical non-Gaussian random variables.5–8 Our results would have a potential to be related to the higher powers of creation and annihilation operators acting on a new class of interacting Fock spaces.