HILBERT SPACE OF ANALYTIC FUNCTIONS ASSOCIATED WITH THE MODIFIED BESSEL FUNCTION AND RELATED ORTHOGONAL POLYNOMIALS

Abstract

Let μ be a probability measure on I ⊂ ℝ with finite moment of all orders and γα, c1, 2 be a probability measure on ℂ which is expressed by the modified Bessel function. In this paper, we shall first consider operators B-, B+, B° and express them in terms of bosonic creation and annihilation operators. Secondly, we shall construct the Hilbert space of analytic L2 functions with respect to γα, c1, 2, [Formula: see text]. It will be seen that the Sμ-transform under certain conditions is a unitary operator from L2 (I, μ) onto [Formula: see text]. Moreover, we shall give explicit examples including Laguerre, Meixner and Meixner–Pollaczek polynomials and explain how the Hilbert space [Formula: see text] and B-, B+, B° are related from the probabilistic point of view. It will also be given the relationship between the heat kernel on ℂ and γα, c1, 2 (subordination property).