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Generating functions of orthogonal polynomials and Szegö-Jacobi parameters

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Asai N. , Kubo I. , Kuo H.-H.

Probability and Mathematical Statistics

2003

tom Vol. 23, Fasc. 2

273--291

In this paper, we present a more direct way to compute the Szegö-Jacobi parameters from a generating function than that in [5] and [6]. Our study is motivated by the notions of one-mode interacting Fock spaces defined in[1] and Segal-Bargmann transform associated with non-Gaussian probability measures introduced in [2]. Moreover, we examine the relationships between the representations of orthogonal polynomials in terms of differential or difference operators and our generating functions. The connections provide practical criteria to determine when functions of a certain form are orthogonal polynomials.

Interpolations of Bargmann type measures

100%

Asai N. , Krystek A. D. , Wojakowski Ł. J.

2016

tom Vol. 49, nr 3

293--301

In this paper, we shall discuss Bargmann type measures on C for several classes of probability measures on R. The unified interpolation expressions include not only the classical Bargmann measure and its q-deformation, but also their t-deformations and dilations. As a special case, we get conditions on existence and an explicit form of the Bargmann representation for the free Meixner family of probability measures.