Quantum probabilistic approach to the Segal-Bargmann transform and infinitely divisible laws

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From the point of multiplicative renormalization method, we examined the Brenke-Chihara problem, which is to determine all orthogonal polynomials associated with the Brenke type generating functions. As a result, all of the Jacobi-Szego parameters and q-hypergeometric series representations for all MRM-factors are obtained, explicitly. Our paper was published from IDAQP, an international mathematics journal. We have begun to consider if there would exist q-Fourier or q-Segal-Bargmann transforms and associated convolution products in a meaningful way from the quantum(algebraic) probabilistic view point.