MRM-factors for the probability measures in the meixner class

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It is known that the gamma distribution γκ is MRM-applicable for h(x) = ex and for some hypergeometric functions also. We are interested in the problem to determine all possible MRM-factors of probability measures which are MRM-applicable for ex. We may say that the measures are in Meixner class. Such typical measures are Gaussian, Poisson, gamma, negative binomial and Meixner distributions and others are obtained from their modifications by affine transforms. We will give the complete list of MRM-factors different from ex up to trivial deformation: (1) h̃(x) = 0F1(-;κx) for gamma distribution γκ. (2) h̃(x) = 1F1(c;κ x) for gamma distribution γκ. (3) h̃(x) = 1F1(c/2;1/2;-x2) + 1F 1(c+1/2;3/2;-x2), x for standard Gaussian distribution. (4) h̃(x) = 1F1(1;2;x) = 1/x(ex-1) for shifted negative binomial distribution. σβNegBin(κ, p) with κ = 2, β = 1, for Meixner distribution M κ,η with κ = 2 and for gamma distribution γκ with κ = 2, which is a special case of (2) with c = 1. © 2010 World Scientific Publishing Company.

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Infinite Dimensional Analysis, Quantum Probability and Related Topics

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