MRM-FACTORS FOR THE PROBABILITY MEASURES IN THE MEIXNER CLASS

Abstract

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) [Formula: see text] for gamma distribution γκ. (2) [Formula: see text] for gamma distribution γκ. (3) [Formula: see text] for standard Gaussian distribution. (4) [Formula: see text] for shifted negative binomial distribution. σβ NegBin (κ,p) with κ = 2, β = 1, for Meixner distribution Mκ,η with κ = 2 and for gamma distribution γκ with κ = 2, which is a special case of (2) with c = 1.