MRM-applicable measures for the power function of the second order
Izumi Kubo, Hui-Hsiung Kuo, Suat Namli · Communications on Stochastic Analysis · 2011
The authors have studied multiplicative renormalization method (MRM) for generating functions of orthogonal polynomials.They determined all MRM-applicable measures for renormalizing functions h(x) = exp[x], h(x) = (1 -x) -κ , κ = 1/2 and 1.For the cases of h(x) = exp[x] MRM-applicable measures are Meixner class and very rich.For the case h(x) = (1 -x) -κ , κ = 1, MRM-applicable measures are rather rich.For the case κ = 1/2, MRMapplicable measures are only uniform measures on intervals.In this paper, they determine all MRM-applicable measures for h(x) = (1 -x) -2 , whose typical cases are three special beta distributions.