MRM-applicable orthogonal polynomials for certain hypergeometric functions

The multiplicative renormalization method (MRM) is intorduced to obtain generating functions of orthogonal polynomials of given probabil- ity measures. Complete lists of MRM-applicable measures for MRM-factors h(x) = e x and (1 i x) i• were obtained recently. On the other hand, it is known that gamma distributions have at least two types of MRM-factors h(x) = 0F1(i;•;x) and h(x) = 1F1(c;•;x). The usual MRM-factor e x is a special case of 1F1(c;•;x) when c = •. We flrst determine all MRM- applicable measures for h(x) = 0F1(i;•;x). Then we determine all possible MRM-factors of gamma distributions. 1. MRM-applicability of Orthogonal Polynomials A probability measure " on R with density f"(x) is said to be applicable to the multiplicative renormalization method for h(x) (or simply MRM-applicable), if there exists a suitable analytic function ‰(t) around t = 0 with ‰(0) = 0; r1 = ‰ 0 (0) 6= 0 such that ˆ(t;x) = h(‰(t)x) '(t) with '(t) = µ(‰(t)); µ(t) =