Probability distributions with binomial moments

Abstract

Abstract. We prove that if p ≥ 1 and −1 ≤ r ≤ p − 1 then the binomial sequence np+r n , n = 0, 1, . . ., is positive definite and is the moment sequence of a probability measure ν(p, r), whose support is contained in 0, p p (p − 1) 1−p . If p > 1 is a rational number and −1 < r ≤ p − 1 then ν(p, r) is absolutely continuous and its density function V p,r can be expressed in terms of the Meijer G-function. In particular cases V p,r is an elementary function. We show that for p > 1 the measures ν(p, −1) and ν(p, 0) are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence np+r n is positive definite if and only if either p ≥ 1, −1 ≤ r ≤ p − 1 or p ≤ 0, p − 1 ≤ r ≤ 0. The measures corresponding to the latter case are reflections of the former ones.

Free Self-decomposability and Unimodality of the Fuss–Catalan Distributions

“…In order to check free self-decomposability of µ(p, r) we should check whether or not xW p−r,r (x)dx is unimodal with mode 0. (16). Since 0 < r ≤ min{p/2, p − 1}, we have p − r ≥ r. Hence k ′ p,r (x) ≥ 0 for x ∈ (0, ǫ), where ǫ > 0 is sufficiently small.…”

“…In particular, A k (2, 1) is the famous Catalan sequence, see [20]. It is known that the sequence A k (p, r) is positive definite if and only if either p ≥ 1, 0 < r ≤ p or p ≤ 0, p − 1 ≥ r or else if r = 0, see [15,16,17,9,14] for various proofs. The corresponding probability measure we will call the Fuss-Catalan distribution and denote µ(p, r), so that A k (p, r) = ∫ R x k µ(p, r)(dx).…”

“…Theorem 3.2. The Fuss-Catalan distribution µ(p, p) is freely infinitely divisible if and only if 1 ≤ p ≤ 2.Note that this case was overlooked in Corollary 7.1 in[16].…”

A note on Lee–Yang zeros in the negative half-plane

“…For q = 1, the m with support on a domain D t ⊆ [0, t t (t − 1) 1−t ]. Here we have t = q ∈ N, r = 0, and h q,0 (η) can be written in terms of the Meijer G-function [38]. This permits, for some q, to express the limiting density of the η α 's in terms of elementary functions.…”

Loop vertex expansion for higher-order interactions

“…The corresponding action is indeed a sum over single loops of arbitrary order decorated by trees. It is closely related to the generating function of the cumulants in the Gallavotti field-theoretic representation of classical dynamical systems [22], and it can be explicitly written in terms of the Fuss-Catalan [23] generating function of order p. Notice however that such functions cannot be expressed in terms of radicals of the initial fields for p > 4. Nevertheless Fuss-Catalan functions are shown rather easily to have bounded derivatives of all orders (see Theorem III.1 below).…”