Beta distribution and associated Stirling numbers of the second kind

TL;DR

The paper studies $S_r(p,m)$ and derives a probabilistic representation via the $p-rm$-th moment of a sum of i.i.d. Beta$(1,r)$ variables. This connects combinatorics with probability and recovers the classical case $r=1$ with the Irwin–Hall distribution and the generating function $\sum_{p\ge m} S_1(p,m)t^p/p!=(e^t-1)^m/m!$. The authors develop sharp bounds for $\mathcal{M}_r(k,m)$ in three regimes (large $m$, large $k$, large $r$) and show asymptotics such as $S_r(p,m)\sim m^{p}/m!$ as $p\to\infty$ with $m,r$ fixed. Numerical experiments compare the new bounds to classical bounds for the Bell numbers and Stirling numbers, demonstrating tighter estimates in key parameter ranges.

Abstract

This article gives a formula for associated Stirling numbers of the second kind based on the moment of a sum of independent random variables having a beta distribution. From this formula we deduce, using probabilistic approaches, lower and upper bounds for these numbers.

Figures (5)

• Figure 1: This figure report $\ln(L(p,m))-\ln(L_{\rm rd}(p,m))$ as a function of $m$ (on the $x-$axis) and $p$ (on the $y-$axis). One may observe that for most integers the lower bound $L(p,m)$ is a better approximation of $S_1(p,m)$ than $L_{\rm rd}(p,m)$ (as $\ln(L(p,m))-\ln(L_{\rm rd}(p,m))>0$).
• Figure 2: This figure report $\ln(S_1(p,m))-\ln(L(p,m))$ (on the left) and $\ln(U(p,m))-\ln(S_1(p,m))$ (on the right) as a function of $m$ and $p$. These numerical experiments comply with Propositions \ref{['prop:bounds_Jansen']} and \ref{['prop:asym']} since both lower and upper bounds accurately approximate $S_1(p,m)$ when $p$ is large and $m$ is small or when $m$ is large and $p-m$ is small.
• Figure 3: This figure report $\ln(U_{bt}(p))-\ln(U(p))$ as a function of $p$. One may observe that when $p\ge 13$, $U(p)$ is more accurate upper bound for $S_1(p,m)$ than $U_{\rm bt}(p)$ (as $\ln(U_{bt}(p))-\ln(U(p))>0$ for $p\ge 13$).
• Figure 4: This figure report $U(p)/B(p)$ as a function of $p$. One may observe that $U(p)/B(p)$ is approximately equal to $e$ when $p$ is large.
• Figure 5: Illustration of the inequality given in Lemma \ref{['lem:c']}.

Theorems & Definitions (16)

• Theorem 2.1
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 5.1
• proof
• proof
• Lemma 5.2
• proof
• proof