Balls in Constrained Urns and Cantor-Like Sets

Abstract

Let $A_n(k)$ denote the number of different ways to distribute $k$ indistinguishable balls into $n$ constrained urns, with capacities $c_1,\cdots, c_n$. We consider the $normalized \ counting \ functions \ \varphi_n(x) = \gamma_nA_n(|\varrho x|)$, where $\varphi_n, \varrho_n > 0$ are appropriate constants such that supp$(\varphi_n) = [0,1]$ and $\int^1_0 \varphi_n (x)dx = 1$. It is shown here that, if $(c_n)_{n \in \mathbb N}$ is $asymptotically \ geometric \ with \ weight \ q > \frac{3}{2}$, i.e. if $q^{–n}c_n$ converges to some positive real number, then the functions $\varphi_n$ converge to some $C^{\infty}$-function $\varphi$ on $\mathbb R$. This function $\varphi$ is the unique solution of the integral equation $\varphi(x) = \frac{q}{q–1} \int^{qx}{qx–q+1} \varphi (t)dt$ satisfying supp $\varphi \in [0,1]$ and $\int^1_0 \varphi(t)dt = 1$. Moreover, if $q > 2$, it is shown that $\varphi$ is a polynomial on each interval outside a Cantor-like set in the interval [0, 1].