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 $normalizedcountingfunctions{\phi }_n(x)={\gamma }_n{A}_n(|\varrho x|)$, where ${\phi }_n,{\varrho }_n>0$ are appropriate constants such that supp$({\phi }_n)=[0,1]$ and ${\int }_0^1{\phi }_n(x)dx=1$. It is shown here that, if $(c_n{)}_{n\in \mathbb{N}}$ is $asymptoticallygeometricwithweightq>\frac{3}{2}$, i.e. if $q^{–n}{c}_n$ converges to some positive real number, then the functions ${\phi }_n$ converge to some $C^{\infty }$-function $\phi$ on $\mathbb{R}$. This function $\phi$ is the unique solution of the integral equation $\phi (x)=\frac{q}{q–1}{\int }^{qx}qx–q+1\phi (t)dt$ satisfying supp $\phi \in [0,1]$ and ${\int }_0^1\phi (t)dt=1$. Moreover, if $q>2$, it is shown that $\phi$ is a polynomial on each interval outside a Cantor-like set in the interval [0, 1].