# Balls in Constrained Urns and Cantor-Like Sets

### Günther J. Wirsching

Katholische Universität Eichstätt, Germany

## Abstract

Let $A_{n}(k)$ denote the number of different ways to distribute $k$ indistinguishable balls into $n$ constrained urns, with capacities $c_{1},⋯,c_{n}$. We consider the $normalizedcountingfunctionsφ_{n}(x)=γ_{n}A_{n}(∣ϱx∣)$, where $φ_{n},ϱ_{n}>0$ are appropriate constants such that supp$(φ_{n})=[0,1]$ and $∫_{0}φ_{n}(x)dx=1$. It is shown here that, if $(c_{n})_{n∈N}$ is $asymptoticallygeometricwithweightq>23 $, i.e. if $q_{–n}c_{n}$ converges to some positive real number, then the functions $φ_{n}$ converge to some $C_{∞}$-function $φ$ on $R$. This function $φ$ is the unique solution of the integral equation $φ(x)=q–1q ∫_{qx}qx–q+1φ(t)dt$ satisfying supp $φ∈[0,1]$ and $∫_{0}φ(t)dt=1$. Moreover, if $q>2$, it is shown that $φ$ is a polynomial on each interval outside a Cantor-like set in the interval [0, 1].

## Cite this article

Günther J. Wirsching, Balls in Constrained Urns and Cantor-Like Sets. Z. Anal. Anwend. 17 (1998), no. 4, pp. 979–996

DOI 10.4171/ZAA/862