Balls in constrained urns and Cantor-like sets
Summary: Let \(A_n(k)\) denote the number of different ways to distribute \(k\) indistinguishable balls into \(n\) constrained urns, with capacities \(c_1,\dots, c_n\). We consider the normalized counting functions \(\varphi_n(x)= \gamma_n A_n([\varrho_nx])\), where \(\varphi_n,\varrho_n>0\) are appropriate constants such that \(\text{supp}(\varphi_n)= [0,1]\) 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 \(q>{3\over 2}\), 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 \(\varphi(x)= {q\over q-1}\int^{qx}_{qx-q+1}\varphi(t) dt\) satisfying \(\text{supp }\varphi\subset[0,1]\) 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 \([0,1]\).
scientific article; zbMATH DE number 1270173