An algebraic approach to Polya processes

P'olya processes are natural generalization of P'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\le 1\mathrm{/}2$; otherwise, it is called large).