Many discrete-time pure birth processes are symmetrical in the following sense: The probability that the process is in a fixed state is independent of the sequence of transitions inducing . This is always the case whenever a transition is a time-dependent or a state-dependent random variable or the product of such independent variables. We use this property in order to derive algebraic descriptions in terms of symmetric polynomials. Besides new solutions, our approach offers a uniform point of view on a large class of often considered distributions.
In this paper we provide an algebraic approach to the generalized Stirling numbers (GSN). By defining a group that contains the GSN, we obtain a unified interpretation for important combinatorial functions like the binomials, Stirling numbers, Gaussian polynomials. In particular we show that many GSN are products of others. We provide an explanation for the fact that many GSN appear as pairs and the inverse relations fulfilled by them. By introducing arbitrary boundary conditions, we show a Chu-Vandermonde type convolution formula for GSN.Using the group we demonstrate a solution to the problem of finding the connection constants between two sequences of polynomials with persistent roots.
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