Non-central generalized q-factorial coefficients and q-Stirling numbers

AbstractThe qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r, at t=0, are thoroughly examined. These numbers for s→0 and s→∞ converge to the non-central q-Stirling numbers of the first and the second kind, respectively. Explicit expressions, recurrence relations, generating functions and other properties of these q-numbers are derived. Further, a sequence of Bernoulli trials is considered in which the conditional probability of success at the nth trial, given that k successes occur before that trial, varies geometrically with n and k. Then, the probability functions of the number of successes in n trials and the number of trials until the occurrence of the kth success are deduced in terms of the qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r

Normally ordered forms of powers of differential operators and their combinatorics

We investigate the combinatorics of the general formulas for the powers of the operator h∂k, where h is a central element of a ring and ∂ is a differential operator. This generalizes previous work on the powers of operators h∂. New formulas for the generalized Stirling numbers are obtained.Ministerio de Economía y competitividad MTM2016-75024-PJunta de Andalucía P12-FQM-2696Junta de Andalucía FQM–33