Polynomials of binomial type and compound Poisson processes

Let q_nand s_n, n ε N, respectively, be a set of polynomials of binomial type and a Sheffer set related to it, both having positive coefficients. Then q_n(x), x > 0 is connected with the probability that a compound Poisson process starting at zero is in state n at time τx andq_n(x)/q_n(1) is the probability generating function of the number of jumps of this process in 0, τ] given that it is in state n at time τ. The s_n admit similar interpretations when the initial distribution of the compound Poisson process is not concentrated at zero. The possible limits for n → ∞ ofq_n(x)/q_n(1)ands_n(x)/s_n(1) are studied.