A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite-time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times, τi ∼ Erlang (gi, λi) , i = 1, 2, . . ., any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when τi ∼ Erlang (gi, λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for τi ∼ Erlang (1, λi) , i = 1, 2, . . ., i.e., when the inter-arrival times are non-identically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τi ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time non-ruin probability formulae
