The solution of the recurrence relation f_n(t)=a_n(t)f_n-1(t)-b_n(t)(∂/∂t)f_n-1(t)

Many linear recurrence relations for combinatorial numbers depending on two indices - like, e.g. the Stirling numbers - can be transformed into a sequence of linear differential equations (of first order) for the corresponding generating functions. In this paper, the most general sequence of such differential equations is considered where the coefficient functions are assumed to be analytic. Using an Ansatz of factoring the sought-for solution into the product of two functions each satisfying a particular associated differential equation, an explicit solution is derived. Some concrete examples are treated. Furthermore, first results for sequences of linear differential equations of second-order are presented and the difficulties of treating the higher order case within the above mentioned method are discussed.