A stable recurrence for the incomplete gamma function with imaginary second argument

Even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction, for an imaginary second argument there can be a considerable loss of correct digits before stability sets in. We present an approach to compute the recurrence relation to full precision, also for small values of the arguments, when the first argument is negative and the second one is purely imaginary. Furthermore, the method we present is valid for the computation of solutions of general first order difference equations.