Uniform Complexity of Approximating Hypergeometric Functions with Absolute Error

The approximation of the general hypergeometric function H(a;b;x) = pFq(a;b;x) to any specified absolute error bound is shown to be solvable. In other words, we provide an algorithm that is uniform in the hypergeometric parameters a = (a1, . . . , ap),b = (b1, . . . , bq). An explicit bound for the complexity of our algorithm is given when the input numbers are rational. We further address the problem of evaluating H when x is a “blackbox number”, i.e., x is represented by a procedure that returns an approximation of x to any specified absolute precision. This generalization allows us to extend our approximability results to most of the familiar transcendental functions of classical analysis that are derived from H. In particular, this solves the so-called Table Maker’s Dilemma for such functions. Our algorithm has been implemented in our open-source Core Library.