Computation of special functions by Padé approximants with orthogonal polynomial denominators (Q1911445)

This paper deals with three large families of special functions having continued fraction (Padé approximant) representations with three key properties: (a) the approximants converge rapidly; (b) the backward recurrence algorithm for computing the approximants is reasonably stable, and (c) one can compute the exact number of significant decimal digits of any given approximant. The three families of continued fractions are Stieltjes fractions, real \(J\)-fractions and non-negative \(T\)-fractions with denominators as orthogonal polynomials or Laurent polynomials. Orthogonality of these denominators plays an important role in the analysis of errors due to numerical round off and truncation of infinite sequences of approximants. From the rigorous error bounds described one can determine the exact number of significant decimal digits contained in the approximation of a given function value. Few results from computational experiments are given to illustrate the methods.