Quasipower and hypergeometric series — construction and evaluation

Abstract

It is proved that some power series converging very slowly in a neighbourhood of the point 1 can be transformed intoquasipower series. The latter converge faster but are more complicated because they contain some hypergeometric series₂ F ₁. Standard methods of the values evaluation for needed hypergeometric series with the aid of recurrence relations are not sufficiently efficient for some variable values. Therefore a new method, formally similar to Levin's transforms, is proposed. More generally, this is a method of approximative evaluating of such a solution of an inhomogeneous recurrence relation of order one which has some particular asymptotic properties.

The efficacity of the proposed methods is analyzed in detail for Euler's dilogarithm. This is a typical function whose power series is approached with difficulties ifz≈1. In particular, its Padé approximants are sufficiently accurate only for, sayx∈[−1, 1/2]. Hermite-Padé approximation is more effective. Resulting irrational approximants generalize in some sense partial sums of the quasipower series introduced here.