A Rational Approximant for the Digamma Function

TL;DR

The paper addresses the slow convergence of power-series representations for the digamma function by exploiting explicit truncation-error information to build an explicit rational approximant for $\mathcal{Z}(z)$. It introduces the $T_n^{(k)}$ transformation, derived from transforming the error term via a Dirichlet-series rearrangement, which yields a closed-form with a known transformation-error series and reproduces the leading poles. Numerical tests show that the new rational approximant performs comparably to Wynn's epsilon algorithm and is numerically stable, validating its utility for computing $\psi(1+z)$. The work suggests further enhancements, such as asymptotic expansions of the transformation error and Levin-type acceleration strategies, to improve efficiency for larger $k$.

Abstract

Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Padé approximants or other rational functions constructed from sequence transformations. However, neither Padé approximants nor sequence transformation utilize the information which is avaliable in the case of a special function -- all power series coefficients as well as the truncation errors are explicitly known -- in an optimal way. Thus, alternative rational approximants, which can profit from additional information of that kind, would be desirable. It is shown that in this way a rational approximant for the digamma function can be constructed which possesses a transformation error given by an explicitly known series expansion.