A Rational Approximant for the Digamma Function

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 Pad\'{e} approximants or other rational functions constructed from sequence transformations. However, neither Pad\'{e} 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.