  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.
