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A rational approximant for the digamma function. (English) Zbl 1045.33005
Summary: 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 Padé approximants or other rational functions constructed from sequence transformations instead. 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.
MSC:
33B15Gamma, beta and polygamma functions
41A20Approximation by rational functions
65D20Computation of special functions, construction of tables