Values of the polygamma functions at rational arguments

Abstract

Gauss in 1812, in his celebrated memoir on the hypergeometric series, presented a remarkable formula for the psi (or digamma) function, ψ(z), at rational arguments z, which can be expressed in terms of elementary functions. Davis in 1935 extended Gauss's result to the polygamma functions by using a known series representation of ψ(n)(z) in an elementary yet technical way. Kölbig in 1996, in his CERN technical report, also gave two extensions to ψ(n)(z) by using the series definition of polylogarithm function and the above-known series representation. Here we aim at deriving general formulae expressing ψ(n)(z) as rational arguments in terms of other functions, which will be obtained in two ways. In addition, several special cases are also considered and, as a by-product of our main results, we derive, in a simple and unified manner, all formulae given by Gauss, Davis and Kölbig. Finally, it should be noted that all our results, in view of the relationship between ψ(n)(z) and the Hurwitz zeta function, ζ(s, a), could be rewritten in the representation of ζ(s, a).