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, psi(z), at rational arguments z, which can be expressed in terms of elementary functions. Davis in 1935 extended Gausss result to the polygamma functions psi((n))(z)(n is an element of N) by using a known series representation of psi((n))(z) in an elementary yet technical way. Kolbig in 1996, in his CERN technical report, also gave two extensions to psi((n))(z) by using the series definition of polylogarithm function and the above-known series representation. Here we aim at deriving general formulae expressing psi((n))(z) (n is an element of N-0) 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 Kolbig. Finally, it should be noted that all our results, in view of the relationship between.( n)( z) and the Hurwitz zeta function, zeta(s, a), could be rewritten in the representation of zeta(s, a).