A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations

Abstract

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the $\Gamma$-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of $\gamma_1(1/2)$, $\gamma_1(1/3)$, $\gamma_1(2/3)$, $\gamma_1(1/4)$, $\gamma_1(3/4)$, $\gamma_1(1/6)$, $\gamma_1(5/6)$, which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formul{\ae} involving the first generalized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the $\Gamma$-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form.