Balanced polygamma function
Introduction In mathematics, the '''generalized polygamma function''' or '''balanced negapolygamma function''' is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
Definition The generalized polygamma function is defined as follows::     or alternatively,: where  is the polygamma function and , is the Hurwitz zeta function.The function is balanced, in that it satisfies the conditions:.
Relations Several special functions can be expressed in terms of generalized polygamma function. :where  are the Bernoulli polynomials:where  is the -function and  is the Glaisher constant.
Special values The balanced polygamma function can be expressed in a closed form at certain points (where  is the Glaisher constant and  is the Catalan constant)::
References