arXiv subjects
Publications and source records attributed to Victor H. Moll.
We study the properties of a function $ψ(z, q)$ (the generalized polygamma function), intimately connected with the Hurwitz zeta function and defined for complex values of the variables $z$ and $q$, which is entire in the variable $z$ and reduces to the usual polygamma function $ψ^{(m)}(q)$ for $z$ a non-negative integer $m$, and to the balanced negapolygamma function $ψ^{(-m)}(q)$ for $z$ a negative integer $-m$.
We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin (πq), ln Gamma(q) and the polygamma functions. Many of the results are most conveniently formulated in terms of a family of functions A_k(q):=k zeta'(1-k,q), where k is a natural number, and a family of polygamma functions of negative order, whose properties we study in some detail.
We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, log Gamma(q) and log sin(q).