We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series in hypergeometric functions on each side. These equalities capture basic analytic properties of the Hurwitz zeta function like meromorphic extension and the functional equation. We give rates of convergence for the involved series. As a special case we find that the exponential rate of convergence for the deformation of the log-sine to the fractional part in the imaginary component is carried over to these equalities.
Linear Operators, the Hurwitz Zeta Function and Dirichlet $L$-Functions
C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function
Zeros of the Hurwitz zeta function in the interval (0,1)
![QR code for arXiv:2009.07134 [math.NT]](http://oss.arxitics.com/images/favicon.svg)