An Interpolating Function for Power Series Coefficients and the Mellin Transform of the Sum

We consider a one-dimensional power series with a non-zero radius of convergence, centered at zero. A connection is established between the analytic continuation of the power series via coefficient interpolation and its Mellin transform. For rational functions, the Mellin transform can be represented as a product of gamma functions and an entire function of exponential type. Using an interpolating function, this class can be extended.