The calculus of finite differences gives the formal sum \[ f'(0)=\Delta (0)-\Delta '(0)+\sum^{\infty}_{k=2}\frac{(-1)^ k}{k!}B_ k\Delta^{(k)}(0), \] where \(\Delta (t)=f(t+1)-f(t)\) and the \(B_ k\) are Bernoulli numbers. The author applies this to the function \(f(t)=\zeta (s,a+t)\), where \(\zeta\) (s,a) is the Hurwitz zeta function, and derives asymptotic series for \(\zeta\) (s,a), its derivative with respect to a, and related functions. Explicit formulas are also given when \(s=0\) or a negative integer. Some results are new and some are simpler derivations of known formulas.