Psi Function

$$\Psi(z,s,v)\equiv \sum_{n=0}^\infty {z^n\over (v+n)^s}$$

for $\vert z\vert<1$ and $v\not = 0, -1$, ... (Gradshteyn and Ryzhik 1980, pp. 1075-1076).

See also Hurwitz Zeta Function, Ramanujan Psi Sum, Theta Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.