# Expressing a series involving the Riemann zeta function in terms of known functions

We have the series:

$$\sum_{n=1}^{\infty}\sin\left(\frac{\pi n}{2}\right)\frac{\zeta(n+1)}{(2\pi)^{n+1}}\frac{\Gamma(z)}{\Gamma(z-n)}\left[\psi^{0}(z-n)-\psi^{0}(z)\right]$$

Where $\psi^{0}(\cdot)$ is the digamma function, and $z$ is a complex parameter.

Is there a way to express this series in terms of other known functions ?

EDIT

if we write : $$f(z)=\frac{1}{2}\sum_{m=1}^{\infty}\binom{z-1}{2m}\frac{B_{2m}}{z-2m}$$ Where $B_{n}$ is the nth Bernoulli number, then our series is just : $$\frac{df(z)}{dz}$$

• Are you sure that the series even converges ? – Lucian Jul 30 '15 at 23:35

By making a few reductions the series can be seen in the form \begin{align} S = \sum_{n=1}^{\infty} \sin\left(\frac{\pi n}{2}\right) \, \frac{\zeta(n+1)}{(2\pi)^{n+1}} \, \frac{\Gamma(z)}{\Gamma(z-n)} \, \left[\psi^{0}(z-n)-\psi^{0}(z)\right] = \sum_{n=0}^{\infty} \frac{\zeta(2n+2) \, (1-z)_{2n+1}}{(2\pi)^{2n+2}} \, \left(\sum_{k=1}^{2n+1} \frac{1}{z-k}\right) \end{align} where $\Gamma(a) \, (a)_{n} = \Gamma(a+n)$ is Pochhammer's notation. The inner summation can be expressed in terms of Hurwitz zeta functions and is given by \begin{align} \sum_{k=1}^{2n+1} \frac{1}{z-k} = \zeta(1, z+1) - \zeta(1,z+2n). \end{align} One can define the two series \begin{align} f_{1}(a,t) &= \sum_{n=0}^{\infty} \zeta(2n+2) \, (a)_{2n+1} \, t^{n} \tag{1}\\ f_{2}(a,t) &= \sum_{n=0}^{\infty} \zeta(2n+2) \, \zeta(1,z+2n) \, (a)_{2n+1} \, t^{n} \tag{2} \end{align} for which \begin{align} S = \frac{1}{(2 \pi)^{2}} \left[ f_{1}\left(1-z, \frac{1}{2\pi}\right) - f_{2}\left(1-z, \frac{1}{2 \pi} \right) \right] \end{align}
• $$\frac{\Gamma(z)}{\Gamma(z-n)} \, \left[\psi^{0}(z-n)-\psi^{0}(z)\right]=\frac{d}{dz}\frac{\Gamma(z)}{\Gamma(z-n)}$$ it should make your expression much easier – Mohammad Al Jamal Jul 30 '15 at 20:16