Solution to simple functional equations

Question

What is $\psi$ in functional equation: $$\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{n+x}}=1/2\,\Psi \left( 1/2+x/2 \right) -1/2\,\Psi \left( x/2 \right)?$$

Answer 1

Hint. Here $\psi$ is the digamma function defined as the logarithmic derivative of the $\Gamma$ function: $$ \psi(x):=\frac{d}{dx}\log \Gamma(x)=\frac{\Gamma'(x)}{\Gamma(x)}, \quad x>0. \tag1 $$ You may recall the Weierstrass infinite product representation of the Gamma function

$$ \Gamma(x) = \frac{e^{-\gamma x}}{x} \prod_{k=1}^{\infty} \frac{e^{\frac{x}{k}}}{1+\frac{x}{k}}, \quad x>0. \tag2 $$ Then, from $(1)$ and $(2)$, you obtain $$\begin{equation} \psi(x+1) = -\gamma + \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{x+k} \right), \quad x >-1, \tag3 \end{equation} $$ where $\gamma$ is the Euler-Mascheroni constant, and from $(3)$ you get $$\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{n+x}}=1/2\,\psi \left( 1/2+x/2 \right) -1/2\,\psi \left( x/2 \right).$$

Answer 2

$\Psi$ is the Digamma function.