Proving Identity for Sum

I am tasked to show that the following identity holds: $$\sum_{n \geq 1}\frac{1}{n^2-z} = \frac{1-\pi \sqrt{z}cot(\pi\sqrt{z})}{2z}$$ for $z \in \mathbb{C} \setminus \mathbb{N}$

This is for a class in functional analysis using Gerald Teschl's "Topics in Linear and Nonlinear Functional Analysis."

The hint is to use the trace formula:

$$\int_0^1G(z, x, x) = \sum_{j \geq 0}\frac{1}{E_j-z}$$ where $G(z, x, x)$ is the Green's function, and $E_j$ are the eigenvalues of some operator.

Unfortunately I have no experience with Green's functions and so I'm note sure how to go about constructing one for this particular problem. Any help would be appreciated.

Solution 1:

Without using the hint.

Write $$\frac 1{n^2-z}=\frac{1}{2 \sqrt{z}}\Big[\frac{1}{n-\sqrt{z}}-\frac{1}{n+\sqrt{z}} \Big]$$ Now, using generalized harmonic numbers $$\sum_{n=1}^p \frac{1}{n-\sqrt{z}}=H_{p-\sqrt{z}}-H_{-\sqrt{z}}$$ $$\sum_{n=1}^p \frac{1}{n+\sqrt{z}}=H_{p+\sqrt{z}}-H_{\sqrt{z}}$$ $$H_{\sqrt{z}}-H_{-\sqrt{z}}=\frac{1}{\sqrt{z}}-\pi \cot \left(\pi \sqrt{z}\right)$$

For the other terms, using the asymptotics of generalized harmonic numbers $$H_{p-\sqrt{z}}-H_{p+\sqrt{z}}=-\frac{2 \sqrt{z}}{p}+\frac{\sqrt{z}}{p^2}+O\left(\frac{1}{p^3}\right)$$

$$\sum_{n=1}^p\frac 1{n^2-z}=\frac{1}{2 \sqrt{z}}\Big[\frac{1}{\sqrt{z}}-\pi \cot \left(\pi \sqrt{z}\right)-\frac{2 \sqrt{z}}{p}+O\left(\frac{1}{p^2}\right) \Big]$$ $$\sum_{n=1}^\infty\frac 1{n^2-z}=\frac{1-\pi \sqrt{z} \cot \left(\pi \sqrt{z}\right)}{2 z}$$