# Eigenvalues of a specific Hankel matrix

I have an $$\frac{N}{2} \times \frac{N}{2}$$ matrix $$G$$ with entries given by

$$$$G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2},$$$$

where $$N$$ is an even integer. By the Fourier transformation the matrix elements can be expressed as

$$$$\begin{split} G_{ij} &= \mathrm{Im} \left[\sum_{n=0}^{N-1} \exp\left(\mathrm{i}\frac{\pi(2n+1)}{N}\left(i+j-\frac{3}{2}\right)\right) \right]\\ &= \sum_{n=0}^{N-1} \sin\left[\frac{\pi(2n+1)}{N}\left(i+j-\frac{3}{2}\right)\right]. \end{split}$$$$

Another way to write the entries is as an integral:

$$$$G_{ij} = \frac{1}{\pi} \int_0^\infty \frac{x^{\frac{i+j-3/2}{N}-1}}{1+x} \mathrm{d}x.$$$$

Is it possible to find a closed-form expression for the following quantities:

(i) the eigenvalues of $$G$$ or even its full eigendecomposition?

(ii) $$\mathrm{Tr}(G^k)$$, for $$k \in \mathbb{N}$$ (in order to determine, e.g., $$\mathrm{Tr}(\log(G))$$)?

• In other words, $$G_{ij} = \frac1{\sin(\frac{\pi}{N}(i+j-\frac32))}.$$ Feb 17 '18 at 5:16
• That is correct. The entries can also be written as $$G_{ij} = \mathrm{Im} \left[ \sum_{n=0}^{N-1} \exp\left(\mathrm{i} \frac{\pi(2n+1)}{N} (i+j-3/2)\right) \right]$$ My idea was to use the Fourier transformation to obtain the eigendecomposition.
– Marc
Feb 17 '18 at 16:04
• @Marc -- apologies, my answer was mistaken, I have deleted it. Nov 1 '19 at 8:59