Series involving log

Question

Let $\lambda >0$ and $a>0$. I want to evaluate : $$\sum_{\ell=2}^{\infty}\log\left(\frac{ \ell^4-\lambda\ell^2-1+\lambda}{a+\ell^4-\lambda\ell^2+1-\lambda}\right)\\ $$

I am particularly interested by tracking $\lambda$ and $a$.

Answer 1

Hint. By the Weierstrass product for the sine and cosine functions $$ \prod_{n\geq 1}\left(1+\frac{1}{n^2+A^2}\right)=\frac{A\sinh\left(\pi\sqrt{1+A^2}\right)}{\sqrt{1+A^2}\sinh(\pi A)} \tag{1}$$ hence you just have to factor a biquadratic polynomial as $(n^2+A^2)(n^2+B^2)$, apply $(1)$, reindex and take the logarithm.