In this reference (pg. 4) and few others a specific parameter in a recursion formula is solved by setting the determinant of an infinite matrix to $0$. In this precise case we have
$c_{n-1} - D_n c_n + c_{n+1} = 0 $,
where
$D_n = \frac{(\mu + i n)^2+a}{q}$.
In order to find $\mu$, people assume that
$$ \begin{vmatrix} & & \ddots & & & & & \\ \cdots & 0 & 1 & - D_{n+1} & 1 & 0 & \cdots & \\ & \cdots & 0& 1 & - D_n & 1 & 0 & \cdots \\ & & & & & \ddots & & \\ \end{vmatrix} = 0 $$
as if this were a trivial consequence from the finite-dimensional case. In fact I don't know why this has to be true, but my guess is that $\lim_{n \to \pm \infty } c_n = 0$ is supposed. Is this correct? Still I don't follow how from this condition you get that the determinant has to vanish, I would appreciate any insight into that.
Besides that, I would like to know how to calculate determinants as the one given here. In the article I linked before, it is expected some knowledge about the definition and properties of determinants of infinite matrices. I hope that someone would be able to show me how to calculate this determinant, at least in principle, as it looks fairly easy, since only one (the "main") diagonal contains values different from $0$ and $1$.
EDIT:
I tried to understand what is going on by saying that the determinant of the infinite matrix given by the Fibonacci recurrence should be also $0$, as it has a non-trivial solution:
$$ \begin{vmatrix} & & \ddots & & & & & \\ \cdots & 0 & 1 & 1 & -1 & 0 & \cdots & \\ & \cdots & 0& 1 & 1 & -1 & 0 & \cdots \\ & & & & & \ddots & & \\ \end{vmatrix} = 0 $$
but again I'm stuck, because by truncating this to an $n \times n$ matrix, the determinant is $F_{n+1}$, which goes to infinity for increasing $n$.
Moreover, any infinite matrix representation of the recursion $c_{n+1} = a c_n + b c_{n-1}$ should have determinant $0$ by using the same argument, which brings me back to my original concern: Why does the given recursion (at the top of this post) have solutions only for specific values of $\mu$?