I am trying to find the simplest way to get an expression of the determinant of the following infinite matrix as m tends to infinity.
$ \left[\begin{array}{cccccc} 1 & a_{1} & 0 & \cdots & 0 & 0 \\ \beta_{1} & 1 & a_{2} & \cdots & 0 & 0\\ 0 & \beta_{2} & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \cdots & 1 & a_{m}\\ 0 & 0 & 0 & \cdots & \beta_{m} & 1 \end{array} \right]$
I have considered using both the Leibniz formula or the Laplace formula. Leibniz formula required considering sums over permutations which I was hoping to avoid and Laplace formula seems somewhat recursive even though I have only 2-3 elements in each row it. Is there a simpler solution to this problem which I am overlooking ?
Edit: I am just after an algebraic expansion of the determinant into and infinite series of some kind ?
Any help would be much appreciated.