I would like to produce an easily-interpretable explicit upper bound (i.e. no unspecified constants) for the function
$$
f(n) := L_n^{\left(-n-\frac{d}{2}\right)}\left(-\frac{1}{2}\right), \quad n,d \in \mathbb{N}
$$

with $d$ a fixed parameter. This bound needn't be especially sharp (I only need it to characterize $f(n)$'s growth rate up to exponential factors), but for my application I would like it to hold in the *nonasymptotic* regime.

I haven't been able to find much on nonasymptotic bounds for generalized Laguerre polynomials with negative argument (particularly for the case in which both parameters are negative and co-varying), but by applying the recurrence relation: $$ L_n^{(a)}(x) = \frac{\alpha + 1 - x}{n} L_{n-1}^{(\alpha + 1)}(x) - \frac{x}{n} L_{n-2}^{(\alpha + 2)}(x) $$ I have shown that the function $f(n)$ can be described by the following recurrence relation: $$ f(0) = 1 \\ f(1) = \frac{1 - d}{2} \\ f(n) = \left(\frac{3-d}{2n} - 1\right)f(n-1) + \frac{1}{2n} f(n-2), \quad n \ge 2 \\ $$ I'm hoping it might be possible to use this description of $f(n)$ to recover a closed-form expression for it, or failing that, at the very least use it to control f(n)'s growth rate. Given the simple form of the recurrence relation, I've been trying to solve it using generating functions, but haven't met with much success yet (although I am far from an expert in these techniques, so it's entirely possible that there's a much simpler/better way :-P).

Does anyone know of any uniform bounds for Laguerre polynomials that would be relevant for this application (i.e. with x = -1/2)? Or failing that, does anyone have a recommendation for a technique that could be used to either solve the recurrence or control its growth rate?

Thanks!