$\require{\physics}$
Hi,
I am wondering if it is possible to approximate the roots of the generalized Laguerre polynomial $L_n^{(\alpha)}(x)$ not with respect to $x$ but with respect to $n$, i.e. finding or approximating $\tilde{n}_{\alpha,x}$
such that $L_{\tilde{n}_{\alpha,x}}^{(\alpha)}(x) = 0 $.
To put it differently the goal would be to find the roots of the polynomial
$$
f(n,\alpha) =
\sum_{l=0}^{\infty}\,\sum_{j=0}^l \frac{(-1)^l\,x^{l}}{l!\,(l+\alpha) !}
\begin{bmatrix} l \\ j \end{bmatrix}
(-1)^{j}\, n^j \equiv
\frac{n!}{(n+\alpha)!} L^{(\alpha)}_n(x)
$$
where $\begin{bmatrix} l \\ j \end{bmatrix}$ is the unsigned Stirling number of first kind. The polynomial $f$ is explicitly written in terms of the powers of the order $n$ of the Laguerre polynomials. I am particularly interested in the case when $x\in\mathbb{R}_{+}$ and $\alpha\in\mathbb{N}$.
If someone can help me with this or can suggest some paper I would be very thankful!
Best Answer
I solved my problem and I think it might be interesting for other people, so here is how I did it.
Laguerre polynomials can be approximated in various ways but the best approximation in the regime that I am interested in, namely $\alpha\in\mathbb{N}$ and $x\in(0,1)$, is the asymptotic formula of Hilb's type (cf. G. Szegő, "Orthogonal polynomials", p. 199) $$ L^{(\alpha)}_{n}(x)\approx e^{x/2} \,\, x^{-\alpha /2} \,\, \left(n+\frac{\alpha +1}{2}\right)^{-\alpha /2} \,\, \frac{(n+\alpha)!}{n!} \,\, J_{\alpha}\!\left(2 \sqrt{x} \sqrt{n+\frac{\alpha +1}{2}}\right) $$ where $J_{\alpha}$ is the Bessel function. From this expression we notice that the zeros of the Laguerre polynomials with respect to the order $n$ are related to the zeros of the Bessel functions. Denote the latter ones as $\{j_{\alpha,m}\}_{m=1}$ with $j_{\alpha,1}$ being the least zero of $J_{\alpha}$, then $$ \tilde{n}_{\alpha,x,m} \approx \frac{\left(j_{\alpha ,m}\right){}^2-2 \alpha x - 2 x}{4 x}\,. $$ Since $\{j_{\alpha,m}\}_{m=1}$ can be numerically computed (there exist functions in Python and Mathematica to do this) or be read from tables, I consider the problem solved at least for the parameter regime that I am interested. Numerically, this approximation agrees very well with the actual roots of $L^{(\alpha)}_{n}(x)$.
Let me know if I am missing something of if there are other/better methods.