# Roots of reverse Bessel polynomials define asymptotically a lattice

(Title changed after comment from Nemo).

Set $$P_n=\sum_{k=0}^n\frac{(n+k)!x^{n-k}}{k!(n-k)!}\ .$$ (The polynomial $$\theta_n(x)=2^{-n}P_n(2x)$$ is the so-called reverse Bessel polynomial, see comment from Nemo.)

Let $$\mathcal R_n$$ denote the set of all roots of $$P_n$$. The union $$\bigcup_{n=1}^\infty \mathcal R_n$$ intersects a small window centered at a very large negative real number almost in a sort of limit-lattice $$\Lambda$$ . (The sets $$\mathcal R_n$$ are close to parabolas opening to the right.) The real generator of $$\Lambda$$ (defined as $$\lim_{n\rightarrow\infty}r_{2n-1}-r_{2n+1}$$ where $$r_{2n-1}=\mathcal R_{2n-1}\cap \mathbb R$$ is the real root of $$P_{2n-1}$$) seems to be given by $$4\lambda$$ where $$\lambda=0.662743419349\ldots$$ is the real solution of $$\lambda e^{\sqrt{1+\lambda^2}}=1+\sqrt{1+\lambda^2}\ .$$ (The constant $$\lambda$$, sometimes called "Lagrange's constant", is apparently related to celestial mechanics.)

One seems to have $$\lim_{n\rightarrow \infty}\frac{r_{2n-1}}{4n-1}=-\lambda\ .$$ Experimentally, there exists seemingly a sequence \begin{align*} a_2&=0.05490008226\ldots,\\ a_4&=-0.01648163\ldots,\\ a_6&=0.021761\ldots,\\ &\ \ \vdots \end{align*} such that we have asymptotic expansions $$-\frac{r_{2n-1}}{(4n-1)\lambda}=1+\sum_{n=1}^N\frac{a_{2n}}{(4n-1)^{2n}}+O(n^{-2N-2})$$ involving the unique real root $$r_{2n-1}=\mathcal R_{2n-1}\cap\mathbb R$$ of $$P_{2n-1}$$.

I noticed nothing special for "the" other generator of $$\Lambda$$.

Are there any explanations for this curious behaviour?

(Added after comment by Nemo: Since Bessel polynomials are orthogonal polynomials for a suitable measure on the complex unit-circle and are useful for filters, an explanation involves probably properties of Bessel polynomials.)

• en.wikipedia.org/wiki/Bessel_polynomials
– Nemo
Jul 31, 2021 at 19:32
• Nemo, thanks for this useful information. I have changed the title of the question in order to reflect the link with reverse Bessel polynomials. Jul 31, 2021 at 20:16

Regarding asymptotics of real zero $$\alpha_n$$ ($$2/r_{2n-1}$$ in OP's notation) of generalized Bessel polynomials (with an extra parameter $$a$$ specialized in the current case to $$a=2$$), see M.G. de Bruin, E.B. Saff’ and R.S. Varga, On the zeros of generalized Bessel polynomials, Proceedings A, 84 (l), 20, (1981). In theorem $$7.3$$ they state Obviously, $$2/\alpha_n$$ will be the real zero of the inverse polynomial. Here $$\hat{r}=\lambda$$ is Laplace limit in OP's notation, also changing $$n\to 2n-1$$ (since the numeration used in the paper is different), and noting $$K(\hat{r};2)=\hat{r}$$, we recover OP's formula.
To account for the zeros of the inverse polynomial one takes the conformal map $$2/z$$.