Asymptotics of an entire function with real zeroes on the real line

Question

Define $ F(x) := A x^m e^{B x} \prod_{k \geqslant 1} (1 - x/\alpha_k)e^{x/\alpha_k} $ where we suppose that $ \alpha_k \in \mathbb{R} $. This function is defined on the whole complex plane and is entire under suitable conditions. I am interested in its asymptotic behaviour when $ x \to \pm \infty $, $ x \in \mathbb{R} $ and in particular the link between these asymptotics and the properties of the zeroes.

My case study is the Airy function that has such a product form and that is equivalent to $ C e^{-\frac{2}{3} x^{3/2} } x^{-1/4} $ when $ x \to +\infty $ and $ C'x^{-1/4} \cos \phi(x) $ for a certain $ \phi $ when $ x \to -\infty $. The zeroes are known to satisfy some asymptotics of the type $ |\alpha_k| \sim C'' k^{2/3} $ when $ k \to +\infty $ and my guess would be that this $ 2/3 $ has something to do with the $ 3/2 $ in the exponential.

My true motivation is the integrability on $ \mathbb{R} $ of $ |F(x)|^p $ with $ p \geq 1 $ when the zeroes are real. I am in particular looking for a condition on the zeroes that would ensure the function $ F $ to be in $ L^p(\mathbb{R}) $. Is there a general theory studying this, with some nice reference ?

Answer 1

Asymptotics for $|\alpha_k|$ is not sufficient to make conclusions about asymptotics of $f$. You need to know separately, the asymptotics on the positive and negative ray. Let us enumerate the roots $\alpha_k$ by all integers for convenience, and suppose that $\alpha_k\sim ak^{2/3},\; k\to +\infty$ and $\alpha_k\sim b(-k)^{2/3}, k\to-\infty$. Then there is a formula for asymptotics of $\log|f|$ of the form $$\log|f(x)|\sim h_1x^{3/2},\quad \log|f(-x)|\sim h_2|x|^{3/2},$$ as $ x\to+\infty$ outside a small exceptional set near the zeros.

Here $h_1$ and $h_2$ are constants which can be explicitly computed in terms of $a$ and $b$. See, for example, B. Levin, Distribution of zeros of entire functions, AMS, 1980, Chap. II. The condition $h_1<0,h_1<0$ is then sufficient for $f\in L^p$.

Actually it is easy to see that when both $a,b$ are positive, then both $h_j<0$, so $f\in L^p$ for all $p$.

The necessary conditions are of course $h_1\leq 0$ and $h_2\leq 0$. But when $h_j=0$ for some $j$, (or one of the $a,b$ $=0$), as is the case for the Airy function, the question becomes more subtle, and more precise asymptotics of zeros is needed to conclude $f\in L^p$.

If this is really what you need, I can compute $h_1$ and $h_2$ for you.

Remark. Your conditions imply that the Phragmén-Lindelöf indicator is $$h_f(\theta)=A\sin((3/2)(|\theta|-\alpha)),\quad |\theta| \leq\pi,$$ where $A>0$ and $0\leq\alpha\leq \pi/3$. This makes my statement $h_j<0$ evident.