Asymptotics bound on Jacobi polynomials in the complex plane and for large $n$

Question

Dear mathematicians and theoretical physicists,

I am a theoretical physicist and I am bothering to you since I need to know some asymptotic and analytical properties of Jacobi polynomials $P^{(A,B)}_n(z)$ (where the discrete index $n$ is related to the angular momentum and, in "physical variables", $z=\cos\Theta$) in the complex plane as well as for large $n$.

Before writing down my two questions ($Q_1$ and $Q_2$), let me explain a simpler case in which the answer is known for the Legendre polynomial $P_n(z)$. Namely, in a simpler (rather standard) problem it would enter the Legendre function of order $n$: $P_n(z)$ where, in Physical coordinates, $z=\cos(\Theta)$ and $n$ is the "orbital angular momentum". Two properties which are of great help in the standard case are bounds on the Legendre function when it is analytically continued to the complex $z$-plane and when $n$ is very large. The properties which are very useful are:

• The first property has to do with a bound on the Legendre function as order $n \to \infty$): $$| P_n(\cos \Theta) | < n^{-1/2}e^{ n | \text{Im }\Theta | }$$ where $| x |$ is the absolute value of $x$. So when $n$ is very large, one gets an exponential bound in terms of the order $n$ of the Jacobi polynomial and the (absolute value of) the imaginary part of $\Theta$ (as well as the square root of 1/$n$).

• The second has to do with the behavior of $P_n(z)$ as $|z|\to\infty$: $P_n(z) \sim c_n z^n$

I would really need similar properties for $P^{(-q-p,-q+p)}_n(z)$ where $p$ is an integer and $q$ can be both integer and half-integer (the most important situation for me being $q$=1/2 ). Namely:

• Is there a bound, when $n \to \infty$ , on $P^{(-q-p,-q+p)}_n(z)$ such as $$| P^{(-q-p,-q+p)}_n(\cos \Theta) | < n^{-1/2}e^{n|\text{Im }\Theta|} ?$$
• Is there a bound on $P^{(-q-p,-q+p)}_n(z)$ as $|z|\to\infty$ such as $P^{(-q-p,-q+p)}_n(z) \sim c^{(p,q)}_n z^n$ ? If yes, how does $c^{(p,q)}_n$ behave asymptotically?

I would really appreciate if you could help me even with some references (which a physicist is able to understand).

Answer 1

For your second question, by the second formula under "Definitions" on the wikipedia page, the term with the largest power $n$ in $P_n^{(\alpha,\beta)}(z)$ is

$$ \frac{\Gamma(\alpha + \beta + 2n + 1)}{n! \Gamma(\alpha + \beta + n + 1)} \left(\frac{z-1}{2}\right)^n \sim \frac{\Gamma(\alpha + \beta + 2n + 1)}{2^n n! \Gamma(\alpha + \beta + n + 1)} z^n $$

Taking $\alpha = -q-p$ and $\beta = -q+p$ as in your question, we obtain

$$ P_n^{(-q-p,-q+p)}(z) \sim \frac{\Gamma(-2q + 2n + 1)}{2^n n! \Gamma(-2q + n + 1)} z^n $$

as $|z| \to \infty$. Stirling's formula tells us that the coefficients behave asymptotically like

$$ \frac{\Gamma(-2q + 2n + 1)}{2^n n!\Gamma(-2q + n + 1)} \sim 2^{n-2q} \sqrt{\frac{1}{\pi n}} $$ as $n \to \infty$.