On the asymptotic behavior of Jacobi polynomials with first varying parameter

Abstract

We investigate the large $n$ behavior of Jacobi polynomials with varying parameters $P_n^{(an+\alpha ,bn+\beta )}(1-2{\lambda }^2)$ for $a,b>-1$ and $\lambda \in (0,1)$. This is a well-studied topic in the literature but some of the published results appear to be discordant. To address this issue we provide an in-depth investigation of the case $b=0$, which is most relevant for our applications. Our approach is based on a new and surprisingly simple representation of $P_n^{(an+\alpha ,\beta )}(1-2{\lambda }^2),a>-1$ in terms of two integrals. The integrals' asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. As a consequence we prove that if $a\in (\frac{2\lambda }{1-\lambda },\infty )$ then ${\lambda }^{an}{P}_n^{(an+\alpha ,\beta )}(1-2{\lambda }^2)$ shows exponential decay and we derive simple exponential upper bounds in this region. If $a\in (\frac{-2\lambda }{1+\lambda },\frac{2\lambda }{1-\lambda })$ then the decay of ${\lambda }^{an}{P}_n^{(an+\alpha ,\beta )}(1-2{\lambda }^2)$ is $\mathcal{O}(n^{-1/2})$ and if $a\in \{\frac{-2\lambda }{1+\lambda },\frac{2\lambda }{1-\lambda }\}$ then ${\lambda }^{an}{P}_n^{(an+\alpha ,\beta )}(1-2{\lambda }^2)$ decays as $\mathcal{O}(n^{-1/3})$. A new phenomenon occurs in the parameter range $a\in (-1,\frac{-2\lambda }{1+\lambda })$, where we find that the behavior depends on whether or not $an+\alpha$ is an integer: If $a\in (-1,\frac{-2\lambda }{1+\lambda })$ and $an+\alpha$ is an integer then ${\lambda }^{an}{P}_n^{(an+\alpha ,\beta )}(1-2{\lambda }^2)$ decays exponentially. If $a\in (-1,\frac{-2\lambda }{1+\lambda })$ and $an+\alpha$ is not an integer then ${\lambda }^{an}{P}_n^{(an+\alpha ,\beta )}(1-2{\lambda }^2)$ may increase exponentially depending on the proximity of the sequence $(an+\alpha {)}_n$ to integers.

Revision of 2016 preprint with more technical details and numerical experiments

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