On the asymptotic behavior of Jacobi polynomials with first varying parameter

TL;DR

This paper analyzes the asymptotic behavior of Jacobi polynomials with varying parameters, providing new integral representations and detailed decay rate classifications depending on parameter ranges and integer conditions.

Contribution

Introduces a novel integral representation for Jacobi polynomials with varying parameters and thoroughly characterizes their asymptotic decay behavior across different parameter regimes.

Findings

Exponential decay in certain parameter ranges.

Polynomial decay of order n^{-1/2} in others.

Behavior depends on whether a linear combination of parameters is integer.

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.