On the asymptotic behavior of Jacobi polynomials with varying parameters
Résumé
We study the large $n$ behavior of Jacobi polynomials with varying parameters $P_{n}^{(an+\alpha,\beta)}(1-2\lambda^{2})$ for $a>-1$ and $\lambda\in(0,\,1)$. This appears to be a well-studied topic in the literature but some of the published results are unnecessarily complicated or incorrect. The purpose of this paper is to provide a simple and clear discussion and to highlight some flaws in the existing literature. Our approach is based on a new representation for $P_{n}^{(an+\alpha,\beta)}(1-2\lambda^{2})$ 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. In summary 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 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 $\cO(n^{-1/2})$ 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 $\cO(n^{-1/3})$. Lastly we find that if $a\in(-1,\frac{-2\lambda}{1+\lambda})$ then $\lambda^{an}P_{n}^{(an+\alpha,\beta)}(1-2\lambda^{2})$ decays exponentially iff $an+\alpha$ is an integer and increases exponentially iff it is not.
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