%0 Unpublished work
%T On the asymptotic behavior of Jacobi polynomials with varying parameters
%+ no affiliation
%+ Institut de Mathématiques de Marseille (I2M)
%A Szehr, Oleg
%A Zarouf, Rachid
%8 2016-05-07
%D 2016
%Z 1605.02509
%K Jacobi polynomials
%K integral representation
%K method of stationary phase
%K Laplace's method
%Z 2010 Mathematics Subject Classification: Primary: 33C45; Secondary: 30E15
%Z Mathematics [math]/Classical Analysis and ODEs [math.CA]
%Z Mathematics [math]/Complex Variables [math.CV]Preprints, Working Papers, ...
%X 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.
%G English
%2 https://hal.science/hal-01312436/document
%2 https://hal.science/hal-01312436/file/DRAFT_OS_RZ_6_Mai.pdf
%L hal-01312436
%U https://hal.science/hal-01312436
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ ESPE-AMU_PUBLICATIONS