Refined estimates on the growth rate of Jacobi polynomials (Q865371)

Let \(P_{n}^{(\alpha,\beta)}(x)\) be the Jacobi polynomials of degree \(n\) and parameters \((\alpha,\,\beta)\). In this paper the author considers the problem of the decay rate for integer \(n\) of \(\theta^{k}\,P_{n}^{(k,\beta)}(1-2\theta^2)\) as \(n\) and \(k\) are simultaneously increased along lines in the \(kn\) plane. In particular, it is shown that for \(0<\theta<1\), \(\theta^{an}\,P_{n}^{(\alpha+an,\beta)}(1-2\theta^2)\) decays exponentially as \(n\rightarrow \infty\) when \(a>2\theta/(1-\theta)\). It is also shown that for \(-1<a<2\theta/(1-\theta)\), the decay of \(\theta^{an}\,P_{n}^{(\alpha+an,\beta)}(1-2\theta^2)\) is \(O(n^{-1/2})\).