Asymptotics of some ultra-spherical polynomials and their extrema (Q2458960)

Motivated by questions related to the preconditioning of spectral methods and proving order and consistency of the residual smoothing scheme, the author considers the asymptotics of the Legendre polynomial and its derivatives. The starting point is the integral representation of the Gegenbauer ultraspherical polynomial given by \[ C_N^{(\lambda)}(x)=A_{\lambda,N}\int_0^\pi(x+i\sqrt{1-x^2} \cos \phi)^N (\sin \phi)^{2\lambda-1}d\phi, \] where \[ A_{\lambda,N}=\frac{2^{1-2\lambda} \Gamma(N+2\lambda)}{N! (\Gamma(\lambda))^2}, \] for fixed \(\lambda\) and \(N\to\infty\). For \(\lambda=1/2, 3/2, \ldots\) the above polynomial reduces to the Legendre polynomial \(P_N(x)\) and its derivatives up to multiplicative constants. Let \(x=\cos (z/N)\) where \(z\) is real. The asymptotics of \(C_N^{(\lambda)}(\cos (z/N))\) when \(2\lambda-1\) is restricted to be an even integer are examined in the intermediate range \(z\in [\pi K, \pi\Lambda N]\), where \(K\) is a suitably defined integer and \(\Lambda\in (0,1/2)\). The standard method of stationary phase is not applicable in this case, since the phase is not of the form of \(N\) multiplied by a function of the integration variable. The author develops a stationary phase-like approach for dealing with this situation and shows that in the intermediate region \[ C_N^{(\lambda)}(\cos \frac{z}{N})\sim 2\surd\pi A_{\lambda,N} \text{ Re} \left(ie^{iz} \sum_{k=\lambda-\frac{1}{2}}^\infty \chi_N^{-k-\frac{1}{2}}Q_{k,\lambda}(\chi_N/N)\right) \] as \(N\rightarrow\infty\), where \(\chi_N=-iNe^{-iz/N}\sin (z/N)\) and \(Q_{k,\lambda}\) are polynomials of degree \(k\). Explicit representations of these polynomials are given for \(\lambda=1/2\), \(3/2\), \(5/2\) and \(7/2\) Asymptotic formulas for the zeros of the Legendre polynomial \(P_N(\cos(z/N))\) and its first derivative are obtained.