The bounds for the error term of an asymptotic approximation of Jacobi polynomials

Fast and Accurate Computation of Gauss--Legendre and Gauss--Jacobi Quadrature Nodes and Weights

“…Unfortunately, there is no asymptotic expansion of the Legendre polynomials involving only elementary functions that is valid near x = ±1 [33]. The boundary asymptotic formula we use was obtained by Baratella and Gatteschi [4], based on the method described by Olver [34], and is an expansion in Bessel functions of the first kind. It is derived by considering the second order differential equation that are satisfied by Legendre polynomials, and is given by (3.16) where ρ = n + 1 2 and…”

Szegő’s conjecture on Lebesgue constants for Legendre series

“…It is this result that has led to the four-term asymptotic expansion of L n given in (1.10). Motivated by Theorem A, Baratella and Gatteschi [2] showed that Pi a '^(cos0) also has the Cherry-type approximation [3] given in Theorem B below, complete with an explicit error bound. Let As we shall see in this paper, it is this latter result which has led to the error estimate (1-14).…”

“…To prove (1.14), we shall make use of some recent results of Baratella and Gatteschi [2] concerning asymptotic approximations of Jacobi polynomials and their zeros. Although these results are in a sense refinements of the asymptotic approximations obtained by Frenzen and Wong [5], they are of quite different nature from those given in [5].…”

“…For the Lebesgue constant (1.4), we need a = 1 and P = 0, a case not included in Theorem B. Nevertheless, by a slight modification of the argument given in [2], we have the following corollaries. A precise estimate for the root 6* can be obtained as follows.…”

“…We would like to express our sincere thanks to Professor L. Gatteschi for presenting us a copy of Reference [2] prior to its publication, and to Mr. Tom Lang for carrying out the numerical computations of L n (1 < n < 50) given in §3. We would also like to thank Professor L. Lorch for providing us with some historical background to the problem.…”