Abstract
We present a geometric approach to the asymptotics of the Legendre polynomials Pk,n+1, based on the Szegö kernel of the Fermat quadric hypersurface, leading to complete asymptotic expansions holding on expanding subintervals of [-1,1].
Highlights
The search for asymptotic expansions and approximations of special functions is a very classical vein of research and is of great relevance in pure mathematics, in numerical analysis, mathematical physics, and the applied sciences.The goal of this paper is to develop a geometric approach to the asymptotics of the Legendre polynomial Pk,n+1(t) for k → +∞, with t = cos(θ) ∈ [−1, 1] and n ≥ 1 fixed; as is well-known, Pk,n+1(t) is the restriction to Sn of the Legendre harmonic, expressed in polar coordinates on the sphere
We present a geometric approach to the asymptotics of the Legendre polynomials Pk,n+1, based on the Szegokernel of the Fermat quadric hypersurface, leading to complete asymptotic expansions holding on expanding subintervals of [−1, 1]
We obtain an asymptotic expansion holding on expanding subintervals of [−1, 1], rather than on fixed subintervals of the form [−1 + δ, 1 − δ] for some given δ > 0, as one typically finds in the literature
Summary
The search for asymptotic expansions and approximations of special functions is a very classical vein of research and is of great relevance in pure mathematics, in numerical analysis, mathematical physics, and the applied sciences (see, for instance, with no pretence of completion [1,2,3,4]). In recent years, a considerable amount of work and attention has been devoted to algebro-geometric Szegokernel asymptotics, which have played a fundamental role in complex geometry It seems per se very natural and interesting to illustrate the important conceptual juncture between spherical harmonics and Szegokernels, by revisiting classical results on Legendre asymptotics in view of these recent developments. For n = 2, we obtain the formula of Laplace (cfr [12], Section 4.6; [4], (8.01) of Ch. 4; [13], Theorem 8.21.2), but as a full asymptotic expansion holding uniformly on expanding subintervals converging to [−1, 1] at a controlled rate, as above: Pk,. We obtain for Pk(n/2−1,n/2−1)(cos(θ)) as asymptotic expansion with leading order term. 1 cos (θ/2)(n−1)/2 sin (θ/2)(n−1)/2 cos (αk,n (θ)) , in agreement with (10) on page 198 of [14]
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