Asymptotic Identities for Jacobi Polynomials via Spectral Geometry of Rank-One Symmetric Spaces
Ankita Sharma
TL;DR
This work derives asymptotic identities for Jacobi polynomials by exploiting the spectral geometry of compact rank-one symmetric spaces, where radial Laplace-Beltrami eigenfunctions are expressed via Jacobi polynomials $\mathcal{P}_\ell^{(\alpha,\beta)}$ with $\alpha=(p+q-1)/2$ and $\beta=(q-1)/2$. By mapping radial geometry through $\Pi(u)=\cos(2\omega \rho(u,e))$ and applying results of Minakshisundaram-Pleijel and Zelditch, the author obtains a precise asymptotic identity for Jacobi polynomials: $$\lim_{m\to\infty} \frac{1}{m} \sum_{\ell=0}^m \frac{(2\ell+\alpha+\beta+1)\Gamma(\ell+1)\Gamma(\ell+\alpha+\beta+1)}{\Gamma(\ell+\alpha+1)\Gamma(\ell+\beta+1)} \left|\mathcal{P}_\ell^{(\alpha,\beta)}(x)\right|^2 = \frac{2^{\alpha+\beta+1}}{\pi (1-x)^{(2\alpha+1)/2} (1+x)^{(2\beta+1)/2}}$$, with a parameter identification $p=2(\alpha-\beta)$, $q=2\beta+1$. The paper also establishes a $k$-codimension endpoint identity at $x=-1$ for non-spherical spaces, relating sums of squared Jacobi polynomials to $2/k$ and providing explicit constants for projective spaces, highlighting a deep link between Jacobi polynomials and the Fourier-analytic structure of measures on submanifolds. These results contribute new asymptotic formulas in the theory of special functions with potential implications for spectral geometry and harmonic analysis on symmetric spaces.
Abstract
Radial eigenfunctions of the Laplace-Beltrami operator on compact rank-one symmetric spaces may be expressed in terms of Jacobi polynomials. We use this fact to prove an identity for Jacobi polynomials which is inspired by results of Minakshisundaram-Pleijel and Zelditch on the Fourier coefficients of a smooth measure supported on a compact submanifold of a compact Riemannian manifold.