On Jacobi polynomials ${\mathscr {P}}_{k}^{(\alpha ,\beta )}$ and coefficients $c_{j}^{\ell }(\alpha ,\beta )$ $\left( k\ge 0, \ell =5,6; 1\le j\le \ell ; \alpha ,\beta > -1\right)$

Abstract

In their paper, Awonusika and Taheri proved a spectral identity that relates the even ($2\ell$)th, $\ell \ge 1$, derivatives of Jacobi polynomials $P_{k}^{(\alpha ,\beta )}(\cos \theta )$, $k\ge 0; \alpha ,\beta >-1$ (evaluated at $\theta =0$), to the $\ell$th-degree polynomials ${\mathscr {R}}_{\ell }^{(\alpha ,\beta )}\left( \lambda _{k}\right) :=R_{\ell}( \lambda _{k}^{\alpha ,\beta })$ with constant coefficients $c^{\ell }_{j}(\alpha ,\beta )$ ($1\le j\le \ell$; $\alpha ,\beta >-1$), called Jacobi coefficients; the numbers $\lambda _{k}^{\alpha ,\beta }=k(k+\alpha +\beta +1)$ ($k\ge 0$) are the eigenvalues of the associated Jacobi operator. These Jacobi coefficients appear in the Maclaurin spectral expansion of the Schwartz kernels of functions of the Laplacian on rank one compact symmetric spaces. The first Jacobi coefficients $c^{\ell }_{j}(\alpha ,\beta )$ ($1\le j\le \ell \le 4$) were computed using the qth, $1\le q\le \ell ,$ derivative formula for Jacobi polynomials $P_{k}^{(\alpha ,\beta )}(t)$ (evaluated at $t=1$) and the basic properties of the Gamma function. In this paper, we use the Jacobi differential equation to generate a recursion formula that explicitly computes the polynomials ${\mathscr {R}}_{\ell }^{(\alpha ,\beta )}\left( \lambda _{k} \right)$ and the coefficients $c^{\ell }_{j}(\alpha ,\beta )$. To illustrate the recursion formula we compute the higher coefficients $c^{\ell }_{j}(\alpha ,\beta )$ for $\ell =5,6;1\le j\le \ell$. Remarkably, the local Jacobi coefficients ${\mathsf {c}}^{q}_{j}(\alpha ,\beta )$ $(1\le j\le q\le \ell )$ (which appear in the computation of $c^{\ell }_{j}(\alpha ,\beta )$) coincide with the Jacobi–Stirling numbers of the first kind ${{\mathsf {j}}}{{\mathsf {s}}}_{j}^{q}(\alpha ,\beta )$, and this is a new phenomenon in the Maclaurin spectral analysis of the Laplacian on rank one compact symmetric spaces. The Jacobi coefficients $c^{\ell }_{j}(\alpha ,\beta )$ are useful in the description of constants appearing in the power series expansion of any spectral functions involving Jacobi polynomials.