Integral Representations and Identities on Rank One Symmetric Spaces of Compact Type

Abstract

The Jacobi coefficients $c_{j}^{\ell}(\alpha,\beta)$ ($1\leq j\leq \ell ; \alpha,\beta>-1$) associated with the normalised Jacobi polynomials $\mathscr{P}_k^{(\alpha, \beta)}$ ($k\geq 0$; $\alpha,\beta>-1$) describe the Maclaurin heat coefficients $b^{N}_{2\ell}$ ($N,\ell\geq 1$) and the associated spectral polynomials $\widetilde{\mathscr{R}}^{(\alpha,\beta)}_{\ell}$ of $N$-dimensional compact rank one symmetric spaces. In this paper, apart from constructing a spectral polynomial $\mathscr{R}^{(\alpha,\beta)}_{\ell}$ associated with the product $\left[ \mathscr{P}_{k}^{(\alpha, \beta)}\right]^{2} $ we develop integral representations (involving Gegenbauer polynomials and Jacobi polynomials) for $\mathscr{R}^{(\alpha,\beta)}_{\ell}$ in terms of the spectral sum of integer powers of eigenvalues of the corresponding Gegenbauer and Jacobi operators. These integrals apart from being interesting in their own right lead to integral representations and identities for these eigenvalues and their multiplicities.