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

Authors

  • R. O. Awonusika Adekunle Ajasin University

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.

 

Author Biography

R. O. Awonusika, Adekunle Ajasin University

Lecturer II

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Published

2021-08-31

How to Cite

Awonusika, R. O. (2021). Integral Representations and Identities on Rank One Symmetric Spaces of Compact Type. Journal of the Nigerian Mathematical Society, 40(2), 129–148. Retrieved from https://ojs.ictp.it/jnms/index.php/jnms/article/view/459

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Articles