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# A spectral identity on Jacobi polynomials and its analytic implications

Published:2017-11-10

• Richard Awonusika,
Department of Mathematics, University of Sussex, Brighton, UK
• Ali Taheri,
Department of Mathematics, University of Sussex, Brighton, UK
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## Abstract

The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq j\leq \ell$, $\alpha,\beta\gt -1$) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt -1$) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented.
 Keywords: Jacobi coefficient, Laplace-Beltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial
 MSC Classifications: 33C05 - Classical hypergeometric functions, ${}_2F_1$ 33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 35A08 - Fundamental solutions 35C05 - Solutions in closed form 35C10 - Series solutions 35C15 - Integral representations of solutions

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