Geodesic
A Spectral Identity on Jacobi Polynomials and its Analytic Implications
Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 473-482

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The Jacobi coefficients $c_{j}^{\ell }\left( \alpha ,\,\beta\right)\,\left( 1\,\le \,j\,\le \,\ell ,\,\alpha ,\,\beta \,>\,-1 \right)$ 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 Jacobi polynomials $P_{k}^{\left( \alpha ,\,\beta\right)}\,\left( k\,\ge \,0,\,\alpha ,\,\beta \,>\,-1 \right)$ 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.
DOI : 10.4153/CMB-2017-056-8
Mots-clés : 33C05, 33C45, 35A08, 35C05, 35C10, 35C15, Jacobi coefficient, Laplace–Beltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial 
Awonusika, Richard; Taheri, Ali. A Spectral Identity on Jacobi Polynomials and its Analytic Implications. Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 473-482. doi: 10.4153/CMB-2017-056-8
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     title = {A {Spectral} {Identity} on {Jacobi} {Polynomials} and its {Analytic} {Implications}},
     journal = {Canadian mathematical bulletin},
     pages = {473--482},
     year = {2018},
     volume = {61},
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     doi = {10.4153/CMB-2017-056-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-056-8/}
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