Advances in Operator Theory

A formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomials

Stuart Day and Ali Taheri

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The Jacobi polynomials $(\mathscr{P}^{(\alpha, \beta)}_k: k \ge 0, \alpha, \beta >-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the spherical or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.

Article information

Adv. Oper. Theory, Volume 2, Number 4 (2017), 506-515.

Received: 13 May 2017
Accepted: 28 July 2017
First available in Project Euclid: 4 December 2017

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Zentralblatt MATH identifier

Primary: 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)
Secondary: 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] 35C05: Solutions in closed form 35C10: Series solutions 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Jacobi coefficients Symmetric spaces Bell polynomials Spectral functions Laplace-Beltrami operator Schwartzian kernels Jacobi polynomials


Day, Stuart; Taheri, Ali. A formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomials. Adv. Oper. Theory 2 (2017), no. 4, 506--515. doi:10.22034/aot.1705-1163.

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