On Jacobi polynomials $${\mathscr {P}}_{k}^{(\alpha ,\beta )}$$ P k
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ORIGINAL RESEARCH PAPER
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On Jacobi polynomials Pk and coefficients cj‘ ða,b) (k ‡ 0,‘ = 5,6; 1 £ j £ ‘;a,b > -- 1) Richard Olu Awonusika1 Received: 6 November 2018 / Accepted: 8 September 2020 Forum D’Analystes, Chennai 2020
Abstract In their paper, Awonusika and Taheri proved a spectral identity that relates the even ða;bÞ (2‘)th, ‘ 1, derivatives of Jacobi polynomials Pk ðcos hÞ, k 0; a; b [ 1 ða;bÞ
(evaluated at h ¼ 0), to the ‘th-degree polynomials R‘ ðkk Þ :¼ R‘ ðka;b k Þ with constant coefficients c‘j ða; bÞ (1 j ‘; a; b [ 1), called Jacobi coefficients; the numbers ka;b k ¼ kðk þ a þ b þ 1Þ (k 0) are the eigenvalues of the associated Jacobi operator. These Jacobi coefficients appear in the Maclaurin spectral expansion of the Schwartz kernels of functions of the Laplacian on rank one compact symmetric spaces. The first Jacobi coefficients c‘j ða; bÞ (1 j ‘ 4) were comða;bÞ
puted using the qth, 1 q ‘; derivative formula for Jacobi polynomials Pk ðtÞ (evaluated at t ¼ 1) and the basic properties of the Gamma function. In this paper, we use the Jacobi differential equation to generate a recursion formula that ða;bÞ explicitly computes the polynomials R‘ ðkk Þ and the coefficients c‘j ða; bÞ. To illustrate the recursion formula we compute the higher coefficients c‘j ða; bÞ for ‘ ¼ 5; 6; 1 j ‘. Remarkably, the local Jacobi coefficients cqj ða; bÞ ð1 j q ‘Þ (which appear in the computation of c‘j ða; bÞ) coincide with the Jacobi–Stirling numbers of the first kind jsqj ða; bÞ, and this is a new phenomenon in the Maclaurin spectral analysis of the Laplacian on rank one compact symmetric spaces. The Jacobi coefficients c‘j ða; bÞ are useful in the description of constants appearing in the power series expansion of any spectral functions involving Jacobi polynomials. Keywords Orthogonal polynomials Jacobi coefficients Maclaurin expansion Schwartz kernel Jacobi–Stirling numbers
Mathematics Subject Classification 33C05 33C45 35A08 35C05 35C10 35C15
Extended author information available on the last page of the article
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R. O. Awonusika
1 Introduction ða;bÞ
The Jacobi polynomials Pk (k 0; a; b [ 1) play a fundamental role in the analysis of the Laplacians on rank one compact symmetric spaces and are natural and key ingredients in the representations of the associated heat kernels ([12]). It is well-known that the spherical functions (normalised eigenfunctions) of the Laplacians on these symmetric spaces are given in terms of the Jacobi polynomials, and recently, it has been shown that the Maclaurin expansion of the heat kernels on rank one compact symmetric spaces are explicitly described by the Jacobi coefficients c‘j ða; bÞ, 1 j ‘; a; b [ 1 (see [6, 8]). Jacobi polynomials are therefore known as eigenfunctions of a Sturm-Liouville operator and their expansions are useful in solving problems arising from mathematical and physical applications ([13, 16–19, 47]). In fact, this class of polynomials co
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