Laguerre polynomials as Jensen polynomials of Laguerre–Pólya entire functions
Copyright policy )Laguerre polynomials as Jensen polynomials of Laguerre–Pólya entire functions
The question if there exist entire special functions whose Jensen polynomials are orthogonal is investigated. Let \(\varphi(x)\) be an entire function from the Laguerre-Pólya class \(\varphi\in\mathcal L\mathcal P\) [see \textit{G. Pólya}, Über die algebraisch-funktionentheoretischen Untersuchungen von J. L. W. Jensen. Meddelelser Kobenhavn 7, Nr.~17, S. 3--33 (1927; JFM 53.0309.01)] with Maclaurin expansion \(\varphi(x)= \sum_{k=0}^\infty \gamma_kx^k/k!\) then polynomials \(g_n(\varphi;x) =\sum_{k=0}^n\binom{n}{k} \gamma_kx^k\) are called Jensen polynomials associated to \(\varphi(x)\). The main result of the paper is Theorem 1: The only Jensen polynomials that are orthogonal are the Laguerre polynomials. Four proofs for Theorem 1 are given. Moreover a new proof of the fact that all zeros of the Bessel function \(J_\alpha(z)\) are real when \(\alpha>-1\) is provided.
Applied Mathematics, Other special orthogonal polynomials and functions, Zeros, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Bessel functions, Computational Mathematics, Laguerre polynomial, Laguerre–Pólya class, Jensen polynomials, zeros, Laguerre polynomials, Laguerre-Polya class, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Laguerre-Pólya class
Applied Mathematics, Other special orthogonal polynomials and functions, Zeros, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Bessel functions, Computational Mathematics, Laguerre polynomial, Laguerre–Pólya class, Jensen polynomials, zeros, Laguerre polynomials, Laguerre-Polya class, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Laguerre-Pólya class
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