A Characterization of Bessel and Jacobi Orthogonal Polynomials via a Three-Parameter Differential Raising Operator

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DOI: https://doi.org/10.28924/2291-8639-24-2026-351
Published: Oct 15, 2026
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Khalid Ali Alanezy, Jihad Souissi, A Characterization of Bessel and Jacobi Orthogonal Polynomials via a Three-Parameter Differential Raising Operator, Int. J. Anal. Appl., 24 (2026), 351.

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Khalid Ali Alanezy, Jihad Souissi

Abstract

Let \(\mathcal A_{\xi_1,\xi_2,\mu} :=x(xD+\mu\mathbb I)+\xi_1\mathbb I+\xi_2D\) be a first-order differential raising operator, where \((\xi_1,\xi_2)\in\mathbb C^2\setminus\{(0,0)\}\), \(D=d/dx\), and \(\mathbb I\) is the identity operator on the polynomial space. We characterize all monic orthogonal polynomial sequences for which the normalized sequence generated by \(\mathcal A_{\xi_1,\xi_2,\mu}\) is also orthogonal. Under the appropriate regularity conditions, the resulting sequences are scaled Bessel polynomials when \(\xi_2=0\) and scaled Jacobi polynomials when \(\xi_2\neq0\), with the Gegenbauer polynomials arising precisely in the subcase \(\xi_1=0\). Some structural properties and connection relations are also established.

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