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<p>In this work, we develop a constructive method for deriving four structure relations and a fourth-order homogeneous linear differential equation satisfied by Laguerre–Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre–Hahn polynomials of class zero, analogous to Hermite. In addition, we present results for a semiclassical example of class&#xa0;1.</p>

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On a general method for deriving a fourth-order differential equation satisfied by Laguerre–Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite

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  • 发表日期：

    2026年7月21日

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  • Mohamed Khalfallah,
  • Pascal Maroni,
  • Zélia da Rocha

摘要

In this work, we develop a constructive method for deriving four structure relations and a fourth-order homogeneous linear differential equation satisfied by Laguerre–Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre–Hahn polynomials of class zero, analogous to Hermite. In addition, we present results for a semiclassical example of class 1.