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<p>This paper is dedicated to the investigation of polynomial sequences, which are not necessarily orthogonal, generated by integral composite powers of a first order differential–difference Dunkl operator and intimately linked to the Dunkl-classical polynomials. This link is established through the canonical element of their dual sequences. For those sequences that satisfy orthogonality, the resulting polynomials coincide with the generalized Hermite polynomials. Additionally, we derive new Rodrigues-type formulas for the generalized Hermite polynomials.</p>

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Generalized Hermite polynomial sequences generated by integral powers of a first order differential–difference Dunkl operator

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Mohamed Khalfallah

摘要

This paper is dedicated to the investigation of polynomial sequences, which are not necessarily orthogonal, generated by integral composite powers of a first order differential–difference Dunkl operator and intimately linked to the Dunkl-classical polynomials. This link is established through the canonical element of their dual sequences. For those sequences that satisfy orthogonality, the resulting polynomials coincide with the generalized Hermite polynomials. Additionally, we derive new Rodrigues-type formulas for the generalized Hermite polynomials.