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<p>This study presents an extensive generalization of Legendre–Laguerre polynomials along with their Appell-type counterparts. Using the quasi-monomiality approach, we establish core analytical features, including recurrence relations, associated multiplicative and differential operators, and the governing differential equation. We further derive series representations and determinantal expressions for this newly defined polynomial family. Within this framework, several noteworthy subclasses are introduced and analyzed, such as the generalized Legendre–Laguerre–Gould–Hopper–Appell polynomials. The formulation is extended through fractional operator techniques to explore their inherent structural attributes. Moreover, we construct and investigate new families namely, the generalized Legendre–Laguerre–Gould–Hopper–Bernoulli, Legendre–Laguerre–Gould–Hopper–Euler, and Legendre–Laguerre–Gould–Hopper–Genocchi polynomials highlighting their operational and algebraic properties. Collectively, these results advance the theory of special functions and provide a foundation for potential applications in mathematical physics and the study of differential equations.</p>

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Extended forms of Legendre-Laguerre-based hybrid polynomials and their characteristics via fractional operator approach

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月27日

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  • Waseem Ahmad Khan,
  • Shahid Ahmad Wani,
  • Mohammad Ayman-Mursaleen,
  • Ketan Kotecha,
  • Prakash Jadhav

摘要

This study presents an extensive generalization of Legendre–Laguerre polynomials along with their Appell-type counterparts. Using the quasi-monomiality approach, we establish core analytical features, including recurrence relations, associated multiplicative and differential operators, and the governing differential equation. We further derive series representations and determinantal expressions for this newly defined polynomial family. Within this framework, several noteworthy subclasses are introduced and analyzed, such as the generalized Legendre–Laguerre–Gould–Hopper–Appell polynomials. The formulation is extended through fractional operator techniques to explore their inherent structural attributes. Moreover, we construct and investigate new families namely, the generalized Legendre–Laguerre–Gould–Hopper–Bernoulli, Legendre–Laguerre–Gould–Hopper–Euler, and Legendre–Laguerre–Gould–Hopper–Genocchi polynomials highlighting their operational and algebraic properties. Collectively, these results advance the theory of special functions and provide a foundation for potential applications in mathematical physics and the study of differential equations.