An-Najah University Journal for Research - A (Natural Sciences)

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Keywords

• fractional calculus
• Hermite polynomials
• Legendre polynomials
• Symbolic operators
• fractional kinetic equation.
• Mittag-Leer function
• Gegenbauer polynomials
• Chebyshev polynomials

Abstract

n this paper, we employ the symbolic operator approach, a versatile tool for studying and generalizing special functions, to introduce a novel class of polynomials, the Two-Variable Mittag-Le er-Gegenbauer polynomials. This family generalizes several classical polynomials, including Laguerre, Hermite, and Gegenbauer olynomials, providing a unifying framework for their analysis.We investigate the main properties of our polynomials, including series representations, generating functions, operational rules, and relations via fractional integrals and derivatives. The practical relevance is illustrated through numerical examples and graphical demonstrations. Additionally, we explore an application to fractional kinetic equations, highlighting how these polynomials naturally model memory-dependent processes and reveal new features in fractional dynamics. Overall, this work demonstrates that the combination of symbolic operators and a two-variable structure provides a powerful framework for generating, analysing, and applying new classes of special polynomials in both theoretical and applied settings

معلومات المقال

قيد النشر

الكلمات الإفتتاحية

• fractional calculus
• Hermite polynomials
• Legendre polynomials
• Symbolic operators
• fractional kinetic equation.
• Mittag-Leer function
• Gegenbauer polynomials
• Chebyshev polynomials

الملخص

n this paper, we employ the symbolic operator approach, a versatile tool for studying and generalizing special functions, to introduce a novel class of polynomials, the Two-Variable Mittag-Le er-Gegenbauer polynomials. This family generalizes several classical polynomials, including Laguerre, Hermite, and Gegenbauer olynomials, providing a unifying framework for their analysis.We investigate the main properties of our polynomials, including series representations, generating functions, operational rules, and relations via fractional integrals and derivatives. The practical relevance is illustrated through numerical examples and graphical demonstrations. Additionally, we explore an application to fractional kinetic equations, highlighting how these polynomials naturally model memory-dependent processes and reveal new features in fractional dynamics. Overall, this work demonstrates that the combination of symbolic operators and a two-variable structure provides a powerful framework for generating, analysing, and applying new classes of special polynomials in both theoretical and applied settings