Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations

Zayed, Mohra; Ben Omrane, Ines; Costabile, Francesco Aldo; Jeelani, Mdi Begum; Wani, Shahid Ahmad

doi:10.3390/math14111907

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Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations

by

Mohra Zayed

¹

,

Ines Ben Omrane

²,

Francesco Aldo Costabile

^3,*,

Mdi Begum Jeelani

²

and

Shahid Ahmad Wani

^4,*

¹

Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia

²

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia

³

Department of Mathematics and Computer Science, University of Calabria, 87036 Rende, CS, Italy

⁴

Symbiosis Institute of Technology Pune, Symbiosis International (Deemed) University, Pune 412115, India

^*

Authors to whom correspondence should be addressed.

Mathematics 2026, 14(11), 1907; https://doi.org/10.3390/math14111907 (registering DOI)

Submission received: 4 May 2026 / Revised: 19 May 2026 / Accepted: 27 May 2026 / Published: 30 May 2026

(This article belongs to the Special Issue Advances in Operator Theory and Nonlinear Evolution Equations)

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Abstract

This paper introduces a new family of extended degenerate Bell-based Appell polynomials by applying Euler’s integral as a fractional operator to the Appell-type degenerate Bell polynomials. Beginning with the exponential operational rule and employing the fractional operator framework, this paper derives the operational connection, generating function, explicit summation formula, determinant representation, complete four-step proof via Cramer’s rule, recurrence relations, and the monomiality principle with raising and lowering operators and rigorously verifies the commutation relation. Applications to the extended degenerate Bell–Bernoulli, Bell–Euler, and Bell–Genocchi polynomials are presented as three numbered examples, each with generating functions; operational connections; determinant representations; numerical first values; and graphical illustrations including surface plots, zero diagrams, and stacked zero figures.

Keywords: fractional operators; Euler’s integral; degenerate Bell polynomials; Appell polynomials; extended degenerate Bell-based Appell polynomials; determinant representation; monomiality principle; Stirling numbers; zeros; graphical analysis

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MDPI and ACS Style

Zayed, M.; Ben Omrane, I.; Costabile, F.A.; Jeelani, M.B.; Wani, S.A. Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations. Mathematics 2026, 14, 1907. https://doi.org/10.3390/math14111907

AMA Style

Zayed M, Ben Omrane I, Costabile FA, Jeelani MB, Wani SA. Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations. Mathematics. 2026; 14(11):1907. https://doi.org/10.3390/math14111907

Chicago/Turabian Style

Zayed, Mohra, Ines Ben Omrane, Francesco Aldo Costabile, Mdi Begum Jeelani, and Shahid Ahmad Wani. 2026. "Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations" Mathematics 14, no. 11: 1907. https://doi.org/10.3390/math14111907

APA Style

Zayed, M., Ben Omrane, I., Costabile, F. A., Jeelani, M. B., & Wani, S. A. (2026). Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations. Mathematics, 14(11), 1907. https://doi.org/10.3390/math14111907

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Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations

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