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<p>This paper presents a novel and unifying approach in the theory of special functions by proposing an extended formulation of the generalized <i>M</i>-series defined through a beta function with general kernel. The newly developed structure further broadens the generalized <i>M</i>-series, offering a comprehensive and integrative form that encompasses all its existing variants. Integral representations, fundamental analytic properties, and the behaviors of the series under the Riemann–Liouville and Caputo fractional operators are investigated in detail. Moreover, closed-form expressions are derived for the beta, Laplace, generalized Laplace, Sumudu, Elzaki, and general integral transforms. The results demonstrate that the proposed framework enhances analytical flexibility and provides an original and comprehensive theoretical foundation for the analysis of fractional systems.</p>

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A general kernel-based beta function approach to the forward expansion of generalized M-series with applications to fractional operators and integral transforms

  • 期刊论文

  • 发表日期：

    2026年3月3日

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  • Enes Ata

摘要

This paper presents a novel and unifying approach in the theory of special functions by proposing an extended formulation of the generalized M-series defined through a beta function with general kernel. The newly developed structure further broadens the generalized M-series, offering a comprehensive and integrative form that encompasses all its existing variants. Integral representations, fundamental analytic properties, and the behaviors of the series under the Riemann–Liouville and Caputo fractional operators are investigated in detail. Moreover, closed-form expressions are derived for the beta, Laplace, generalized Laplace, Sumudu, Elzaki, and general integral transforms. The results demonstrate that the proposed framework enhances analytical flexibility and provides an original and comprehensive theoretical foundation for the analysis of fractional systems.