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<p>This paper introduces a novel class of kernels for <i>q</i>-integral transforms, called symmetrical Sonin–Luchko <i>q</i>-kernels, with the goal of advancing the theory of general fractional <i>q</i>-calculus. Building upon the classical Sonin method and the recent results of Luchko on symmetrical Sonin kernels in the continuous setting, we develop a <i>q</i>-analogue using the <i>q</i>-Laplace integral transform. As a result, we derive new kernels that generalize the Riemann–Liouville <i>q</i>-integral and <i>q</i>-derivative kernels. In particular, we construct explicit Sonin <i>q</i>-kernels in terms of the <i>q</i>-Mittag–Leffler and <i>q</i>-Wright functions, thereby extending Luchko’s framework to the <i>q</i>-calculus setting.</p>

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General fractional q-integrals and q-derivatives: a Sonin–Luchko q-kernel approach

  • 期刊论文

  • 发表日期：

    2025年7月31日

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  • Fethi Bouzeffour

摘要

This paper introduces a novel class of kernels for q-integral transforms, called symmetrical Sonin–Luchko q-kernels, with the goal of advancing the theory of general fractional q-calculus. Building upon the classical Sonin method and the recent results of Luchko on symmetrical Sonin kernels in the continuous setting, we develop a q-analogue using the q-Laplace integral transform. As a result, we derive new kernels that generalize the Riemann–Liouville q-integral and q-derivative kernels. In particular, we construct explicit Sonin q-kernels in terms of the q-Mittag–Leffler and q-Wright functions, thereby extending Luchko’s framework to the q-calculus setting.