AI 中文总结
本文定义了一类核含多元q-米塔格-莱夫勒函数的新型q-分数阶积分算子，推导了对应的q-分数阶导数，探讨了其与经典理论算子的异同，并将其应用于求解q-分数阶微分方程，为q-微积分研究提供了新方向。
AI 中文摘要
涉及特殊函数作为核的分数阶积分算子长期以来一直是分数阶微积分的主要研究主题之一。作为经典分数阶微积分的离散版本，q-分数阶微积分不仅常带来新现象和新视角，还为描述和研究应用领域的新问题提供了有用的新方向。最近，Luo和Zhou基于一套基本通用理论，开发了构建性能良好的q-分数阶积分算子的方法。本文的目的是定义并引入一类新型q-分数阶积分算子，其核包含新的多元q-米塔格-莱夫勒函数。我们还将定义对应的q-分数阶导数（作为q-分数阶积分算子的逆），以及q-Caputo分数阶导数算子。我们详细讨论了这些q-算子与经典理论中对应算子的相似性和差异，特别是在Prabakar分数阶微积分方面。此外，还考虑了所定义算子在求解某些q-分数阶微分方程中的应用。本文全程提及我们建立的结果与某些已知重要情形的有用关联。在结语中，我们简要指出了q-微积分当前主题进一步研究的几个可能建议（或方向）。
英文摘要
Fractional integral operators involving special functions as kernels have long been one of the main subjects of fractional calculus. As a discrete version of classical fractional calculus, $q$-fractional calculus not only brings often new phenomena and new perspectives but also provides new useful areas for describing and studying new problems of applied nature. Recently, Luo and Zhou (see below) developed a methodology for formulating well-behaved $q$-fractional integral operators steming from a basic general theory. Our aim in this paper is to define and introduce a new class of $q$-fractional integral operators whose kernel involves a new multivariate $q$-Mittag-Leffler function. We shall also define the corresponding $q$-fractional derivatives (as the inverses of $q$-fractional integral operators) and define also the $q$-Caputo fractional derivative operators. We discuss in detail the similarities and variations between the $q$-operators and their counterparts in classical theory, particularly in Prabakar fractional calculus. Further, applications of the defined operators in solving certain $q$-fractional differential equations are also considered. We also mention throughout this paper some useful relevances of our established results with certain known important cases. In the concluding remarks, we point out briefly few possible proposals (or directions) for further work on the present subject matter of $q$-calculus.
