Scholarlink

<p>Till recently, the operators of <i>Generalized Fractional Calculus (GFC)</i> with the Fox <i>H</i>- and Meijer <i>G</i>-functions as singular kernels (V. Kiryakova, <i>Generalized Fractional Calculus and Applications</i>, Longman-J. Wiley, Harlow-New York, 1994), have been considered as one of the most general case of integrations and differentiations of fractional orders and multi-orders. The <i>H</i>-function (as an extension of the <i>G</i>-function) is a generalized hypergeometric function including practically almost all Special Functions (SF) related to classical Calculus and to fractional order calculus (FC). However, it happened that <i>still there are important classes of SF that cannot be represented in terms of the </i><i>H</i><i>-function</i>, like the polylogarithms, Riemann Zeta-function, Feynman integrals, Le Roy function and their recent multi-index versions. That is why, the so-called <i>Rathie </i><i>I</i><i>-function</i> has attracted our attention as a more general analogue of the <i>H</i>-function, since it encompasses also the mentioned other classes of special functions. Now, we introduce <i>GFC operators involving as kernels the </i><i>I</i><i>-function</i>, and prove that they satisfy all the basic axioms of FC, extend the GFC with <i>H</i>- and <i>G</i>-functions (Kiryakova, 1994) and provide relations to new classes of SF. It happens that such SF, for example the multi-index Mittag-Leffler-Le Roy type functions, appear <i>as eigen functions for the new GFC operators</i>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Generalized fractional integrals based on the Rathie I-function as a kernel

  • 期刊论文

  • 发表日期：

    2025年9月28日

查看全文
  • Virginia Kiryakova

摘要

Till recently, the operators of Generalized Fractional Calculus (GFC) with the Fox H- and Meijer G-functions as singular kernels (V. Kiryakova, Generalized Fractional Calculus and Applications, Longman-J. Wiley, Harlow-New York, 1994), have been considered as one of the most general case of integrations and differentiations of fractional orders and multi-orders. The H-function (as an extension of the G-function) is a generalized hypergeometric function including practically almost all Special Functions (SF) related to classical Calculus and to fractional order calculus (FC). However, it happened that still there are important classes of SF that cannot be represented in terms of the H-function, like the polylogarithms, Riemann Zeta-function, Feynman integrals, Le Roy function and their recent multi-index versions. That is why, the so-called Rathie I-function has attracted our attention as a more general analogue of the H-function, since it encompasses also the mentioned other classes of special functions. Now, we introduce GFC operators involving as kernels the I-function, and prove that they satisfy all the basic axioms of FC, extend the GFC with H- and G-functions (Kiryakova, 1994) and provide relations to new classes of SF. It happens that such SF, for example the multi-index Mittag-Leffler-Le Roy type functions, appear as eigen functions for the new GFC operators.