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Transmuted fractional calculus with analytic kernels

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Abstract

Several kinds of fractional calculus are defined by operators with different kernel functions, among which one important class is with analytic kernels: namely, where a singular power function is multiplied by an analytic function of a fractional power to form a kernel. Other kinds arise from transmuting standard fractional calculus through invertible linear operators, such as composition and multiplication operators. This paper surveys a general model of fractional calculus given by transmutations via arbitrary linear maps of the operators with analytic kernels. We establish several fundamental results for transmuted fractional calculus with analytic kernels, including function space mappings, series formulae, and semigroup properties. To further illustrate its significance, we examine several special cases, many of which are already known in the literature: right-sided fractional calculus, fractional calculus with respect to functions, and weighted fractional calculus, all with analytic kernels – by which we mean that the kernel functions are analytic power series modified by fractional power function substitutions and singular power function multipliers.

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Authors and Affiliations

  1. Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, 44000, Islamabad, Pakistan

    Haniyyah Ul Irshad, Hafiz Muhammad Fahad & Khadija Bibi

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  1. Haniyyah Ul Irshad
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  2. Hafiz Muhammad Fahad
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  3. Khadija Bibi
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Correspondence to Hafiz Muhammad Fahad.

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Irshad, H.U., Fahad, H.M. & Bibi, K. Transmuted fractional calculus with analytic kernels. Fract Calc Appl Anal 28, 3117–3151 (2025). https://doi.org/10.1007/s13540-025-00462-w

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