# Adyghe Int. Sci. J. Vol. 22, No 4. P. 29-38. ISSN 1726-9946

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DOI: https://doi.org/10.47928/1726-9946-2022-22-4-29-38

MATHEMATICS

 MSC 32А10; 32А37 Original Article

# On some formulas for fractional integration of one Fox function with four parameters

Fatima Gidovna Khushtova
Researcher of Department of Fractional Calculus, Institute of Applied Mathematics and Automation of KBSC RAS, (360017, 89 A Shortanova St., Nalchik, Russia), Ph.D., http://orcid.org/0000-0003-4088-3621, khushtova@yandex.ru

Abstract. Solutions to many problems of mathematical physics, engineering and economics arc expressed through the so-called special functions. In the theory of special functions an important place is occupied by functions of the hypergeometric type. Many of them can be written in terms of the Meyer G-function. A generalization of the Meyer function is the Fox H-function. Some properties of this function can be obtained from its representation using the Mellin – Barnes integral. When deriving some formulas for this function for particular values of its parameters, due to the cumbersome writing of the Fox function, it is more convenient to use simplified notation. In this paper, we consider a special case of such a Fox function containing four parameters. For this function, Riemann-Liouville and Erdelyi-Kober fractional integration formulas arc obtained. An integral representation of the considered function h through the Mellin – Burns integral, we write out the conditions under which it converges absolutely, and the asymptotic expansions for this function for large and small values of the argument. The formulas proved in the paper are obtained using the indicated Mellin – Burns integral representation and the well-known integration formulas from power functions. For particular values of the parameters, the function under consideration yields some well-known elementary and special functions, and from the obtained formulas of fractional integration – the known integral values of these functions.
Keywords: Fox function, Mittag-Leffler type function, hypergeometric function, incomplete gamma function, Mellin – Barnes integral, degenerate hypergeometric Kummer function, Riemann-Liouville fractional integration, Erdelyi-Kober fractional integration

Acknowledgments: the author are thankful to the anonymous reviewer for his valuable remakes.

For citation. F. G. Khushtova On some formulas for fractional integration of one Fox function with four parameters. Adyghe hit. Sei. J. 2022. Vol. 22, No. 4. P. 29-38.
DOI: https://doi.org/10.47928/1726-9946-2022-22-4-29-38

The author has read and approved the final version of the manuscript.
Submitted 13.12.2022; approved after reviewing 21.12.2022; accepted for publication 24.12.2022

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