Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 140-149. ISSN 2079-6641

Research Article
Full text in Russian
MSC 33С60

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To the Properties of One Fox Function

F. G. Khushtova^*

Institute of Applied Mathematics and Automation KBSC RAS, 89А Shortanova St., Nalchik, 360000, Russia

Abstract. The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.

Key words: Fox function, Mellin-Barnes integral, Euler gamma function, Macdonald function, hypergeometric function.

Received: 29.11.2022; Revised: 16.03.2023; Accepted: 29.03.2023; First online: 16.04.2023

For citation. Khushtova F. G. To the properties of one fox function. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 140-149. EDN: FXXPSA.

Funding. The study was carried out without financial support from foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for
submitting the final version of the article to print.

^*Correspondence: E-mail:

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Khushtova F. G., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)


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Information about the author

Khushtova Fatima Gidovna – PhD (Math. & Phys.), Professor, Researcher, Department of Fractional Calculus, Institute of Applied Mathematics and Automation RAS, Nalchik, Russia,