Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 9 – 32. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-1-9-32
Research Article
Full text in English
MSC 33C65

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Tables of the Lauricella Hypergeometric Functions F^{(3)}_A

M. O. Abbasova^{\ast}¹, T. G. Ergashev²

¹Namamgan State University, Namangan, 160119, 316 Uychi St., Uzbekistan
²National Research University “Tashkent Institute of Irrigation and Agricultural Mechanization Engineers Tashkent, 100000, Kari Niyazi str. 39, Uzbekistan

Abstract. The generalized hypergeometric function _qF_p is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions _2F_1 and _3F_3 are most common special cases of the generalized hypergeometric function _qF_p. The Appell hypergeometric functions F_1-F_4 are product of two hypergeometric functions _2F_1 that appear in many areas of mathematical physics and hypergeometric Appell functions are most common special cases of the Kamp´e de F´eriet hypergeometric function which is a power double series. As demonstrated by Opps, Saad and Srivastava, in 2005, the double integral integral representation of F_2 can be reduced to a single integral that can be easily evaluated for certain values of the parameters in terms of _2F_1 and _3F_3. Using many of the reduction formulas of _2F_1 and _3F_3 and representation of F_2in terms of a single integral, in 2008, Murley and Saad tabulated new reduction formulas for F_2.The Lauricella hypergeometric function F^{(3)}_A is a power triple series which is a product of the Appell function F_2 and the Gaussian hypergeometric function _2F_1. Following the work of Opps, Saad and Srivastava, we establish that the triple integral integral representation of F^{(3)}_A can be reduced to a single integral, under which is the Appell function F_2. Applying the known reduction formulas of _2F_1, _3F_3 and F_2 and the representation of F^{(3)}_A in terms of a single integral for the Appell function F_2, we have begun to tabulate new reduction formulas for the Lauricella function F^{(3)}_A .

Key words: multiple hypergeometric functions, Lauricella function, Appell series, Gauss hypergeometric
function, reduction formulas.

Received: 30.12.2025; Revised: 28.01.2026; Accepted: 30.01.2026; First online: 29.03.2026

For citation. Abbasova M. O., Ergashev T. G. Tables of the Lauricella hypergeometric functions F^{(3)}_A . Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 9-32. EDN: PRDPSC. https://doi.org/10.26117/2079-6641-2026-54-1-9-32.

Funding. Not funding

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

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: abbasovamunira21@gmail.com

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

© Abbasova M. O, Ergashev T. G., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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Information about the authors

Abbasova Munira Obudjonovna – Assistant of the Namangan State University, Namangan, Uzbekistan, ORCID 0009-0001-4511-7304.

Ergashev Tuhtasin Gulamjanovich – D. Sci.(Phys. Math.), Professor, Department of Higher Mathematics, National Research University “Tashkent Institute of Irrigation and Agricultural Mechanization Engineers (TIIAME)” , Tashkent, Uzbekistan, ORCID 0000-0003-3542-8309.

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