Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 52. no. 3. P. 24 – 43. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-52-3-24-43
Research Article
Full text in English
MSC 33C15

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New Extended Three-Variable Mittag-Leffler Type Functions

A. Hasanov, H. A. Yuldashova^{\ast}

V. I. Romanovskiy Institute of Mathematics, 9 University str., Tashkent, 100174, Uzbekistan

Abstract. This article presents a systematic investigation of a new class of Mittag-Leffler-type functions in three variables. These functions are a natural and significant extension of the classical Mittag-Leffler function, and are constructed to correspond analogously to the well-known Lauricella hypergeometric functions of three variables. Our study comprehensively explores the fundamental properties and analytical characteristics of these threevariable functions. A primary focus is the establishment of their precise interrelationships with other existing extensions and generalizations of the classical Mittag-Leffler function, thereby situating them within the broader landscape of special functions. Key analytical findings presented in this work include: The derivation of the exact three-dimensional regions of convergence for the series defining these functions. The formulation of elegant Euler-type integral representations, which provide a powerful tool for further analysis. A detailed exploration of their integral transforms, specifically the derivation of both one-dimensional and three-dimensional Laplace transforms.The examination of their intimate connections with fractional calculus, demonstrating their natural emergence as kernels and solutions in the context of the Riemann-Liouville fractional integral and differential operators. Furthermore, we delve into the associated differential equations, showing that these Mittag-Lefflertype
functions serve as solutions to specific systems of partial differential equations. This work not only enriches the theory of special functions but also provides a robust mathematical framework for potential applications in fractional differential equations, anomalous diffusion, and other areas of mathematical physics.

Key words: Extended Mittag-Leffler type function; Hypergeometric function; Special (or higher transcendental) function; Lauricella function; Integral representation; System of partial differential equation; One- and threedimensional Laplace transform; Riemann-Liouville fractional integral; Riemann-Liouville fractional derivative; Appell and Kamp´e de F´e riet functions; Srivastava-Daoust hypergeoemetric function.

Received: 16.10.2025; Revised: 06.11.2025; Accepted: 08.11.2025; First online: 11.11.2025

For citation. Hasanov A., Yuldashova H. A. New extended three-variable Mittag-Leffler type functions. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 52: 3, 24-43. EDN: XQKWGW. https://doi.org/10.26117/2079-6641-2025-52-3-24-43.

Funding. The study was conducted without the support of foundations.

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: hilolayuldashova77@gmail.com

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

© Hasanov A., Yuldashova H. A., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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