On the five-parameter Mittag-Leffler matrix function and its properties

Salma Aljawi; Vinod Kumar Jatav; Ankit Pal; Salma Aljawi; Vinod Kumar Jatav; Ankit Pal

doi:10.3934/math.2026711

AIMS Mathematics

2026, Volume 11, Issue 6: 17382-17398. doi: 10.3934/math.2026711

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Research article

On the five-parameter Mittag-Leffler matrix function and its properties

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia; Email: snaljawi@pnu.edu.sa
2.
Division of Mathematics, School of Advanced Sciences & Languages, VIT Bhopal University, Sehore-466 114, Madhya Pradesh, India; Email: vinodkumarjatav@vitbhopal.ac.in
3.
Division of Mathematics, School of Advanced Sciences & Languages, VIT Bhopal University, Sehore-466 114, Madhya Pradesh, India; Email: ankit.pal@vitbhopal.ac.in

Received: 10 March 2026 Revised: 21 May 2026 Accepted: 08 June 2026 Published: 15 June 2026
MSC : 15A15, 26A33, 33C15, 33E12

In this paper, we introduce and study a matrix analog of the five-parameter Mittag-Leffler function. We establish the absolute convergence of the series defining this function on the unit circle $ |\varpi| = 1 $ under certain spectral conditions. Several fundamental properties are derived, including integral representations, derivative formulas, and differential recurrence relations. Furthermore, we obtain a variety of finite summation formulas for this matrix function and its related Fox-Wright matrix analog. Finally, we investigate the composition of the function with generalized fractional calculus operators introduced by Katugampola, deriving closed-form expressions for both fractional integrals and derivatives. The results presented here extend the theory of special matrix functions and contribute to the field of fractional calculus.
- Mittag-Leffler functions,
- matrix functional calculus,
- generalized fractional calculus
Citation: Salma Aljawi, Vinod Kumar Jatav, Ankit Pal. On the five-parameter Mittag-Leffler matrix function and its properties[J]. AIMS Mathematics, 2026, 11(6): 17382-17398. doi: 10.3934/math.2026711

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Abstract

In this paper, we introduce and study a matrix analog of the five-parameter Mittag-Leffler function. We establish the absolute convergence of the series defining this function on the unit circle $ |\varpi| = 1 $ under certain spectral conditions. Several fundamental properties are derived, including integral representations, derivative formulas, and differential recurrence relations. Furthermore, we obtain a variety of finite summation formulas for this matrix function and its related Fox-Wright matrix analog. Finally, we investigate the composition of the function with generalized fractional calculus operators introduced by Katugampola, deriving closed-form expressions for both fractional integrals and derivatives. The results presented here extend the theory of special matrix functions and contribute to the field of fractional calculus.

References

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