GENERALIZED-TWO-VARIABLE-MITTAG-LEFFLER FUNCTION AND ATANGANA-BALEANU RIEMANN-LIOUVILLE DERIVATIVES OF GENERALIZED MITTAG-LEFFLER FUNCTIONS
By
Madan Lal1 , Amit Punia2,* and Rajeev Kumar Gupta3
1,2,3Department of Mathematics and Statistics, Jai Narain Vyas University,
Jodhpur, Rajasthan, India-342001
Email: madan.lakhani288@gmail.com, * rs_maths amitpunia@jnvu.edu.in,drrkbgupta@yahoo.co.in * corresponding author
(Received: July 23, 2025; In format: September 08, 2025; Revised: November 17, 2025; Accepted: November 20, 2025)
DOI: https://doi.org/10.58250/jnanabha.2025.55223
Abstract
This paper studies the fractional differentiation of generalized Mittag-Leffler functions using the Atangana–Baleanu Riemann–Liouville (AB–RL) operator. Exact series representations are derived for the AB–RL derivative of these functions, including some special cases. The analysis leads to a compact reformulation involving a two-variable function, termed the Generalized–Two–Variable–Mittag–Leffler (GML) function, which includes several classical Mittag-Leffler families. We establish its structural properties and limiting cases, and provide graphical illustrations to demonstrate the impact of key parameters such as the memory index µ. These results enrich the analytical structure of fractional calculus involving generalized special functions and highlight the relevance of the GML function in memory-influenced systems.
2020 Mathematical Sciences Classification: 26A33, 33E12, 34A08.
Keywords and Phrases: Fractional calculus, AB–RL derivative, Mittag-Leffler function, Generalized special functions, Two–Variable-Generalized–Mittag–Leffler function