GENERALIZED-TWO-VARIABLE-MITTAG-LEFFLER FUNCTION AND ATANGANA-BALEANU RIEMANN-LIOUVILLE DERIVATIVES OF GENERALIZED MITTAG-LEFFLER FUNCTIONS

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.