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We define an analogue of the classical Mittag-Leffler function which isapplied to two variables, and establish its basic properties. Using acorresponding single-variable function with fractional powers, we define anassociated fractional integral operator which has many interesting properties.The motivation for these definitions is twofold: firstly their link with somefundamental fractional differential equations involving two independentfractional orders, and secondly the fact that they emerge naturally fromcertain applications in bioengineering.

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators