Fractional calculus containing certain bivariate Mittag-Leffler kernel with respect to function

Abstract In the present study, we introduce a general integral operator containing bivariate Mittag-Leffler (M-L) kernel with respect to a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>τ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \tau \left(z) . This general family includes the usual, Hadamard, Katugampola, Erdélyi-Kober, and Tempered versions for specific choices of the function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>τ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \tau \left(z) . We investigate the main properties of the general family by using series representation and conjugation relation.