On Fractional Calculus Operators and the Basic Analogue of Generalized Mittag-Leffler Function - Bhadana, Krishna Gopal; Meena, Ashok Kumar (Communications in Mathematics and Applications, 2022)
In the present paper, we have derived some unified image formulas of the generalized \(q\)-Mittag-Leffler function under fractional calculus operators. We have derived the integral and derivative formulas of Saigo's for the generalized \(q\)-Mittag-Leffler function in terms of basic hypergeometric series \(_2\Phi_1 [a,b;c \, | \, q,z]\) and with the help of main results we have obtained the known formulas of the generalized \(q\)-Mittag-Leffler function such as Riemann-Liouville fractional integral & derivatives. The Kober and Weyl integrals of the generalized \(q\)-Mittag-Leffler function are also obtained as special cases.
Keywords: Saigo’s fractional q-calculus operator; Generalized q-Mittag-Leffler function; q-gamma function; q-shifted factorial and basic hypergeometric series