# On Fractional Calculus Operators and the Basic Analogue of Generalized Mittag-Leffler Function

## DOI:

https://doi.org/10.26713/cma.v13i3.1854## Keywords:

Saigo’s fractional q-calculus operator, Generalized q-Mittag-Leffler function, q-gamma function, q-shifted factorial and basic hypergeometric series## Abstract

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.

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*Communications in Mathematics and Applications*,

*13*(3), 835–842. https://doi.org/10.26713/cma.v13i3.1854

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