Some properties relating to the Mittag–Leffler function of two variables

Abstract

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered.

Keywords:

• Mittag–Leffler type functions
• fractional calculus operators
• integral transforms
• integral relations
• operational relations

2010 Mathematics Subject Classifications:

• 33E12
• 65R10
• 26A33

Acknowledgements

We thank the anonymous reviewers for their valuable suggestions, which made the presentation of the paper more readable.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The third author is supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021), as well as by EPSRC [grant EP/R003025/2].