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On a new \((p,q)\)-Mathieu-type power series and its applications. (English) Zbl 1499.31010

Summary: Dedicated to Prof. Gradimir Milovanovic on the occasion on his 70th birthday. Our aim in this paper, is to establish certain new integral representations for the \((p,q)\)-Mathieu-type power series. In particular, we investigate the Mellin-Barnes type integral representations for a particular case of these special function. Moreover, we introduce the notion of the \((p,q)\)-Mittag-Leffler functions and we present a relationships between these two functions. Some other applications are proved, in particular two Turán type inequalities for the \((p,q)\)-Mathieu-type series are derived.

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
33C20 Generalized hypergeometric series, \({}_pF_q\)
33E20 Other functions defined by series and integrals
33E12 Mittag-Leffler functions and generalizations
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References:

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[19] Faculté des Sciences de Tunis, Université Tunis El Manar, Tunisia. Département de Mathématiques, Issat Kasserine, Université de Kairouan, Tunisia E-mail: k.mehrez@yahoo.fř Zivorad Tomovski, University “ St. Cyril and Methodius”, Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Repubic of Macedonia. E-mail: tomovski@pmf.ukim.edu.mk, tomovski33@gmail.com, zivoradt@yahoo.com
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