The Mittag-leffler and related functions

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Abstract
In our attempts to find solutions of fractional differential equations we are led, in a natural fashion, to the study of certain transcendental functions. These functions may be defined by a fractional integral or combinations of such integrals. Further investigations show that these new functions are intimately related to the classical Mittag-Leffler function. Our objective in this brief paper is to make a modest generalization of these new functions, study some of their elementary properties, and show how they too are related to the Mittag-Leffler function.

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