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We introduce and study a new function called

The Mittag-Leffler function has gained importance and popularity during the last one and a half decades due to its direct involvement in the problems of physics, biology, engineering, and applied sciences. Mittag-Leffler function naturally occurs as the solution of fractional order differential equations and fractional order integral equations.

In 1903, the Swedish mathematician Mittag-Leffler [

The generalization of (

It is an entire function of order

A detailed account of Mittag-Leffler functions and their applications can be found in the monograph by Haubold et al. [

An interesting generalization of (

An interesting and useful generalization of both the Riemann-Liouville and Erdélyi-Kober fractional integration operators is introduced by Saigo [

The

(i) If we set

(ii) In the special case of (

(iii) If we set

(iv) If we put

(v) If we take

(vi) If we take

(vii) If we set

(viii) If we set

(ix) If we set

In this section we derive two theorems relating to generalized fractional integrals and derivative of the

Let

Following the definition of Saigo fractional integral [

This completes the proof of Theorem

If we put

Let

Let

Following the definition of Saigo fractional derivative as given in (

By virtue of (

This completes the proof of Theorem

If we put

Let

A number of known and new results can be obtained as special cases of Theorems

In this paper we derive a new generalization of Mittag-Leffler function and obtain the relations between the

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors wish to thank the referees for their useful suggestions for the improvement of the paper. The authors are thankful to Professor R. K. Saxena for giving useful suggestions, which led to the present form of the paper.

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