Theorem 1 (Derivative of the Mittag-Leffler function)
If , , , , and , then
Proof Using definition (3), we have that
The Wright function is expressed with help of the Mittag-Leffler function:
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Theorem 2 (Integration of the Mittag-Leffler function)
If , , , , and , then
Proof According to the Mittag-Leffler function, we have
Integrating both sides gives
Relation with the Wright functions is as follows:
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Theorem 3 Let , , , , , , then
Proof By definition (3), we have that
Relation with the Wright functions is as follows:
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Theorem 4 For , , , , , . ‘Note that’
Proof We have
Equation (6) can be written as follows:
(7)
We find from equation (7) that
(8)
or
We now say each summation on the right-hand side of equation (9) is as follows:
(10)
or
We find from equation (11) that
(12)
Using equations (9), (11), and (12), we get
Relation with the Wright functions is as follows:
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Theorem 5 If , , , , , , then
(13)
Proof We have from (13)
For , we obtain
Relation with the Wright functions is as follows:
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