Multiparameter K-Mittag-Leffler Function

In this paper author introduce Multiparameter K-Mittag-Leffler Function definded as, pK (β,η)m q,k [z] = pK (β,η)m q,k [a1, .., ap; b1, .., bq , (β1, η1), .., (βm, ηm); z], pK (β,η)m q,k [z] = ∞ ∑ n=0 ∏p j=1(aj)n,k z n ∏q r=1(br)n,k ∏m i=1 Γk(ηin + βi) , where k ∈ R+ = (0,∞); aj, br, βi ∈ C; ηi ∈ R (j = 1, 2, .., p; r = 1, 2, .., q; i = 1, 2, ..,m). Certain relations that exist between pK (β,η)m q,k [z] function and Riemann-Liouville fractional integral and derivatives has been evaluated. It has been shown that the fractional integration and differentation of pK (β,η)m q,k [z] function with power multipliers into the function of the same form. Also deduce Mittag-Leffler functions introduced by [1],[3],[4],[5],[6],[7],[8],[13],[14],[15] and [16] are particular cases of Multiparameter K-Mittag-Leffler function for some particular values of parameters and deduce some particular cases. Mathematics Subject Classification: 26A33, 33E12, 33C20


Introduction
In [9] the author introduce the generalized K-Gamma Function Γ k (x) as where (x) n,k is the k-Pochhammer symbol and is given by K-Gamma function is given by, and it follows easily that The Fractional Integral operators for α > 0 ( [10], Definition 2.1, Page 33) are defined as, The Fractional Derivative for α > 0 ([10], Definition 2.2, Page 35) are defined as, where [α] means the maximal integer not extending α and {α} is the fractional part of α.

Main result
In this section we introduce Multiparameter K-Mittag-Leffler function and for particular values of parameter we can deduce Multiparameter K-Mittag-Leffler function in to eariler known Mittag-Leffler functions.Finally we calculate Fractional integral and derivative of Multiparameter K-Mittag-Leffler function.

Multiparameter K-Mittag-Leffer Function
Definition: Where Γ k (x) is the K-Gamma function given by ( 1) and (γ) n,k is the K-Pochhammer symbol given by (2).The series (11) is definded when none of the parameter b r (r = 1, 2, .., q) is negative integer or zero.If any parameter a j (j = 1, 2, .., p) in ( 11) is zero or negative, the series terminates into polynomial in z.
Convergent conditions for the series (11) are given by Ratio test, (i) If p < q + m i=1 ( η i k ), then the power series on the right of ( 11) is absolutely convergent for all z ∈ C. (ii) If p = q + m i=1 ( η i k ), then the power series on the right of ( 11) is abso-

Particular cases
By particularizing the parameters in (11), we obtain following known Mittagleffler functions.
. Which is the 3M-Parameter Multi-Index Mittag-Leffler function definded by [5].11),then we obtain Which is the Generalized Mittag-Leffler function studied by [6].11),then we obtain Which is the Multi-Index Mittag-Leffler function studied by [15].
Which is M-Series definded by [14].11),then we obtain Which is the K-Mittag-Leffler function studied by [1].11),then we obtain Which is the Generalized Mittag-Leffler function studied by [8].(11),then we obtain Which is the Mittag-Leffler function studied by [16].11),then we obtain Which is the Mittag-Leffler function studied by [7].

Fractional Claculus of Multiparameter K-Mittag-Leffler Function
We will prove, the fractional integration and differentation of p K (β,η)m q,k [z] function with power multipliers into the function of the same form.
., p; r = 1, 2, .., q; i = 1, 2, .., m).Then there holds the relation, Proff.By virtue of ( 7) and ( 11), we have put t = ux the above expression transforms in to the form using the Beta function formula, we have and rearranging the terms, we obtain which proves the theorem.
Then there holds the relation, Proff.By virtue of ( 8) and ( 11), we have put t = x u and using the Beta function, the above expression transforms in to the form, the reflection formula for gamma function, see ( [10],1.60), and using ( 17) and ( 18) in ( 16), we obtain and rearranging the terms, we obtain which proves the theorem.