A NEW CLASS OF INTEGRALS INVOLVING EXTENDED MITTAG-LEFFLER FUNCTIONS

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1 Journal of Fractional Calculus and Applications Vol. 9(1) Jan. 218, pp ISSN: A NW CLASS OF INTGRALS INVOLVING XTNDD MITTAG-LFFLR FUNCTIONS G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN Abstract. The main aim of this paper is to establish two generalized integral formulas involving the extended Mittag-Leffler function based on the well known Lavoie and Trottier integral formula and the results are express in term of extended Wright-type function. Also, we establish certain special cases of our main result. 1. Introduction The Mittag-Leffler functions α (z) and (z) are defined by the following series: z n α (z), z C, R(α) >, (1) Γ(αn + 1) and (z) z n, z, β C, R(α) >, (2) Γ(αn + β) respectively. For the generalizations and applications of Mittag-Leffler function, the readers can refer to the recent work of researchers ([5, 6, 7, 8, 9, 11, 2, 25, ]), Kilbas et al. ([12], Chapter 1) and Saigo and Kilbas [1]. In recent years, the Mittag-Leffler function and some of its variety generalizations have been numerically established in the complex plane ([1, 27]). A new generalization of the Mittag- Leffler functions (z) of (2) have defined by Prabhakar [2] as follows: γ (z) (γ) n z n, z, β C, R(α) >, () Γ(αn + β) where (γ) n denote the well known Pochhammer Symbol which is defined by: { 1, (n, γ C), (γ) n γ(γ + 1)(γ + 2)...(γ + n 1) (n N, γ C). 21 Mathematics Subject Classification. B2, C2, C5, C6, B15, C5. Key words and phrases. xtended Mittag-Leffler function, Wright-type hypergeometric functions, extended Wright-type hypergeometric functions. Submitted June 5,

2 JFCA-218/9(1) A NW CLASS OF INTGRALS INVOLVING XTNDD In fact, the following special cases are satisfied: 1 (z) (z), 1 α,1(z) (z). () Recently, many researchers have established the significance and great consideration of Mittag-Leffler functions in the theory of special functions for exploring the generalizations and some applications. Various extensions for these functions are found in ([9],[28, 29, ]). Srivastava and Tomovski [2] have defined further generalization of the Mittag-Leffler function 1 (z) of (), which is defined as: γ,k (z) (γ) nk z n Γ(αn + β), (5) where z, α, β C; R(α) > max{, R(k) 1}; R(k) >. Very recently, Ozarslan and Yilmaz [22] have investigated an extended Mittag-Leffler function γ,c (z : p) which is defined as: γ;c (z) B p (γ + n, c γ)(c) n z n B(γ, c γ)γ(αn + β), (6) where p >, R(c) > R(γ) > and B p (x, y) is extended beta function defined in Mainardi as follows: B p (x, y) t t (x 1) (1 t) (y 1) e (p/t(1 t)) dt, (7) where R(p) >, R(x) > and R(y) >. If p, then B p (x, y) reduces to the following beta function: B(x, y) t t (x 1) (1 t) (y 1) dt. (8) The notion of the extended beta functions given by Chaudhry et al. [2] and further analyzed in [16, 18] The gamma function is defined by: Γ(z) t By inspection, we conclude the following relation t (z 1) e ( t) dt; R(z) >. (9) Γ(z + 1) zγ(z). (1) The generalized hypergeometric function p F q (z) is defined in [6] as: pf q (z) p F q (a 1), (a 2 ),..., (a p ); z, p (b 1 ), (b 2 ),..., (b q ); (a 1 ) n (a 2 ) n...(a q ) n z n (b 1 ) n (b 2 ) n...(b p ) n where α i, β i C; i 1, 2,..., p; j 1, 2,..., qand b j, 1, 2,... and (z) n is the Pochhammer symbols. (11)

3 22 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN JFCA-218/9(1) Sharma and Devi [26] defined extended Wright type function in the following series: p+1f q+1 (z; p) p+1 F q+1 (a i, A i ) 1,p, (γ, 1); z (b j, β j ) 1,p, (c, 1); Γ(a 1 + A 1 n)...γ(a p + A p n)b p (γ + n, c γ)z n Γ(b 1 + β 1 n)...γ(b q + β q n)γ(c γ) (12) where i 1, 2,..., p, j 1, 2,..., q, R(p) >, R(c) > R(γ) > and p. Recently, Mittal et al. [17] defined an extended generalized Mittag-Leffer function as: γ,q;c (z; p) B p (γ + nq, c γ)(c) nq z n B(γ, c γ)γ(αn + β), (1) where α, β, γ C, R(α) >, R(β) >, R(γ) >, q > and B p (x, y) is an extended beta function defined in (7). In this paper, we derive new integral formulas involving the Mittag-Leffler function (6) and (1). For the present investigation, we need the following result of Lavoie and Trottier [1] x (a 1) (1 x) (2b 1) ( 1 x ) (2a 1) ( 1 x ) ( ) (b 1) 2a 2 Γ(a)Γ(b) Γ(a + b), (1) where R(a) >, R(b) >. For more details about the recent works in the field of dynamical systems theory, stochastic systems, non-equilibrium statistical mechanics and quantum mechanics, the readers may refer to the recent work of the researchers[1,,, 15, 19, 2, 21] and the references cited therein. 2. Main Result In this section, the generalized integral formulas involving the Mittag-Leffler function (6) and (1), are established here by inserting with the suitable argument in the integrand of (1). Theorem 1. Let y, c, ρ, j, α, β, γ C; with R(α) > max{, R(c) 1}; R(c) >. R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: ( x (ρ+j 1) (1 x) (2ρ 1) 1 x 1 x ( ) 2(ρ+j) 2 Γ(ρ + j) ψ ) (ρ 1) ( ( γ;c y 1 x ) (1 x) 2) (ρ, 1), (c, 1), (γ, 1); y; p Proof. Let S 1 be the left-hand side of (15). Now applying the series representation of the Mittag-Leffler function (6) to the integrand of (1) and by interchanging the order of integration and summation under the given condition in Theorem (1), we. (15)

4 JFCA-218/9(1) A NW CLASS OF INTGRALS INVOLVING XTNDD have S 1 γ;c ( x (ρ+j 1) (1 x) (2ρ 1) 1 x ( ( y 1 x ) (1 x) 2) x (ρ+j 1) (1 x 1 x B p (γ + n, c γ)(c) n B(γ, c γ)γ(αn + β) x (ρ+j 1) (1 x) 2(ρ+n) 1 ( 1 x B p (γ + n, c γ)(c) n y n B(γ, c γ)γ(αn + β). 1 x ) (ρ 1) 1 x ) (ρ 1) ( ( ) ) y 1 x (1 x) 2 n 1 x ) (ρ+n 1) Now using (6) and (1), we get S 1 ( ) 2(ρ+j) 2 Γ(ρ + j)γ(ρ + n) B p (γ + n, c γ)(c) n y n Γ(2ρ + j + n) B(γ, c γ)γ(αn + β) ( ) (2ρ+j) 2 Γ(ρ + j) B p (γ + n, c γ)γ(ρ + n)γ(c + n) y n Γ(c γ)γ(αn + β)γ(2ρ + j + n), which upon using (12), we get S 1 ( ) 2(ρ+j) 2 Γ(ρ + j) ψ (ρ, 1), (c, 1), (γ, 1) (β, α), (2ρ + j, 1), (c, 1) ; y; p. Theorem 2. Let y, c, ρ, j, α, β, γ C; with R(α) > max{, R(c) 1}; R(c) >. R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: ( x (ρ 1) (1 x) 2(ρ+j) 1 1 x ( ) 2(ρ) 2 Γ(ρ + j) ψ 1 x ) ( (ρ+j 1) ( γ;c yx (ρ, 1), (c, 1), (γ, 1); 1 x ) 2 ; p ) ; 9 y; p. (16)

5 226 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN JFCA-218/9(1) Proof. Let S 2 be the left-hand side of (16). Now applying the series representation of the Mittag-Leffler function (6) to the integrand of (1), we have S 2 γ;c ( x (ρ 1) (1 x) 2(ρ+j) 1 1 x ( ( yx 1 x ) ) 2 ; p x (ρ 1) (1 x) 2(ρ+j) 1 ( 1 x ( y ( ) ) 1 x 2 n B p (γ + n, c γ)(c) n B(γ, c γ)γ(αn + β) ( x (ρ+n 1) (1 x) 2(ρ+j) 1 1 x ) 2(ρ+n) 1 ( Now using (6) and (1), we get B p (γ + n, c γ)(c) n y n B(γ, c γ)γ(αn + β). S 2 ( ) 2(ρ+n) 2 Γ(ρ + j)γ(ρ + n) Γ(2ρ + j + n) ( ) 2ρ 2 Γ(ρ + j) B p (γ + n, c γ)(c) n y n B(γ, c γ)γ(αn + β) B p (γ + n, c γ)γ(ρ + n)γ(c + n) Γ(c γ)γ(αn + β)γ(2ρ + j + n) ( 9 y)n, which upon using (12), we get S 2 ( ) 2ρ 2 Γ(ρ + j) ψ (ρ, 1), (c, 1), (γ, 1); ; 9 y; p. Theorem. Let y, c, ρ, j, α, β, γ C; with R(α) > max{, R(c) 1}; R(c) >. R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: ( x (ρ+j 1) (1 x) (2ρ 1) 1 x 1 x ( ) 2(ρ+j) (ρ, 1), (c, q), (γ, 1); 2 Γ(ρ + j) ψ ) (ρ 1) ( ( γ,q;c y 1 x ) ) (1 x) 2 ; p ; y; p Proof. Let S be the left-hand side of (17). Now applying the series representation of the Mittag-Leffler function (6) to the integrand of (1) and by interchanging the order of integration and summation under the given condition in Theorem (1), we (17)

6 JFCA-218/9(1) A NW CLASS OF INTGRALS INVOLVING XTNDD have S γ,q;c ( x (ρ+j 1) (1 x) (2ρ 1) 1 x ( ( y 1 x ) ) (1 x) 2 ; p x (ρ+j 1) (1 x 1 x B p (γ + nq, c γ)(c) nq B(γ, c γ)γ(αn + β) x (ρ+j 1) (1 x) 2(ρ+n) 1 ( 1 x B p (γ + nq, c γ)(c) nq B(γ, c γ)γ(αn + β) 1 x ) (ρ 1) 1 x ) (ρ 1) ( ( ) ) y 1 x (1 x) 2 n y n. 1 x ) (ρ+n 1) Now using (6) and (1), we get S 1 ( ) 2(ρ+j) 2 Γ(ρ + j)γ(ρ + n) Γ(2ρ + j + n) ( ) (2ρ+j) 2 Γ(ρ + j) B p (γ + nq, c γ)(c) nq B(γ, c γ)γ(αn + β) y n B p (γ + nq, c γ)γ(ρ + n)γ(c + nq) y n Γ(c γ)γ(αn + β)γ(2ρ + j + n), which upon using (12), we get S ( ) 2(ρ+j) 2 Γ(ρ + j) ψ (ρ, 1), (c, 1), (γ, 1); ; y; p. Theorem. Let ρ, j, α, β, γ C; with R(α) >, R(c) > R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: ( x (ρ 1) (1 x) 2(ρ+j) 1 1 x ( ) 2(ρ) 2 Γ(ρ + j) ψ Γ(ρ) 1 x (ρ, 1), (c, q), (γ, 1); ) ( (ρ+j 1) ( γ,q;c yx 1 x ) 2 ; p ) ; 9 y; p. (18)

7 228 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN JFCA-218/9(1) Proof. Let S be the left-hand side of (18). Now applying the series representation of the Mittag-Leffler function (6) to the integrand of (1), we have S γ,q;c ( x (ρ 1) (1 x) 2(ρ+j) 1 1 x ( ( yx 1 x ) ) 2 ; p x (ρ 1) (1 x) 2(ρ+j) 1 ( 1 x ( y ( ) ) 1 x 2 n B p (γ + nq, c γ)(c) nq B(γ, c γ)γ(αn + β) ( x (ρ+n 1) (1 x) 2(ρ+j) 1 1 x ) 2(ρ+n) 1 ( Now using (1) and (1), we get B p (γ + nq, c γ)(c) n y n B(γ, c γ)γ(αn + β) S ( ) 2(ρ+n) 2 Γ(ρ + j)γ(ρ + n) Γ(2ρ + j + n) ( ) 2ρ 2 Γ(ρ + j) B p (γ + nq, c γ)(c) nq B(γ, c γ)γ(αn + β) B p (γ + nq, c γ)γ(ρ + n)γ(c + nq) Γ(c γ)γ(αn + β)γ(2ρ + j + n) y n ( 9 y)n, which upon using (12), we get S ( ) 2ρ 2 Γ(ρ + j) ψ (ρ, 1), (c, q), (γ, 1); 9 y; p.. Particular cases In this section, we obtained some particular cases of Theorems 1-. Setting p in (6) and (1), then we get the following well known Mittag-Leffler functions. and γ,c (z) B p (γ + n, c γ)(c) n z n B(γ, c γ)γ(αn + β), (19) γ,q;c (z) B p (γ + nq, c γ)(c) nq z n B(γ, c γ)γ(αn + β). (2)

8 JFCA-218/9(1) A NW CLASS OF INTGRALS INVOLVING XTNDD Corollary.1. Let y, c, ρ, j, α, β, γ C; with R(α) > max{, R(c) 1}; R(c) >. R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: ( x (ρ+j 1) (1 x) (2ρ 1) 1 x 1 x ( ) 2(ρ+j) 2 Γ(ρ + j) 2ψ 2 ) (ρ 1) ( ( γ y 1 x ) (ρ, 1), (γ, 1); (β, α), (2ρ + j, 1); (1 x) 2) Corollary.2. Let y, c, ρ, j, α, β, γ C; with R(α) > max{, R(c) 1}; R(c) >. R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: ( x (ρ 1) (1 x) 2(ρ+j) 1 1 x 1 x ) ( (ρ+j 1) ( γ yx 1 x ) ) 2 ( ) 2(ρ) (ρ, 1), (γ, 1); 2 Γ(ρ + j) 2ψ 2 ; 9 y. (22) (β, α), (2ρ + j, 1); Corollary.. Let y, c, ρ, j, α, β, γ C; with R(α) > max{, R(c) 1}; R(c) >. R(γ) >, R(2ρ + j) > and x >. Then the following formula holds true: y. (21) ( x (ρ+j 1) (1 x) (2ρ 1) 1 x 1 x ) (ρ 1) ( ( γ,q y 1 x ) (1 x) 2) ( ) 2(ρ+j) (ρ, 1), (γ, q); 2 Γ(ρ + j) 2ψ 2 ; y (2) (β, α), (2ρ + j, 1); Corollary.. Let ρ, j, α, β, γ C; with R(α) >, R(c) > R(γ) >, R(2ρ+j) > and x >. Then the following formula holds true: ( x (ρ 1) (1 x) 2(ρ+j) 1 1 x ( ) 2(ρ) 2 Γ(ρ + j) 2ψ 2 Γ(ρ) 1 x ) ( (ρ+j 1) ( γ,q yx (ρ, 1), (γ, q); (β, α), (2ρ + j, 1); ; 9 y 1 x ) 2 ). (2) References [1] D. Baleanu, K. Diethelm,. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Sci.,, 1-16, 212. [2] M.A. Chaudhry, A. Qadir, M. Rafique, S.M. Zubair, xtension of ulers beta function. J. Comput. Appl. Math. 78, 1, 19-2, [] J. Choi, P. Agarwal, Certain unified integrals associated with Bessel functions, Boundary Value Problems 21, 95, 21. [] J. Choi, P. Agarwal, S. Mathur, S.D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bull. Korean Math. Soc. 51, 995-1, 21. [5] M.M. Dzrbasjan, Integral Transforms and Representations of Functions in the Complex Domain, Nauka, Moscow, (in Russian), [6] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (ds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer Series on CSM Courses and Lectures, vol. 78, Springer-Verlag, Wien, , 1997.

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10 JFCA-218/9(1) A NW CLASS OF INTGRALS INVOLVING XTNDD Gauhar Rahman Department of Mathematics, International Islamic University, Islamabad, Pakistan -mail address: gauhar55uom@gmail.com A. Ghaffar Department of Mathematical Science, BUITMS Quetta, Pakistan -mail address: abdulghaffar.jaffar@gmail.com K.S. Nisar Department of Mathematics, College of Arts and Science-Wadi Al-dawaser, 11991, Prince Sattam bin Abdulaziz University, Alkharj, Kingdom of Saudi Arabia -mail address: n.sooppy@psau.edu.sa, ksnisar1@gmail.com Shahid Mubeen Department of Mathematics, University of Sargodha, Sargodha, Pakistan -mail address: smjhanda@gmail.com