Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 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 obtain results are express in term of extended Wright-type function. Also, we establish certain special cases of our main result. and 1. introduction The Mittag-Leffler functions α z) and z) are defined by the following series: z n α z), z C, Rα) >, 1.1) Γαn + 1) z) z n, z, β C, Rα) >, 1.2) Γαn + 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, 1, 2, 22, ]), Kilbas et al. [11], Chapter 1) and Saigo and Kilbas [12]. In recent years, the Mittag-Leffler function and some of its variety generalizations have been numerically established in the complex plane [9, 2]). A new generalization of the Mittag-Leffler functions z) of 1.2) have defined by Prabhakar [21] as follows: γ z) γ) n z n, z, β C, Rα) >, 1.) Γαn + β) where γ) n denote the well known Pochhammer Symbol which is defined by: { 1, n, γ C), γ) n γγ + 1)γ + 2)...γ + n 1) n N, γ C). 21 Mathematics Subject Classification. B2, C2, C5, C6, B15, C5. Key words and phrases. extended Mittag-Leffler function; Wright-type hypergeometric functions; extended Wright-type hypergeometric functions. Corresponding author. 1 217 by the authors). Distributed under a Creative Commons CC BY license.
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 2 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN In fact, the following special cases are satisfied: 1 z) z), 1 α,1z) z). 1.) 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 [8],[25, 26, 27]). Srivastava and Tomovski [29] have defined further generalization of the Mittag-Leffler function 1 z) of ), which is defined as: γ,k z) γ) nk z n Γαn + β), 1.5) where z, α, β C; Rα) > max{, Rk) 1}; Rk) >. Very recently, Ozarslan and Yilmaz [19] 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 Bγ, c γ)γαn + β), 1.6) where p >, Rc) > Rγ) > 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/t1 t)) dt, 1.7) where Rp) >, Rx) > and Ry) >. If p, then B p x, y) reduces to the following beta function: Bx, y) The gamma function is defined by: Γz) t t By inspection, we conclude the following relation t x 1) 1 t) y 1) dt, 1.8) t z 1) e t) dt; Rz) >. 1.9) Γz + 1) zγz). 1.1) The generalized hypergeometric function p F q z) is defined in [6] as: a 1 ), a 2 ),..., a p ); pf q z) p F q 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 1.11)
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 A NW CLASS OF INTGRALS INVOLVING XTNDD MITTAG-LFFLR FUNCTIONS 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. Sharma and Devi [2] 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); b j, β j ) 1,p, c, 1); z Γa 1 + A 1 n)...γa p + A p n)b p γ + n, c γ)z n Γb 1 + β 1 n)...γb q + β q n)γc γ) 1.12) where i 1, 2,..., p, j 1, 2,..., q, Rp) >, Rc) > Rγ) > and p. Recently, Mittal et al. [15] defined an extended generalized Mittag-Leffer function as: γ,q;c z; p) B p γ + nq, c γ)c) nq z n Bγ, c γ)γαn + β), 1.1) where α, β, γ C, Rα) >, Rβ) >, Rγ) >, q > and B p x, y) is an extended beta function defined in 1.7). In this paper, we derive new integral formulas involving the Mittag-Leffler function 1.6) and 1.1). 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 Γa)Γb) Γa + b), 1.1) where Ra) >, Rb) >. 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, 2,, 1, 16, 17, 18] and the references cited therein. 2. Main Result In this section, the generalized integral formulas involving the Mittag-Leffler function 1.6) and 1.1), are established here by inserting with the suitable argument in the integrand of 1.1). Theorem 1. Let y, c, ρ, j, α, β, γ C; with Rα) > max{, Rc) 1}; Rc) >. Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) 1 x ) ρ 1) γ;c y 1 x ) ) 1 x) 2
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN ) 2ρ+j) 2 Γρ + j) Γγ) ψ ρ, 1), c, 1), γ, 1); β, α), 2ρ + j, 1), c, 1); y; p.2.1) Proof. Let S 1 be the left-hand side of 2.1). Now applying the series representation of the Mittag-Leffler function 1.6) to the integrand of 1.1) and by interchanging the order of integration and summation under the given condition in Theorem 1), we have S 1 x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) 1 x ) ρ 1) γ;c x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) 1 x ) ρ 1) B p γ + n, c γ)c) n Bγ, c γ)γαn + β) x ρ+j 1) 1 x) 2ρ+n) 1 1 x B p γ + n, c γ)c) n y n Bγ, c γ)γαn + β). Now using 1.6) and 1.1), we get S 1 ) 2ρ+j) 2 Γρ + j)γρ + n) Γ2ρ + j + n) ) 2ρ+j) 2 Γρ + j) Γγ) which upon using 1.12), we get ) 2ρ+j) 2 Γρ + j) S 1 ψ Γγ) ) ) y 1 x 1 x) 2 n ) 2ρ+j) 1) 1 x ) ρ+n 1) y 1 x B p γ + n, c γ)c) n y n Bγ, c γ)γαn + β) B p γ + n, c γ)γρ + n)γc + n) y n Γc γ)γαn + β)γ2ρ + j + n), ρ, 1), c, 1), γ, 1) β, α), 2ρ + j, 1), c, 1) ; y; p. ) 1 x) 2 ) Theorem 2. Let y, c, ρ, j, α, β, γ C; with Rα) > max{, Rc) 1}; Rc) >. Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ 1) 1 x) 2ρ+j) 1 1 x ) 2ρ 1) 1 x ) ρ+j 1) γ;c ; p ) 2ρ) 2 Γρ + j) Γγ) ψ ρ, 1), c, 1), γ, 1); β, α), 2ρ + j, 1), c, 1); ; 9 y; p. 2.2)
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 A NW CLASS OF INTGRALS INVOLVING XTNDD MITTAG-LFFLR FUNCTIONS 5 Proof. Let S 2 be the left-hand side of 2.2). Now applying the series representation of the Mittag-Leffler function 1.6) to the integrand of 1.1), we have S 2 ; p x ρ 1) 1 x) 2ρ+j) 1 1 x ) 2ρ 1) 1 x ) ρ+j 1) γ;c x ρ 1) 1 x) 2ρ+j) 1 B p γ + n, c γ)c) n Bγ, c γ)γαn + β) 1 x x ρ+n 1) 1 x) 2ρ+j) 1 1 x B p γ + n, c γ)c) n y n Bγ, c γ)γαn + β). ) 2ρ 1) 1 x ) ρ+j 1) y ) ) 1 x 2 n Now using 1.6) and 1.1), we get S 2 ) 2ρ+n) 2 Γρ + j)γρ + n) Γ2ρ + j + n) ) 2ρ 2 Γρ + j) Γγ) which upon using 1.12), we get ) 2ρ 2 Γρ + j) S 2 ψ Γγ) ) 2ρ+n) 1 1 x ) ρ+j 1) B p γ + n, c γ)c) n y n Bγ, c γ)γαn + β) B p γ + n, c γ)γρ + n)γc + n) Γc γ)γαn + β)γ2ρ + j + n) ρ, 1), c, 1), γ, 1); β, α), 2ρ + j, 1), c, 1); ; 9 y; p. 9 y)n, Theorem. Let y, c, ρ, j, α, β, γ C; with Rα) > max{, Rc) 1}; Rc) >. Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) ) 2ρ+j) 2 Γρ + j) ψ Γγ) 1 x ) ρ 1) γ,q;c y 1 x ) ) 1 x) 2 ; p ρ, 1), c, q), γ, 1); β, α), 2ρ + j, 1), c, 1); ; y; p 2.) Proof. Let S be the left-hand side of 2.). Now applying the series representation of the Mittag-Leffler function 1.6) to the integrand of 1.1) and by interchanging the order of
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 6 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN integration and summation under the given condition in Theorem 1), we have S x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) 1 x ) ρ 1) γ,q;c y 1 x x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) 1 x ) ρ 1) B p γ + nq, c γ)c) nq Bγ, c γ)γαn + β) x ρ+j 1) 1 x) 2ρ+n) 1 1 x B p γ + nq, c γ)c) nq y n Bγ, c γ)γαn + β). Now using 1.6) and 1.1), we get S 1 ) 2ρ+j) 2 Γρ + j)γρ + n) Γ2ρ + j + n) ) 2ρ+j) 2 Γρ + j) Γγ) which upon using 1.12), we get ) 2ρ+j) 2 Γρ + j) S ψ Γγ) ) ) y 1 x 1 x) 2 n ) 2ρ+j) 1) 1 x ) ρ+n 1) B p γ + nq, c γ)c) nq y n Bγ, c γ)γαn + β) B p γ + nq, c γ)γρ + n)γc + nq) y n Γc γ)γαn + β)γ2ρ + j + n), ρ, 1), c, 1), γ, 1); β, α), 2ρ + j, 1), c, 1); ; y; p. ) 1 x) 2 ; p Theorem. Let ρ, j, α, β, γ C; with Rα) >, Rc) > Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ 1) 1 x) 2ρ+j) 1 1 x ) 2ρ 1) 1 x ) ρ+j 1) γ,q;c ; p ) 2ρ) 2 Γρ + j) Γρ) ψ ρ, 1), c, q), γ, 1); β, α), 2ρ + j, 1), c, 1); ; 9 y; p. 2.) Proof. Let S be the left-hand side of 2.). Now applying the series representation of the Mittag-Leffler function 1.6) to the integrand of 1.1), we have S x ρ 1) 1 x) 2ρ+j) 1 1 x ) 2ρ 1) 1 x ) ρ+j 1) γ,q;c ; p )
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 A NW CLASS OF INTGRALS INVOLVING XTNDD MITTAG-LFFLR FUNCTIONS 7 x ρ 1) 1 x) 2ρ+j) 1 1 x B p γ + nq, c γ)c) nq Bγ, c γ)γαn + β) x ρ+n 1) 1 x) 2ρ+j) 1 1 x B p γ + nq, c γ)c) n y n Bγ, c γ)γαn + β) ) 2ρ 1) 1 x ) ρ+j 1) y ) ) 1 x 2 n Now using 1.1) and 1.1), we get S ) 2ρ+n) 2 Γρ + j)γρ + n) Γ2ρ + j + n) ) 2ρ 2 Γρ + j) Γγ) which upon using 1.12), we get S ) 2ρ 2 Γρ + j) Γγ) ψ ) 2ρ+n) 1 1 x ) ρ+j 1) B p γ + nq, c γ)c) nq y n Bγ, c γ)γαn + β) B p γ + nq, c γ)γρ + n)γc + nq) Γc γ)γαn + β)γ2ρ + j + n) ρ, 1), c, q), γ, 1); β, α), 2ρ + j, 1), c, 1); 9 y; p 9 y)n,.. Particular cases In this section, we obtained some particular cases of Theorems 1-. Setting p in 1.6) and 1.1), then we get the following well known Mittag-Leffler functions. and γ,c z) B p γ + n, c γ)c) n z n Bγ, c γ)γαn + β),.1) γ,q;c z) B p γ + nq, c γ)c) nq z n Bγ, c γ)γαn + β)..2) Corollary.1. Let y, c, ρ, j, α, β, γ C; with Rα) > max{, Rc) 1}; Rc) >. Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) 1 x ) ρ 1) γ y 1 x ) ) 1 x) 2
Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 8 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN ) 2ρ+j) 2 Γρ + j) Γγ) 2ψ 2 ρ, 1), γ, 1); β, α), 2ρ + j, 1); y..) Corollary.2. Let y, c, ρ, j, α, β, γ C; with Rα) > max{, Rc) 1}; Rc) >. Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ 1) 1 x) 2ρ+j) 1 1 x ) 2ρ 1) 1 x ) ρ+j 1) γ ) 2ρ) 2 Γρ + j) Γγ) 2ψ 2 ρ, 1), γ, 1); β, α), 2ρ + j, 1); ; 9 y..) Corollary.. Let y, c, ρ, j, α, β, γ C; with Rα) > max{, Rc) 1}; Rc) >. Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ+j 1) 1 x) 2ρ 1) 1 x ) 2ρ+j) 1) ) 2ρ+j) 2 Γρ + j) 2ψ 2 Γγ) 1 x ρ, 1), γ, q); ) ρ 1) γ,q y 1 x ) ) 1 x) 2 β, α), 2ρ + j, 1); ; y.5) Corollary.. Let ρ, j, α, β, γ C; with Rα) >, Rc) > Rγ) >, R2ρ + j) > and x >. Then the following formula holds true: x ρ 1) 1 x) 2ρ+j) 1 1 x ) 2ρ 1) 1 x ) ρ+j 1) γ,q ) 2ρ) 2 Γρ + j) Γρ) 2ψ 2 ρ, 1), γ, q); β, α), 2ρ + j, 1); References ; 9 y..6) [1] D. Baleanu, K. Diethelm,. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Sci., 212), 1-16. [2] J. Choi and P. Agarwal, Certain unified integrals associated with Bessel functions, Boundary Value Problems 21 21), 95. [] J. Choi, P. Agarwal, S. Mathur and S.D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bull. Korean Math. Soc. 51 21), 995-1. [] M.M. Dzrbasjan, Integral Transforms and Representations of Functions in the Complex Domain, Nauka, Moscow, in Russian), 1966).
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Preprints www.preprints.org) NOT PR-RVIWD Posted: 1 May 217 doi:1.29/preprints2175.222.v1 1 G. RAHMAN, A. GHAFFAR, K.S. NISAR AND S. MUBN [2] H.J. Seybold, R. Hilfer, Numerical results for the generalized MittagLeffler function, Fract. Calc. Appl. Anal. 8 25), 12719. [25] H.M. Srivastava, A contour integral involving Foxs H-function, Indian J. Math. 1 1972), 16. [26] H.M. Srivastava, A note on the integral representation for the product of two generalized Rice polynomials, Collect. Math. 2 197), 117121. [27] H.M. Srivastava, K.C. Gupta, S.P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982). [28] H.M. Srivastava, C.M. Joshi, Integral Representation For The Product Of A Class Of Generalized Hypergeometric Polynomials, Acad. Roy. Belg. Bull. Cl. Sci. Ser. 5) 6 197), 919926. [29] H.M. Srivastava, Z. Tomovski,? Fractional calculus with an integral operator containing a generalized MittagLeffler function in the kernel, Appl. Math. Comput. 211 29), 189-21. [] V. V. Uchaikin, Fractional Derivatives for Physicists and ngineers, Volume I, Background and Theory Volume II Applications, Nonlinear Physical Science, Springer-Verlag, BerlinHeidelberg, 21). 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 Kottakkaran Sooppy 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: ksnisar1@gmail.com, n.sooppy@psau.edu.sa Shahid Mubeen: Department of Mathematics, University of Sargodha, Sargodha, Pakistan -mail address: smjhanda@gmail.com