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Kyungpook Mathematical Journal 2019; 59(1): 125-134

Published online March 31, 2019

Copyright © Kyungpook Mathematical Journal.

Pathway Fractional Integral Formulas Involving Extended Mittag-Leffler Functions in the Kernel

Gauhar Rahman, Kottakkaran Sooppy Nisar, Junesang Choi*, Shahid Mubeen, Muhammad Arshad

Department of Mathematics, International Islamic University, Islamabad, Pakistan
e-mail : gauhar55uom@gmail.com

Department of Mathematics, College of Arts and Science-Wadi Al dawser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia
e-mail : ksnisar1@gmail.com and n.sooppy@psau.edu.sa

Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea
e-mail : junesang@mail.dongguk.ac.kr

Department of Mathematics, University of Sargodha, Sargodha, Pakistan
e-mail : smjhanda@gmail.com

Department of Mathematics, International Islamic University, Islamabad, Pakistan
e-mail : marshad_zia@yahoo.com

Received: January 23, 2017; Accepted: September 29, 2017

Abstract

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Since the Mittag-Leffler function was introduced in 1903, a variety of extensions and generalizations with diverse applications have been presented and investigated. In this paper, we aim to introduce some presumably new and remarkably different extensions of the Mittag-Leffler function, and use these to present the pathway fractional integral formulas. We point out relevant connections of some particular cases of our main results with known results.

Keywords: gamma function, beta function, extended Mittag-Leffler functions, pathway integral operator.

1. Introduction and Preliminaries

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The Swedish mathematician Gosta Mittag-Leffler [19] introduced the so-called Mittag-Leffler function

Eα(z)=∑n=0∞znΓ(αn+1)         (z∈ℂ;ℜ(α)>0),

where Γ is the familiar gamma function whose Euler’s integral is given by (see, e.g., [34, Section 1.1])

Γ(z)=∫0∞tz-1e-t dt         (ℜ(z)>0).

Here and in the following, let ℂ, ℝ, ℝ+, ℤ0-, and ℕ be the sets of complex numbers, real numbers, positive real numbers, non-positive integers, and positive integers, respectively, and let ℝ0+:=ℝ+∪{0}. Wiman [39] generalized the Mittag-Leffler function (1.1) as follows:

Eα,β(z)=∑n=0∞znΓ(αn+β)         (z∈ℂ;   min{ℜ(α),ℜ(β)}>0).

The Mittag-Leffler function Eα (1.1) and the extended function Eα,β (1.3) have been extended in a number of ways and, together with their extensions, applied in various research areas. For those extensions and applications, we refer the reader, for example, to [1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 21, 25, 27, 30, 31, 32, 33, 35, 36, 37].

Here, for an easier reference, we give a brief history of some chosen extensions of the Mittag-Leffler function Eα (1.1) and the extended function Eα,β (1.3). Prabhakar [25] introduced an extension of the function Eα,β (1.3)

Eα,βγ(z)=∑n=0∞(γ)nn!Γ(αn+β)zn(z∈ℂ;   min{ℜ(α),ℜ(β),ℜ(γ)}>0),

where the familiar Pochhammer symbol (λ)ν is defined (for λ, ν ∈ ℂ) by

(λ)ν:=Γ(λ+ν)Γ(λ)         (λ+ν∈ℂℤ0-)={1(ν=0)λ(λ+1)⋯(λ+n-1)(ν=n∈ℕ).

Shukla and Prajapati [31] (see also [37]) defined and investigated the following extension

Eα,βγ,q(z)=∑n=0∞(γ)qnn!Γ (αn+β)zn(z∈ℂ;   min{ℜ(α),ℜ(β),ℜ(γ)}>0;   q∈(0,1)∪ℕ).

Salim [28] introduced

Eα,βγ,δ(z)=∑n=0∞(γ)nΓ(αn+β)zn(δ)n(z∈ℂ;min{ℜ(α),ℜ(β),ℜ(γ),ℜ(δ)}>0).

Salim and Faraj [29] generalized the function (1.7)

Eα,β,pγ,δ,q=∑n=0∞(γ)qnΓ(αn+β)zn(δ)pn(z∈ℂ;min{ℜ(α),ℜ(β),ℜ(γ),ℜ(δ)}>0;p,q∈ℝ+).

Özarslan and Yilmaz [23] presented the following extension

Eα,βγ;c(z;p)=∑n=0∞Bp(γ+n,c-γ)B(γ,c-γ)(c)nΓ(αn+β)znn!(z∈ℂ;   min{ℜ(α),ℜ(β)}>0;   ℜ(c)>ℜ(γ)>0;p∈ℝ0+).

Here Bp(x, y) is the extended beta function (see [4, 15])

Bp(x,y)=∫01tx-1(1-t)y-1e-pt(1-t)dt(p∈ℝ0+;   min{ℜ(x),ℜ(y)}>0),

whose particular case when p = 0 reduces to the well-known beta function (see, e.g., [34, Section 1.1])

B(x,y)=∫01tx-1(1-t)y-1dt         (min{ℜ(x),ℜ(y)}>0)=Γ(x)Γ(y)Γ(x+y)         (x,y∈ℂℤ0-).

By using the pathway idea in [16] (see also [17, 18]), Nair [20] introduced the following pathway fractional integral operator

(P0+μ,λf)   (x)=xμ∫0[xα(1-λ)][1-α(1-λ)τx]μ1-λf(τ)dτ(ℜ(μ)>0;   α∈ℝ+;   λ<1),

where f ∈ L(a, b) (the set of measurable real or complex valued functions) and λ is a pathway parameter.

Remark 1.1

For a given scalar λ ∈ ℝ, the pathway model for scalar random variables is represented by the following probability density function (see, e.g., [2])

f(x)=c∣x∣ν-1[1-α(1-λ)∣x∣η]μ1-λ(x∈ℝ;μ∈ℝ0+,η,ν,[1-α(1-λ)∣x∣η]∈ℝ+),

where c is the normalizing constant and λ is called the pathway parameter. For λ ∈ ℝ, the normalizing constant c is given as follows (see, e.g., [2]):

c={η[α(1-λ)]νηΓ(νη+μ1-λ+1)2 Γ(νη) Γ(μ1-λ+1)(λ<1),η[α(1-λ)]νη Γ(μλ-1)2 Γ(νη) Γ(μλ-1-νη)(1<λ<1+η/ν),η(αμ)νη2 Γ(νη)(λ→1).

Setting λ = 0, α = 1 and replacing μ by μ−1 in (1.12) reduces to the well-known left-sided Riemann-Liouville fractional integral operator Ia+μ (e.g., [3, 22, 24, 26, 38])

(P0+μ-1,0f)   (x)=x1-μ Γ(μ)   (I0+μf)   (x)         (ℜ(μ)>1),

where Ia+μ is defined by

(Ia+μf)   (x):=1Γ(μ)∫ax(x-τ)μ-1 f(τ) dτ         (x>a;ℜ(μ)>0)

and [a, b] (−∞ < a < b < ∞) is a finite interval on the real line ℝ.

In this paper, we aim to introduce (presumably) new and (remarkably) different extensions of the Mittag-Leffler function, which are also associated with the pathway fractional integral operator (1.12) to present their integral formulas. Relevant connections of some particular cases of the main results presented here with those earlier ones are also pointed out.

2. Pathway Fractional Integration of an Extended Mittag-Leffler Function

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By considering (1.6) and (1.9) together, we begin by defining a (presumably) new extension of the Mittag-Leffler function as follows:

Eρ,β,δγ,q;c(z;p)=∑n=0∞Bp(γ+nq,c-γ)B(γ,c-γ)(c)nqΓ(ρn+β)zn(δ)n(q∈ℝ+;min{ℜ(ρ),ℜ(β),ℜ(δ)}>0;   ℜ(c)>ℜ(γ)>0;p∈ℝ0+),

where Bp(x, y) is the same as in (1.10).

It is easy to see that (2.1) contains the Mittag-Leffler function and each of its extensions (or generalizations) given in Section 1 as in the following remark.

Remark 2.1

  • The particular case of (2.1) when p = 0 and q = 1 reduces to (1.7).

  • The particular case of (2.1) when δ = 1 is a generalization of (1.6) and (1.9).

We establish a pathway integration formula involving the extended Mittag-Leffler function (2.1), which is asserted in Theorem 2.1.

Theorem 2.1

Let ρ, β, γ, δ, c, μ ∈ ℂ with min {ℜ(ρ), ℜ(β), ℜ(δ), ℜ(μ)} > 0 and ℜ(c) > ℜ(γ) > 0. Also, let ω ∈ ℝ, α, q ∈ ℝ+, andp∈ℝ0+. Further, let λ < 1 withℜ(μ1-λ)>-1. Then

P0+μ,λ(τβ-1Eρ,β,δγ,q;c(ωτρ;p))   (x)=Γ(1+μ1-λ)xμ+β[α(1-λ)]βEρ,β+1+μ1-λ,δγ,q;c(ω(xα(1-λ))ρ;p).
Proof

Let ℒ1 be the left-hand side of (2.10). By applying (2.1) to (1.12), and interchanging the order of integral and summation, which is verified under the given conditions in this theorem, we obtain

ℒ1=xμ∑n=0∞Bp(γ+nq,c-γ)B(γ,c-γ)(c)nqΓ(ρn+β)(ω)n(δ)n×∫0[xα(1-λ)]τβ+ρn-1[1-α(1-λ)τx]μ1-λdτ

Setting α(1-λ)τx=t and using (1.11), we get

∫0[xα(1-λ)]τβ+ρn-1 [1-α(1-λ)τx]μ1-λdτ=xβ+ρn[α(1-λ)]β+ρnΓ(β+ρn)Γ(μ1-λ+1)Γ(β+μ1-λ+1+ρn).

Using (2.4) in (2.3), in terms of (2.1), we have

ℒ1=xμ+βΓ(1+μ1-λ)[α(1-λ)]β∑n=0∞Bp(γ+nq,c-γ)B(γ,c-γ)(c)nqΓ(ρn+β+1+μ1-λ)(ω(xα(1-λ))ρ)n(δ)n=xμ+βΓ(1+μ1-λ)[α(1-λ)]βEρ,β+1+μ1-λ,δγ,q;c(ω(xα(1-λ))ρ;p),

which is the right-hand side of (2.10). This completes the proof.

Corollary 2.1

Let ρ, β, γ, c, μ ∈ ℂ with min {ℜ(ρ), ℜ(β), ℜ(μ)} > 0, ℜ(c) > ℜ(γ) > 0, and ω ∈ ℝ. Also, let λ < 1 withℜ(μ1-λ)>-1. Then

P0+μ,λ(τβ-1Eρ,βγ(ωτρ))   (x)=xμ+βΓ(1+μ1-λ)[α(1-λ)]βEρ,β+1+μ1-λγ(ω(xα(1-λ))ρ).
Proof

Setting p = 0, δ = 1, and q = 1 in (2.10) together with (1.4) yields the desired result (2.5).

Corollary 2.2

Let ρ, β, γ, μ ∈ ℂ with min {ℜ(ρ), ℜ(β), ℜ(γ)} > 0 and ℜ(μ) > 1. Also, let ω ∈ ℝ. Then

P0+μ-1,0(τβ-1 Eρ,βγ(ωτρ)) (x)=Γ(μ) xμ-1+β Eρ,βγ(ω xρ).
Proof

Setting p = λ = 0, q = δ = α = 1, and replacing μ by μ − 1 in (2.10), and using (1.4), we are led to (2.6).

Corollary 2.3

Let ρ, β, γ, μ ∈ ℂ with min {ℜ(ρ), ℜ(β), ℜ(γ)} > 0 and ℜ(μ) > 1. Also, let ω ∈ ℝ and q ∈ ℝ+. Then

P0+μ-1,0 (τβ-1 Eρ,βγ,q(ω τρ))   (x)=Γ(μ) xμ-1+β Eρ,βγ,q(ω xρ).
Proof

Setting p = λ = 0, δ = α = 1, and replacing μ by μ − 1 in (2.10), and using (1.6), we obtain the result (2.7).

Corollary 2.4

Let ρ, β, γ, δ, μ ∈ ℂ with min {ℜ(ρ), ℜ(β), ℜ(δ), ℜ(γ)} > 0 and ℜ(μ) > 1. Also, let ω ∈ ℝ. Then

P0+μ-1,0 (τβ-1 Eρ,βγ,δ(ω τρ))   (x)=Γ(μ) xμ-1+β Eρ,β+μγ,δ (ω xρ).
Proof

Setting p = λ = 0, q = α = 1, and replacing μ by μ − 1 in (2.10), and using (1.7), we obtain the result (2.8).

Remark 2.2

For the results (2.5), (2.6), (2.7), and (2.8), we refer the reader, respectively, to [20, 25, 28, 31, 37]. In view of (1.15), the results (2.6), (2.7), and (2.8) can yield the corresponding ones for the left-sided Riemann-Liouville fractional integration operator Ia+μ.

We present a further generalization of the Mittag-Leffler function, which is a slight extension of the extended Mittag-Leffler function in (2.1).

Eρ,β,sγ,δ,q;c(z;p)=∑n=0∞Bp(γ+nq,c-γ)B(γ,c-γ)(c)nqΓ(ρn+β)zn(δ)sn(q,s∈ℝ+; min{ℜ(ρ),ℜ(β),ℜ(δ)}>0; ℜ(c)>ℜ(γ)>0;p∈ℝ0+),

where Bp(x, y) is the same as in (1.10).

It is easy to see that the particular case s = 1 of (2.9) reduces to (2.1). For more particular cases, see Remark 2.1. We present a pathway integration formula involving the extended Mittag-Leffler function (2.9), which is asserted in Theorem 2.2.

Theorem 2.2

Let ρ, β, γ, δ, c, μ ∈ ℂ with min {ℜ(ρ), ℜ(β), ℜ(δ), ℜ(μ)} > 0 and ℜ(c) > ℜ(γ) > 0. Also, let ω ∈ ℝ, α, q, s ∈ ℝ+, andp∈ℝ0+. Further, let λ < 1 withℜ(μ1-λ)>-1. Then

P0+μ,λ (τβ-1Eρ,β,sγ,δ,q;c (ωτρ;p))   (x)=Γ(1+μ1-λ)xμ+β[α(1-λ)]βEρ,β+1+μ1-λ,sγ,δ,q;c(ω(xα(1-λ))ρ;p).
Proof

The proof runs parallel to that of Theorem 2.1. We omit the details.

We can also provide many particular cases of Theorem 2.2, including those results corresponding to Corollaries 2.1–2.4. The details are left to the interested reader.

3. Concluding Remarks

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Among a variety of extensions (or generalizations) of the Mittag-Leffler function, the extension (2.1) (or (2.9)) seems to be a different one.

One of the Erdélyi-Kober type fractional integrals (see [14, p.105, Eq. (2.126.1)]) appears to be closely related to the pathway fractional integration operator (1.12), even though one integral cannot contain the other one as a purely special case. For generalized multi-index Mittag-Leffler functions and their applications, we refer the reader, for example, to [5] and the references cited therein.

The main results presented here, as their special cases, include many earlier ones, in particular, including some of the identities provided by Nair [20] who first introduced the pathway fractional integral operator (1.12).

  1. P. Agarwal, and J. Choi. Certain fractional integral inequalities associated with pathway fractional integral operators. Bull. Korean Math. Soc., 53(1)(2016), 181-193.
    CrossRef
  2. D. Baleanu, and P. Agarwal. A composition formula of the pathway integral transform operator. Note Mat., 34(2014), 145-155.
  3. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo. Fractional calculus. Models and numerical methods, , Series on Complexity, Nonlinearity and Chaos 3, World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ, 2012.
    CrossRef
  4. M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris. Extended hypergeometric and confluent hypergeometric functions. Appl. Math. Comput., 159(2004), 589-602.
  5. J. Choi, and P. Agarwal. A note on fractional integral operator associated with multi-index Mittag-Leffler functions. Filomat., 30(7)(2016), 1931-1939.
    CrossRef
  6. J. Choi, and P. Agarwal. Certain inequalities involving pathway fractional integral operators. Kyungpook Math. J., 56(2016), 1161-1168.
    CrossRef
  7. M. M. Džrbašjan. Integral transforms and representations of functions in the complex domain, , Nauka, Moscow, (in Russian), 1966.
  8. R. Gorenflo, A. A. Kilbas, and S. V. Rogosin. On the generalized Mittag-Leffler type functions. Integral Transforms Spec. Funct., 7(1998), 215-224.
    CrossRef
  9. R. Gorenflo, Y. Luchko, and F. Mainardi. Wright functions as scale-invariant solutions of the diffusion-wave equation. J. Comput. Appl. Math., 118(2000), 175-191.
    CrossRef
  10. R. Gorenflo, and F. Mainardi. Fractional calculus: integral and differential equations of fractional order. Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), 223–276, CISM Courses and Lect, 378, Springer, Vienna, 1997.
    CrossRef
  11. R. Gorenflo, F. Mainardi, and H. M. Srivastava. Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena, Proceedings of the Eighth International Colloquium on Differential Equations (Plovdiv,1997), VSP, Utrecht, (1998), 195-202.
  12. R. Hilfer, and H. Seybold. Computation of the generalized Mittag-Leffler function and its inverse in the complex plane. Integral Transforms Spec. Funct., 17(2006), 637-652.
    CrossRef
  13. A. A. Kilbas, and M. Saigo. On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations. Integral Transforms Spec. Funct., 4(1996), 355-370.
    CrossRef
  14. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and applications of fractional differential equations. North-Holland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
  15. F. Mainardi. On some properties of the Mittag-Leffler function, Eα(−tα), completely monotone for t > 0 with 0 < α < 1. Discrete Contin. Dyn. Syst. Ser. B., 19(7)(2014), 2267-2278.
    CrossRef
  16. A. M. Mathai. A pathway to matrix-variate gamma and normal densities. Linear Algebra Appl., 396(2005), 317-328.
    CrossRef
  17. A. M. Mathai, and H. J. Haubold. Pathway model, superstatistics, Tsallis statistics and a generalized measure of entropy. Phys. A., 375(2007), 110-122.
    CrossRef
  18. A. M. Mathai, and H. J. Haubold. On generalized distributions and pathways. Phys. Lett. A., 372(2008), 2109-2113.
    CrossRef
  19. G. M. Mittag-Leffler. Sur la nouvelle fonction Eα (x). C. R. Acad. Sci. Paris., 137(1903), 554-558.
  20. S. S. Nair. Pathway fractional integration operator. Fract. Calc. Appl. Anal., 12(2009), 237-252.
  21. K. S. Nisar, A. F. Eata, M. Al-Dhaifallah, and J. Choi. Fractional calculus of generalized k-Mittag-Leffler function and its applications to statistical distribution. Adv. Difference Equ., (2016) Paper No. 304, 17 pp.
    CrossRef
  22. K. S. Nisar, S. D. Purohit, and S. R. Mondal. Generalized fractional kinetic equations involving generalized Struve function of the first kind. J. King Saud Univ. Sci., 28(2016), 167-171.
    CrossRef
  23. M. A. Özarslan, and B. Yilmaz. The extended Mittag-Leffler function and its properties. J. Inequal. Appl., 85(2014) 2014, 10 pp.
  24. I. Podlubny. Fractional differential equations An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, , Academic Press, San Diego, CA, 1999.
  25. T. R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J., 19(1971), 7-15.
  26. S. D. Purohit. Solutions of fractional partial differential equations of quantum mechanics. Adv. Appl. Math. Mech., 5(2013), 639-651.
    CrossRef
  27. M. Saigo, and A. A. Kilbas. On Mittag-Leffler type function and applications. Integral Transforms Spec. Funct., 7(1998), 97-112.
    CrossRef
  28. T. O. Salim. Some properties relating to the generalized Mittag-Leffler function. Adv. App.l Math. Anal., 4(2009), 21-30.
  29. T. O. Salim, and A. W. Faraj. A generalization of Mittag-Leffler function and integral operator associated with fractional calculus. J. Fract. Calc. Appl., 3(5)(2012), 1-13.
  30. H. J. Seybold, and R. Hilfer. Numerical results for the generalized Mittag-Leffler function. Fract. Calc. Appl. Anal., 8(2005), 127-139.
  31. A. K. Shukla, and J. C. Prajapati. On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl., 336(2007), 797-811.
    CrossRef
  32. H. M. Srivastava. A contour integral involving Fox’s H-function. Indian J. Math., 14(1972), 1-6.
  33. H. M. Srivastava. A note on the integral representation for the product of two generalized Rice polynomials. Collect. Math., 24(1973), 117-121.
  34. H. M. Srivastava, and J. Choi. Zeta and q-Zeta functions and associated series and integrals, , Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  35. H. M. Srivastava, K. C. Gupta, and S. P. Goyal. The H-functions of one and two variables with applications, , South Asian Publishers, New Delhi and Madras, 1982.
  36. H. M. Srivastava, and C. M. Joshi. Integral representation for the product of a class of generalized hypergeometric polynomials. Acad. Roy. Belg. Bull. Cl. Sci., 5(60)(1974), 919-926.
  37. H. M. Srivastava, and Z. Tomovski. Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput., 211(1)(2009), 198-210.
  38. V. V. Uchaikin. Fractional derivatives for physicists and engineers Volume I Background and Theory, Volume II Applications. Nonlinear Physical Science, , Springer-Verlag, Berlin-Heidelberg, 2013.
    CrossRef
  39. A. Wiman. Über den fundamentalsatz in der teorie der funktionen Eα(x). Acta Math., 29(1905), 191-201.
    CrossRef