The J-Generalized P - K Mittag-Leﬄer Function

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The J-Generalized P - K Mittag-Leffler Function

Kuldeep Singh Gehlot and Anjana Bhandari

Government Bangur P.G. College Pali, JNV University Jodhpur, Rajasthan, India-306401.

Department of Science and Humanities, Swarrnim startup and Innovation University,

Gandhinagar, Gujarat, India- 382420. Email: [email protected]

Abstract

We know that the classical Mittag-Leffler function play an important role as solution of frac-tional order differential and integral equations. We introduce the j-generalized p - k Mittag-leffler function. We evaluate the second order differential recurrence relation and four different inte-gral representations and introduce a homogeneous linear differential equation whose one of the solution is the j-generalized p-k Mittag-Leffler function.

Also we evaluate the certain relations that exist between j-generalized p - k Mittag-leffler func-tion and Riemann-Liouville fracfunc-tional integrals and derivatives.

We evaluate Mellin-Barnes integral representation of j-generalized p-k Mittag-Leffler Function. The relationship of j-generalized p-k Mittag-Leffler Function with Fox H-Function and Wright hypergeometric function is also establish. we obtained its Euler transform, Laplace Transform and Mellin transform.

Finally we derive some particular cases.

MSC(2011): 33E12, 33B10, 26A33.

Keywords: The j-generalized p - k Mittag-Leffler Function, The p - k Mittag-Leffler Function, Generalized k-Mittag-Leffler Function, Two parameter pochhammer symbol, Two parameter Gamma function.

1 Introduction

The two parameter pochhammer symbol is recently introduce by [9], equation 2.1, in the form,

1.1 Definition

Let x ∈ C;k, p ∈ R+− {0} and Re(x) > 0, n ∈ N, the p - k Pochhammer Symbol (i.e. Two Parameter Pochhammer Symbol), p(x)n,k is given by

p(x)n,k = ( xp

k )( xp

k +p)( xp

k + 2p)...( xp

k + (n−1)p). (1.1)

And the Two Parameter Gamma Function is given by [9], some of it’s result are,

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1.2 Definition

For x ∈C/kZ−;k, p∈ R+− {0} and Re(x) >0, n ∈N, the p - k Gamma Function (i.e. Two Parameter Gamma Function),pΓk(x) as

pΓk(x) = 1 knlim→∞

n!pn+1(np)xk

p(x)n+1,k

. (1.2)

or

pΓk(x) = 1 knlim→∞

n!pn+1₍_np₎x_k−1

p(x)n,k . (1.3)

The integral representation of p - k Gamma Function is given by

pΓk(x) =

Z ∞

0

e−tkp_tx−1_dt. _(1.4)

pΓk(x) = (p k)

x

kΓk(x) = p x k

k Γ( x

k). (1.5)

p(x)n,k = ( p k)

n₍_x₎

n,k= (p)n( x

k)n. (1.6)

Also for Generalized p - k Pochhammer Symbol, we have

p(x)nq,k= ( p k)

nq₍_x₎

nq,k = (p)nq( x

k)nq = (pq) nq

q

Y

r=1

( x

k +r−1

q )n. (1.7)

p(x)n,k= p

Γk(x+nk)

pΓk(x) . (1.8)

pΓk(x+k) = xp

k pΓk(x). (1.9)

npp(x)n−1,k = p(x)n,k− p(x−k)n,k. (1.10) And

p(x)n+j,k = p(x)j,k× p(x+jk)n,k. (1.11) The Mittag-Leffler functionEα(z) introduced by Gosta Mittag-Leffler [4] in 1903, defined as

Eα(z) =

∞

X

n=0

zn

Γ(αn+ 1). (1.12)

Here z∈C, α>0.

Wiman [2] generalized Eα(z) in 1905 and gave Eα,β(z) known as Wiman function, defined as

Eα,β(z) =

∞

X

n=0

zn

Γ(αn+β). (1.13)

Here z, α, β∈C;Re(α)>0, Re(β)>0.

Prabhakar [17] in 1971, gave next generalization of Mittag-Leffler function and denoted as E_α,βγ (z) and defined as

E_α,βγ (z) =

∞

X

n=0

(γ)n Γ(αn+β)

zn

n!. (1.14)

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Shukla and Prajapati [1] in 2007, gave second generalization of Mittag-Leffler function and denoted it as E_α,βγ,q(z) and defined as,

E_α,βγ,q(z) =

∞

X

n=0

(γ)nq Γ(αn+β)

zn

n!. (1.15)

Here z, α, β, γ ∈C;Re(α)>0, Re(β)>0, Re(γ)>0 and q∈(0,1)∪N.

The function E_α,βγ,q(z) converges absolutely for all z if q < Re(α) + 1 and for |z| < 1 if q =Re(α) + 1.It is entire function of order _Re1₍_α₎.

Gehlot K.S.[8] introduce Generalized k- Mittag-Leffler function in 2012, denoted as GE_k,α,βγ,q (z) and defined for k∈R;z, α, β, γ ∈C;Re(α)>0, Re(β)>0, Re(γ)>0 andq ∈(0,1)∪N,as,

GE_k,α,βγ,q (z) =

∞

X

n=0

(γ)nq,kzn

Γk(nα+β)(n!), (1.16)

where(γ)nq,k is the k- pochhammer symbol and Γk(x) is the k-gamma function given by [15]. The generalized Pochhammer symbol is given as,

(γ)nq = Γ(γ+nq) Γ(γ) =q

qn q

Y

r=1

(γ+r−1

q )n, if q ∈N. (1.17)

Gehlot K.S.[9], Introduce The p- k Mittag-Leffler function in 2018, denoted by pEk,α,βγ,q (z) and defined fork, p∈R+_−{₀_}_;_{α, β, γ}_∈_C/kZ−_;_Re₍_α₎_>₀_{, Re}₍_β₎_>₀_{, Re}₍_γ₎_>_{0 and}_q _∈₍₀_,₁₎_∪_N.

pE_k,α,βγ,q (z) =

∞

X

n=0

p(γ)nq,k pΓk(nα+β)

zn

n!. (1.18)

Wherep(γ)nq,k is two parameter Pochhammer symbol given by equation (1.1) andpΓk(x) is the two parameter Gamma function given by equation (1.3).

L.L.Luque [14]in the year 2019, introduce the L-mittag-Leffler function defined for α, β, γ ∈

C;Re(α)>0, Re(β)>0, Re(γ)>0, j∈N0 by the series

Lγ,j_α,β(z) =

∞

X

n=0

(γ)n+j Γ(nα+β)

zn

(n+j)!, (z∈C). (1.19)

The Fractional Integral operators ([16], Definition 2.1, Page 33) are defined as,

(I₀₊ϑ f)(z) = 1 Γ(ϑ)

Z z

0

f(t)

(z−t)1−ϑdt,(z >0), (1.20)

and

(I₋ϑf)(z) = 1 Γ(ϑ)

Z ∞

z

f(t)

(t−z)1−ϑdt,(z >0), (1.21) The Fractional Derivative ([16], Definition 2.2, Page 35) are defined as,

(D₀₊ϑ f)(z) = (d dz)

[Re(ϑ)]+1₍_I1−ϑ+[Re(ϑ)] 0+ f)(z)

= 1

Γ(1−ϑ+ [Re(ϑ)])( d dz)

[Re(ϑ)]+1

Z z

0

f(t)

(z−t)ϑ−[Re(ϑ)]dt,(z >0), (1.22)

and

(Dϑ₋f)(z) = (− d

dz)

[Re(ϑ)]+1₍_I1−ϑ+[Re(ϑ)]

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= 1

Γ(1−ϑ+ [Re(ϑ)])(− d dz)

[Re(ϑ)]+1

Z ∞

z

f(t)

(t−z)ϑ−[Re(ϑ)]dt,(z >0). (1.23)

Where ϑ∈C(Re(ϑ)>0) .

Wright generalized hypergeometric function [15];

pψq





(α1, A1), . . . ,(αp, Ap); z (β1, B1), . . . ,(βq, Bq);



=

∞

X

n=0

Qp

i=1Γ(αi+Ain)zn

Qq

j=1Γ(βi+Bin)n!

. (1.24)

pψq





(α1, A1), . . . ,(αp, Ap); z (β1, B1), . . . ,(βq, Bq);



=H

1,p p,q+1



−z|

(1−α1, A1), . . . ,(1−αp, Ap);

(0,1),(1−β1, B1), . . . ,(1−βq, Bq);



.

(1.25) Where Hp,qm,n[.] denotes the Fox H-function.

Euler Beta transform,[6],

B[f(z) :a, b] =

Z 1

0

za−1(1−z)b−1f(z)dz. (1.26)

Laplace transform,([7],equation3.1.1),

L[f(z) :s] =

Z ∞

0

e−szf(z)dz. (1.27)

Mellin transform,([7],equation 4.1.1),

M[f(z) :s] =

Z ∞

0

zs−1f(z)dz=f∗(s), Re(s)>0, (1.28)

then

f(z) =M−1[f∗(s) :x] =

Z

f∗(s)x−sds, (1.29)

Throughout this paper LetC, R+, Re(), Z−, N0 andN be the sets of complex numbers, positive

real numbers, real part of complex number, negative integer, whole number and natural numbers respectively.

2 The j-generelized p - k Mittag-Leffler function

In this section we introduce the j-generelized p - k Mittag-Leffler function and prove some of its properties.

2.1 Definition

Let k, p ∈ R+− {0};α, β, γ ∈ C/kZ−;Re(α) > 0, Re(β) > 0, Re(γ) > 0, j ∈ N0 and q ∈

(0,1)∪N.The j-generelized p - k Mittag-Leffler function denoted byjpE_k,α,βγ,q (z) and defined as

j pE

γ,q

k,α,β(z) =

∞

X

n=0

p(γ)(n+j)q,k pΓk(nα+β)

zn

(n+j)!, z∈C. (2.1)

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Particular cases : For some particular values of the parametersj, p, q, k, α, β, γ we can obtain certain defined and undefined Mittag-Leffler functions:

(a) Forj= 0 equation (2.1), reduces in the p-k Mittag-Leffler functions defined by [10],

0

pE γ,q

k,α,β(z) =

∞

X

n=0

p(γ)nq,k pΓk(nα+β)

zn

n!, z∈C. (2.2)

(b) Forq = 1 equation (2.1), reduces in j form of p-k Mittag-Leffler functions defined as,

j pE

γ,1

k,α,β(z) =

∞

X

n=0

p(γ)(n+j),k pΓk(nα+β)

zn

(n+j)!, z∈C. (2.3)

(c) Forq = 1, p=k equation (2.1), reduces in j form of k- Mittag-Leffler functions defined as,

j kE

γ,1

k,α,β(z) =

∞

X

n=0

(γ)₍_n₊_j₎_,k Γk(nα+β)

zn

(n+j)!, z∈C. (2.4)

(d) For q = 1, j = 0 equation (2.1), reduces in generalized form of k- Mittag-Leffler functions defined as.

pE_k,α,βγ,1 (z) =

∞

X

n=0

p(γ)n,kzn pΓk(nα+β)(n!)

. (2.5)

(e) For p=k, j = 0 equation (2.1), reduces in Generalized k- Mittag-Leffler functions defined by [8].

kEk,α,βγ,q (z) =

∞

X

n=0

k(γ)nq,kzn kΓk(nα+β)(n!)

=GE_k,α,βγ,q (z). (2.6)

(f ) Forp=k, q= 1, j= 0 equation (2.1), reduces in k - Mittag-Leffler functions defined by [3].

kEk,α,βγ,1 (z) =

∞

X

n=0

(γ)n,kzn Γk(nα+β)(n!)

=E_k,α,βγ (z). (2.7)

(g)Forp=kandk= 1, j = 0 equation (2.1), reduces in Mittag-Leffler functions defined by [1].

1E₁γ,q_,α,β(z) =

∞

X

n=0

(γ)nqzn

Γ(nα+β)(n!) =E γ,q

α,β(z). (2.8)

(h)For p=k=q= 1 equation (2.1), reduces in L-Mittag-Leffler functions defined by [14].

1E₁γ,_,α,β1 (z) =

∞

X

n=0

(γ)n+jzn

Γ(nα+β)(n+j)! =L γ,j

α,β(z). (2.9)

(i) Forp=k, q= 1, j= 0 and k= 1 equation (2.1), reduces in Mittag-Leffler functions defined by [17].

1E₁γ,_,α,β1 (z) =

∞

X

n=0

(γ)nzn

Γ(nα+β)(n!) =E γ

α,β(z), (2.10)

(j) Forp =k, q= 1, k = 1, j = 0 and γ = 1 equation (2.1), reduces in Mittag-Leffler functions defined by [3].

1E11,α,β,1 (z) =

∞

X

n=0

zn

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(k) For p = k, q = 1, k = 1, γ = 1, j = 0 and β = 1 equation (2.1), reduces in Mittag-Leffler functions defined by [4].

1E11,α,,11(z) =

∞

X

n=0

zn

Γ(nα+ 1) =Eα(z). (2.12)

Theorem 2.1: The j-generelized p - k Mittag-Leffler function defined by equation (2.1) is an entire function of order

1 ρ =Re(

α

k)−q+ 1. (2.13)

Proof: Let R is the radius of convergence of the j-generelized p - k Mittag-Leffler function. The asymptotic Striling formula for Gamma function and factorial are given by,[5]

Γ(az+b) =√2πe−az(az)az+b−12 h

1 +o(1 z)

i

,(arg(az+b)< π;pzp→ ∞). (2.14)

and

n! =√2πe−n(n)n+12 h

1 +o(1 n)

i

,(n∈N;n→ ∞). (2.15)

From equation (2.1), we have

j pE

γ,q

k,α,β(z) =

∞

X

n=0

zn (n+j)! =

∞

X

n=0

Cnzn,

since

R= lim sup n→∞

Cn Cn+1

, Cn Cn+1

=

1 (n+j)!×

pΓk(nα+α+β)(n+ 1 +j)! p(γ)(n+1+j)q,k

using equations (2.19) and (2.20) of [9], we have

Cn Cn+1

= (n+ 1 +j) p

α−qk k

Γ(nq+jq+γ_k) Γ(nq+jq+q+γ_k)

Γ(nα+_kα+β) Γ(nα_k+β)

,

using equation (2.15), we have

' p

α k−q

q −q ( α k) α k n α k+1−q

→ ∞

when,

Re(α

k + 1−q)>0,

Thus, the j-generelized p - k Mittag-Leffler function is an entire function for q < Re(α_k) + 1 To determine the order ρ,

ρ= lim sup n→∞

nlnn

ln(_|_C1_n|), (2.16)

1 Cn =

pΓk(nα+β)(n+j)! p(γ)(n+j)q,k

,

using theorem 2.19 and 2.20 of [9], we have

1 Cn =

(n+j)! k p γ k+

nα+β k −

γ+(n+j)qk k

Γ(γ_k)Γ(nα_k+β) Γ(γ_k+ (n+j)q)

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By using equation (2.12) and (2.13), we get 1 Cn =k

−1₍₂_π₎1 2 p

(α−_kqk)n+β_k−jq ( α k) nα k+ β k− 1 2 n nα k + β k− γ

k−nq−jq+n+j+

1 2 e

−nRe(α k+1−q)

taking lnof above equation and put in equation (2.13), we have the order of j-generelized p - k Mittag-Leffler function is given by

ρ= k

Re(α)−qk+k. Hence.

Theorem 2.2: The functional relation between the j-generelized p - k Mittag-Leffler func-tion given by equafunc-tion (2.1) with p - k Mittag-Leffler funcfunc-tion defined by [10] and generelized Mittag-Leffler function defined by [1] are given by

j pE

γ,q

k,α,β(z) =

kpjq−βk

j_Eγk,q α k,

β k

(zpq−αk). (2.17)

( d dz)

lh_zj_× j pE

γ,q k,α,β(z)

i

= p(γ)lq,k zj−l jp−lE γ+lqk,q

k,α,β (z), f or l < j. (2.18)

( d dz)

lh_zj_× j pE

γ,q k,α,β(z)

i

= p(γ)lq,k pEk,α,βγ+lqk,q(z), f or l=j. (2.19)

( d dz)

lh_zj_× j pE

γ,q k,α,β(z)

i

= p(γ)lq,k pEk,α,βγ+lqk,q+lα−jα(z), f or l > j. (2.20)

Proof of equation (2.16)

Using equation (1.5) and (1.6), we get the desired result. Proof of equation (2.17), (2.18) and (2.19)

Using the equation (2.1), in right hand side of (2.17), we have

dl dzl

h

zj × j_pE_k,α,βγ,q (z)

i

=

∞

X

n=0

zn+j−l (n+j−l)!,

dl dzl

h

zj× j pE

γ,q k,α,β(z)

i

=

∞

X

n=0

p(γ)lq,k p(γ+lqk)(n+j−l)q,k pΓk(nα+β)

zn+j−l (n+j−l)!,

hence we have,

= p(γ)lq,k zj−l jp−lE γ+lqk,q

k,α,β (z), f or l < j. = p(γ)jq,k pE_k,α,βγ+lqk,q(z), f or l=j.

= p(γ)lq,k pE_k,α,βγ+lqk,q₊_lα₋_jα(z), f or l > j.

Theorem 2.3: The following elementary properties are satisfied by the j-generelized p - k Mittag-Leffler function defined by equation (2.1),

kj_pE_k,α,βγ,q (z) =pβ _pjE_k,α,βγ,q ₊_k(z) +zpα d dz

j pE

γ,q

k,α,β+k(z). (2.21)

pqp(γ)q−1,k jp−1E

γ+kq−k,q k,α,β (z) =

j pE

γ,q

k,α,β(z)− j pE

γ−k,q

k,α,β (z). (2.22)

∞

X

n=0

(x+y)n j_pE_k,nqk₀_,nk+k,q₊_jk₊_k(xy) =

∞

X

r=0

pΓk(rqk+k)(xyp)r pΓk(rk+jk+k)

× j_pE_k,qk,krqk+k,q(x+y

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Proof of equation (2.20)

Consider the right hand side of equation (2.20),

A≡pβ j_pE_k,α,βγ,q ₊_k(z) +zpα d dz

j pE

γ,q

k,α,β+k(z), using equation (2.1),

A≡pβ

∞

X

n=0

p(γ)(n+j)q,k pΓk(nα+β+k)

zn

(n+j)! +zpα

∞

X

n=0

p(γ)(n+j)q,k pΓk(nα+β+k)

nzn−1 (n+j)!,

A≡p

∞

X

n=0

p(γ)(n+j)q,k(nα+β) pΓk(nα+β+k)

zn (n+j)!, using the equation (1.9), we have

A≡kj_pE_k,α,βγ,q (z). Proof of equation (2.21)

Consider the right hand side of (2.21),

A≡ j_pE_k,α,βγ,q (z)− j_pE_k,α,βγ−k,q(z), using equation (2.1), we have

A≡ ∞

X

n=0

zn

pΓk(nα+β)(n+j)!

h

p(γ)(n+j)q,k− p(γ −k)(n+j)q,k

i

,

using equations (1.10) and (1.11), we have

A≡pqp(γ)q−1,k jp−1E

γ+kq−k,q k,α,β (z). Proof of equation (2.22)

Consider the Left hand side of equation (2.22),

A≡ ∞

X

n=0

(x+y)n j_pE_k,nqk₀_,₍+_nk,q₊_j₊₁₎_k(xy),

A≡ ∞

X

n=0

(x+y)n

∞

X

r=0

p(nqk+k)(r+j)q,k pΓk(nk+jk+k)

(xy)r

(r+j)!, (2.24)

now simplifying, by using equation (1.5) and (1.6), we have

p(nqk+k)(r+j)q,k=p(r+j)q(nq+ 1)(r+j)q,

= p(r+j)qΓ(nq+ (r+j)q+ 1) Γ(nq+ 1)) ,

= p(r+j)qΓ((r+j)q+ 1 +nq) Γ((r+j)q+ 1)

Γ((r+j)q+ 1) Γ(nq+ 1) ,

=pΓk(rqk+k)

p(rqk+k)(r+j)q,k pΓk(nqk+k)

,

then equation (2.23) becomes by rearranging the terms, we have

A≡ ∞

X

r=0

pΓk(rqk+k)(xyp)r pΓk(rk+jk+k)

∞

X

n=0

p(rqk+k)(n+j)q,k pΓk(qkn+k)(n+j)!

(x+y p )

n_.

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3 Fractional Integral and Differentiation of the j-generalized

p - k Mittag-Leffler Function

In this section we evaluate certain relations that exist between the j-generalized p - k Mittag-Leffler Function and Riemann-Liouville fractional integrals and derivatives. It has been shown that the fractional integration and differentiation operators of j-generalized p - k Mittag-Leffler Function with power multipliers into the function of the same form. Also point out some special cases.

Theorem 3.1The left-side Riemann-Liouville Fractional Integral OperatorI₀ϑ+of the j-generalized

p - k Mittag-Leffler Function is given by,

(I₀ϑ+[t

β k−1j

pE γ,q k,α,β(t

α

k)])(z) =pϑz β k+ϑ−1j

pE γ,q

k,α,β+kϑ(z

α

k). (3.1)

Where

k, p∈R+−{0};α, β, γ∈C/kZ−;Re(α)>0, Re(β)>0, Re(γ)>0, j ∈N0, q ∈(0,1)∪N and Re(ϑ)>0.

Proof: Consider left hand side,

A≡(I₀ϑ+[t

β k−1j

pE γ,q k,α,β(t

α k)])(z),

by virtue of equation (1.20) and (2.1), we have

A≡ 1

Γ(ϑ)

Z z

0

tβk−1

(z−t)1−ϑ

∞

X

n=0

tnαk

(n+j)dt,

interchanging the order of integration and summation and evaluate the inner integral by substi-tute t=zuand using the beta function formula, it gives

A≡ z β k+ϑ−1

Γ(ϑ)

∞

X

n=0

znαk

(n+j)!

Γ(nα_k+β)Γ(ϑ) Γ(nα+β_k+kϑ) ,

using the equation (1.5), we have

A≡pϑzβk+ϑ−1j

pE γ,q

k,α,β+kϑ(z

α k).

Hence, we get the desired result.

Theorem 3.2The right-side Riemann-Liouville Fractional Integral OperatorI₋ϑof the j-generalized p - k Mittag-Leffler Function is given by,

(I₋ϑ[t−βk−ϑ j

pE γ,q k,α,β(t

−α

k)])(z) =pϑz− β k j

pE γ,q

k,α,β+kϑ(z

−α

k). (3.2)

Where

A≡(I−ϑ[t− β k−ϑ j

pE γ,q k,α,β(t

(10)

A≡ 1

Γ(ϑ)

Z ∞

z

t−βk−ϑ

(t−z)1−ϑ

∞

X

n=0

t−nαk

(n+j)!dt,

interchanging the order of integration and summation and evaluate the inner integral by substi-tute t= _uz and using the beta function formula, it gives

A≡ z −β_k

Γ(ϑ)

∞

X

n=0

z−nαk

(n+j)!

Γ(nα_k+β)Γ(ϑ) Γ(nα+β_k+kϑ) ,

A≡pϑz−βk j

pE γ,q

k,α,β+kϑ(z

−α k).

Theorem 3.3 The left-side Riemann-Liouville Fractional Derivative Operator D₀ϑ+ of the

j-generalized p - k Mittag-Leffler Function is given by,

(D₀ϑ+[t

β k−1j

pE γ,q k,α,β(t

α

k)])(z) =p−ϑz β k−ϑ−1j

pE γ,q

k,α,β−kϑ(z

α

k). (3.3)

Where

A≡(D₀ϑ+[t

β k−1j

pE γ,q k,α,β(t

α k)])(z),

A≡ 1

Γ(1−ϑ+ [Re(ϑ)])( d dz)

[Re(ϑ)]+1

Z z

0

tβk−1

(z−t)ϑ−Re(ϑ)

∞

X

n=0

tnαk

(n+j)!dt,

interchanging the order of integration and summation and evaluate the inner integral by substi-tute t=zuand using the beta function formula, it gives

A≡ z β k−ϑ−1

Γ(ϑ)

∞

X

n=0

znαk

(n+j)!

Γ(nα_k+β)Γ(ϑ) Γ(nα+β_k−kϑ) ,

A≡p−ϑzβk−ϑ−1j

pE γ,q

k,α,β−kϑ(z

α k).

Theorem 3.4 The right-side Riemann-Liouville Fractional Derivative Operator Dϑ

− of the

j-generalized p - k Mittag-Leffler Function is given by,

(Dϑ−[t− β k+ϑ j

pE γ,q k,α,β(t

−α_k_)])(_z_{) =}_p−ϑ_z−β_k j pE

γ,q

k,α,β−kϑ(z

−α_k₎_. _(3.4)

Where

(11)

A≡(Dϑ₋[t−βk+ϑ j

pE γ,q k,α,β(t

−α k)])(z),

A≡ 1

Γ(1−ϑ+ [Re(ϑ)])(− d dz)

[Re(ϑ)]+1

Z ∞

z

z−βk+ϑ

(t−z)ϑ−[Re(ϑ)]

∞

X

n=0

t−nαk

(n+j)!dt,

interchanging the order of integration and summation and evaluate the inner integral by substi-tute t= _uz and using the beta function formula, it gives

A≡ z −β

k

Γ(ϑ)

∞

X

n=0

z−nαk

(n+j)!

Γ(nα_k+β)Γ(ϑ) Γ(nα+β_k−kϑ) , using the equation (1.5), we have

A≡p−ϑz−βk j

pE γ,q

k,α,β−kϑ(z

−α k).

4 Recurrence Relation and Integral Representation of the

j-generalized p - k Mittag-Leffler Function

In this section we evaluate the recurrence relations and integral representations of the j-generalized p - k Mittag-Leffler function.

Theorem 4.1For k, p∈R+− {0};α+r, β+s+k, γ∈C/kZ−; R(α+r)>0, R(β+s+k)> 0, R(γ)>0, q∈(0,1)∪N, j ∈N0,we get

j pE

γ,q

k,α+r,β+s+k(z)−p j pE

γ,q

k,α+r,β+s+2k(z) = p2 k2

h

(α+r)2z2j_pE¨_k,αγ,q₊_r,β₊_s₊₃_k(z)

+n(α+r)2+ (α+r)(2β+ 2s+ 2k)zoj_pE˙_k,αγ,q₊_r,β₊_s₊₃_k(z)

+(β+s)(β+s+ 2k)_pjE_k,αγ,q₊_r,β₊_s₊₃_k(z)

i

, (4.1)

where jpE˙_k,α,βγ,q (z) = d dz

j

pE_k,α,βγ,q (z) and jpE¨_k,α,βγ,q (z) = d2

dz2

j

pE_k,α,βγ,q (z).

Proof: The j-generalized p-k Mittag-Leffler function, from equation (2.1),

j pE

γ,q

k,α+r,β+s+k(z) =

∞

X

n=0

p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+k)((n+j)!) ,

j pE

γ,q

k,α+r,β+s+k(z) =

∞

X

n=0

k p

p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s){n(α+r) +β+s)}((n+j)!)

, (4.2)

and

j pE

γ,q

k,α+r,β+s+2k(z) =

∞

X

n=0

p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 2k)((n+j)!)

(12)

j pE

γ,q

k,α+r,β+s+2k(z) =

∞

X

n=0

p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s)((n+j)!)

×k

2

p2

1

{n(α+r) +β+s)}{n(α+r) +β+s+k)},

j pE

γ,q

k,α+r,β+s+2k(z) =

∞

X

n=0

k p2[

1

(n(α+r) +β+s) −

1

(n(α+r) +β+s+k)]

× p(γ)(n+j)q,kz

n

pΓk(n(α+r) +β+s)((n+j)!) ,

j pE

γ,q

k,α+r,β+s+2k(z) = k p2[

p k j pE

γ,q

k,α+r,β+s+k(z)−S],

S = p k j pE

γ,q

k,α+r,β+s+k(z)− p2

k j pE

γ,q

k,α+r,β+s+2k(z), (4.4) where

S=

∞

X

n=0

p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s){n(α+r) +β+s+k)}((n+j)!)

, (4.5)

applying the simple identity 1 u =

k u(u+k) +

1

u+k; for u = n(α+r) +β+s+k to (4.5),we obtain,

S =

∞

X

n=0

kp(γ)(n+j)q,kzn

× 1

{n(α+r) +β+s+k}{n(α+r) +β+s+ 2k}

+

∞

X

n=0

p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s){n(α+r) +β+s+ 2k)}((n+j)!) ,

S =

∞

X

n=0

k{n(α+r) +β+s}_p(γ)₍_n₊_j₎_q,kzn pΓk(n(α+r) +β+s)((n+j)!)

× 1

{n(α+r) +β+s}{n(α+r) +β+s+k}{n(α+r) +β+s+ 2k}

+

∞

X

n=0

{n(α+r) +β+s}{n(α+r) +β+s+k}_p(γ)₍_n₊_j₎_q,kzn pΓk(n(α+r) +β+s)((n+j)!)

× 1

{n(α+r) +β+s}{n(α+r) +β+s+k}{n(α+r) +β+s+ 2k},

using equation (1.9), we obtain

S =

∞

X

n=0

k{n(α+r) +β+s}p(γ)(n+j)q,kzn k3

p3 pΓk(n(α+r) +β+s+ 3k)((n+j)!)

+

∞

X

n=0

{n(α+r) +β+s}{n(α+r) +β+s+k}p(γ)(n+j)q,kzn k3

p3 pΓk(n(α+r) +β+s+ 3k)((n+j)!)

,

Sk

3

p3 =

∞

X

n=0

p(γ)(n+j)q,kzn

pΓk(n(α+p) +β+s+ 3k)((n+j)!)

(13)

We now express each summation in the right hand side of (4.6) as follows:

d dz[z

j pE

γ,q

k,α+r,β+s+3k(z)] =

∞

X

n=0

(n+ 1)p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 3k)((n+j)!) ,

zj_pE˙_k,αγ,q₊_r,β₊_s₊₃_k(z) + j_pE_k,αγ,q₊_r,β₊_s₊₃_k(z) =

∞

X

n=0

(n+ 1)p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 3k)((n+j)!) ,

zj_pE˙_k,αγ,q₊_r,β₊_s₊₃_k(z) =

∞

X

n=0

np(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 3k)((n+j)!)

. (4.7)

Again

d2 dz2[z

2j pE

γ,q

k,α+r,β+s+3k(z)] =

∞

X

n=0

(n+ 1)(n+ 2)p(γ)(n+j)q,kzn pΓk(n(α+r) +β+s+ 3k)((n+j)!)

, (4.8)

and

d2 dz2[z

2j pE

γ,q

k,α+r,β+s+3k(z)]

=z2j_pE¨_k,αγ,q₊_r,β₊_s₊₃_k(z) + 4zj_pE˙_k,αγ,q₊_r,β₊_s₊₃_k(z) + 2j_pE_k,αγ,q₊_r,β₊_s₊₃_k(z), (4.9) from equation (4.8) and (4.9) we have

∞

X

n=0

{n2}p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 3k)((n+j)!)

=z2j_pE¨_k,αγ,q₊_r,β₊_s₊₃_k(z)

+ 4zj_pE˙_k,αγ,q₊_r,β₊_s₊₃_k(z)−3

∞

X

n=0

{n}p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 3k)((n+j)!) ,

∞

X

n=0

{n2}p(γ)(n+j)q,kzn

pΓk(n(α+r) +β+s+ 3k)((n+j)!)

=z2j_pE¨_k,αγ,q₊_r,β₊_s₊₃_k(z) +zj_pE˙γ,q_k,α₊_r,β₊_s₊₃_k(z), (4.10) applying equation (4.4), (4.7) and (4.10) to (4.6), we get

k3

p3S = (α+r) 2_z2j

pE¨ γ,q

k,α+r,β+s+3k(z) +

h

(α+r)2+ (α+r)(2β+ 2s+ 2k)iz

×j_pE˙γ,q_k,α₊_r,β₊_s₊₃_k(z) + (β+s)(β+s+ 2k)j_pEγ,q_k,α₊_r,β₊_s₊₃_k(z), Hence.

Theorem 4.2For k, p∈R+− {0};α+r, β+s+k, γ∈C/kZ−; R(α+r)>0, R(β+s+k)> 0, R(γ)>0, q∈(0,1)∪N, j ∈N0,we get

Z 1

0

tβ+s+k−1j_pEγ,q_k,α₊_r,β₊_s(tα+r)dt= p k j pE

γ,q

k,α+r,β+s+k(1)− p2

k j pE

γ,q

k,α+r,β+s+2k(1). (4.11)

Proof: Putz= 1 in equation (4.4) and (4.5), we have

S= p k j pE

γ,q

k,α+r,β+s+k(1)− p2

k j pE

γ,q

k,α+r,β+s+2k(1)

=

∞

X

n=0

p(γ)(n+j)q,k

pΓk(n(α+r) +β+s){n(α+r) +β+s+k)}((n+j)!)

(14)

now consider the integral,

A≡

Z 1

0

tβ+s+k−1j_pE_k,αγ,q₊_r,β₊_s(tα+r)dt,

A≡ ∞

X

n=0

p(γ)(n+j)q,k

Z 1

0

tn(α+r)+β+s+k−1dt,

A≡ ∞

X

n=0

p(γ)(n+j)q,k

pΓk(n(α+r) +β+s){n(α+r) +β+s+k)}((n+j)!) ,

from equation (4.12), we have the desired result.

Theorem 4.3 For k, p ∈ R+_{− {}₀_}_;_{α, β, γ, δ} _∈ _C_;_R₍_α₎ _> ₀_{, R}₍_β₎ _> ₀_{, R}₍_γ₎ _> ₀_{, R}₍_δ₎ _>

0and q∈(0,1)∪N, j ∈N0 then

pδ j_pE_k,α,βγ,q ₊_δk(z) = 1 Γ(δ)

Z 1

0

uβk−1(1−u)δ−1j

pE γ,q k,α,β(z u

α

k)du. (4.13)

Proof : Consider the right side integral and using equation (2.1), we have

A≡ 1

Γ(δ)

Z 1

0

uβk−1(1−u)δ−1j

pE γ,q k,α,β(z u

α k)du,

A≡ 1

Γ(δ)

∞

X

n=0

p(γ)(n+j)q,kzn pΓk(nα+β)((n+j)!)

Z 1

0

uαnk+β−1(1−u)δ−1du,

using the definition of Beta function, we have

A≡ 1

Γ(δ)

∞

X

n=0

Γ(αn_k+β)Γ(δ) Γ(αn_k+β +δ), applying equation (1.5), we have

A≡ ∞

X

n=0

pδp(γ)(n+j)q,kzn pΓk(nα+β+δk)((n+j)!)

=pδ j_pE_k,α,βγ,q ₊_δk(z),

Hence.

Theorem 4.4 For k, p ∈ R+ − {0};β, γ ∈ C;R(β) > 0, R(γ) > 0and α, q ∈ N, j ∈ N0,

then

j pE

γ,q

k,kα,β(z) =

p(γ)jq,k pΓk(β)

q

Y

i=1

α

Y

l=1

Γ(bl) Γ(ai)Γ(bl−ai)

Z 1

0

uai−1(1−u)bl−ai−1_E

1,j+1

uzp q−α_qq

αα

du.

(4.14)

Where ai = γ

k+qj+i−1

q and bl=

β

k +l−1

α .

Proof : Using definition of j-generalized p-k Mittag- Leffler function, from equation (2.1),

A≡ j pE

γ,q

k,kα,β(z) =

∞

X

n=0

p(γ)(n+j)q,kzn pΓk(nkα+β)((n+j)!)

(15)

using relation (1.11),we have

A≡ ∞

X

n=0

p(γ)jq,k p(γ+jqk)nq,kzn p(β)nα,k pΓk(β)((n+j)!)

=

∞

X

n=0

D p(γ)jq,kz n

pΓk(β)((n+j)!)

. (4.15)

Where D≡ p(γ+jqk)nq,k

p(β)nα,k ,

D≡ p(γ+jqk)nq,k

p(β)nα,k

= p

n(q−α)₍γ+jqk k )qn (β_k)nα , using the relation given by equation (1.7), we have

D≡ p

(q−α)n_qqnQq

i=1(

γ

k+jq+i−1

q )n ααnQα

l=1(

β k+l−1

α )n ,

let ai = γ

k+jq+i−1

q and bl =

β

k +l−1

α ,

D≡(p

(q−α)_qq αα )

n q

Y

i=1

α

Y

l=1

(ai)n (bl)n ,

D≡(p

(q−α)_qq αα )

n q

Y

i=1

α

Y

l=1

Γ(ai+n)Γ(bl) Γ(bl+n)Γ(ai) ,

D≡(p

(q−α)_qq αα )

n q

Y

i=1

α

Y

l=1

Γ(bl) Γ(bl−ai)Γ(ai)

Γ(ai+n)Γ(bl−ai) Γ(bl−ai+ai+n)

,

using the definition of Beta function, we have

D≡(p

(q−α)_qq αα )

n q

Y

i=1

α

Y

l=1

Γ(bl) Γ(bl−ai)Γ(ai)

Z 1

0

uai+n−1₍₁₋_u₎bl−ai−1_du, _(4.16)

from equation (4.15) and (4.16), we have

A≡ p(γ)jq,k

pΓk(β) q

Y

i=1

α

Y

l=1

Γ(bl) Γ(bj−ai)Γ(ai)

Z 1

0

uai−1₍₁₋_u₎bj−ai−1 ∞

X

n=0

(uz)n (n+j)!(

p(q−α)_qq αα )

n_du,

Hence.

Theorem 4.5 For k, p ∈ R+ − {0};α, β, γ ∈ C;R(α) > 0, R(β) > 0, R(γ) > 0 and q ∈

(0,1)∪N, j∈N0 then

j pE

γ,q

k,α,β(z) = pjq Γ(γ_k)

Z ∞

0

e−tt(γk+jq−1)j

pE

1,0

k,α,β(zt

q_pq₎_dt. _(4.17)

Proof: Using definition of j-generalized p-k Mittag-Leffler function, equation (2.1), we have

j pE

γ,q

k,α,β(z) =

∞

X

n=0

,

using equation (1.5) and (1.6), we have

j pE

γ,q

k,α,β(z) =

∞

X

n=0

zn

pΓk(nα+β)((n+j)!)

(16)

j pE

γ,q

k,α,β(z) =

∞

X

n=0

zn

pΓk(nα+β)((n+j)!)

pq(n+j) Γ(γ_k)

Z ∞

0

e−tt(γk+q(n+j)−1)dt,

j pE

γ,q

Z ∞

0

e−tt(γk+jq−1) ∞

X

n=0

znpqntqn

pΓk(nα+β)((n+j)!) dt,

j pE

γ,q

Z ∞

0

e−tt(γk+jq−1)j

pE

1,0

k,α,β(zt q_pq₎_dt.

Hence.

5 Integral Transform of the j-generalized P-K Mittag-Leffler

Function

In this section we evaluate Mellin-Barnes integral representation of j-generalized p-k Mittag-Leffler function, relationship with Fox H-function and Wright hypergeometric function. Also evaluate Euler Beta Transform, Laplace Transform and Mellin Transform of j-generalized p-k Mittag-Leffler function.

Mellin-Barnes integral representation of the j-generalized p-k Mittag-Leffler func-tion.

Theorem 5.1Letk, p∈R+− {0};Re(α)>0, Re(β)>0, Re(γ)>0, and q∈(0,1)∪N, j∈N0,

then the j-generalized p-k Mittag-Leffler function is represented by the Mellin-Barnes integral as,

j pE

γ,q

k,α,β(z) =

kpjq−βk

2πiΓ(γ_k)

Z

L

Γ(s)Γ(1−s)Γ(γ_k +jq−qs) Γ(β_k− αs_k)Γ(1 +j−s) (−zp

q−α_k₎−s_ds. _(5.1)

Where|argz |< π; the contour integration beginning at−i∞and ending at +i∞,and indented to separate the poles of the integrand as s = −n for every n ∈ N0(to the left) from those at

s=

γ k+n

q for every n∈N0 (to the right).

Proof Consider the integral on right side of equation(5.1) and use the theorem of calculus of residues,

A≡ kp

jq−β_k

2πiΓ(γ_k)

Z

L

Γ(s)Γ(1−s)Γ(γ_k+jq−qs) Γ(β_k−αs

k)Γ(1 +j−s)

(−zpq−αk)−sds

= 2πi[sum of the residues at the poles s= 0,−1,−2, ...]

A≡ kp

jq−β k

Γ(γ_k)

∞

X

n=0

Re

s=−n (s+n)

hΓ(s)Γ(1−s)Γ(γ

k+jq−qs) Γ(β_k− αs

k)Γ(1 +j−s)

i

(−zpq−αk)−s

A≡ kp

jq−β k

Γ(γ_k)

∞

X

n=0

lim s→−n

hπ(s+n)

sinπs

Γ(γ_k +jq−qs) Γ(1 +j−s)Γ(β_k−αs_k)

i

(−zpq−αk)−s

A≡ kp

jq−β k

Γ(γ_k)

∞

X

n=0

h Γ(γ

k +jq+qn) Γ(1 +j+s)Γ(β_k+ αn_k )

i

(zpq−αk)n

using equations (1.5) and (1.6), we have,

A≡ j_pE_k,α,βγ,q (z).

Hence.

Relationship with Fox H-function

(17)

then the j-generalized p-k Mittag-Leffler function is given in the form of Fox H-function, as.

j pE

γ,q

k,α,β(z) =

kpjq−βk

Γ(γ_k) H

1,2 2,3



−zpq

−α k

(0,1),(1−γ_k −jq, q);

(0,1),(1−β_k,α_k),(−j,1);



. (5.2)

Proof. Using the equations (5.1) and (1.25), we get the desired result.

Relationship with Wright hypergeometric function

Theorem 5.3Letk, p∈R+− {0};Re(α)>0, Re(β)>0, Re(γ)>0, and q∈(0,1)∪N, j∈N0,

then the j-generalized p-k Mittag-Leffler function is given in the form of Wright hypergeometric function, as.

j pE

γ,q

k,α,β(z) =

kpjq−βk

Γ(γ_k) 2Ψ2





(1,1) (γ_k +jq, q); zpq−αk

(β_k,α_k)(1 +j,1);



. (5.3)

Proof. Using the equations (5.1) and (1.24), we get the desired result.

Euler Beta Transform, Laplace Transform and Mellin Transform of j-generalized p-k Mittag-Leffler function

Theorem 5.4Letk, p∈R+− {0};a, b, σ∈C;Re(α)>0, Re(β)>0, Re(γ)>0, Re(σ)>0 and q ∈(0,1)∪N, j ∈N0, then Euler Beta Transform of j-generalized p-k Mittag-Leffler function,

is given by,

Z 1

0

za−1(1−z)b−1j_pE_k,α,βγ,q (xzσ)dz= kp jq−β

kΓ(b)

Γ(γ_k) 3ψ3





(1,1),(γ_k+qj, q),(a, σ); xpq−αk

(β_k,α_k),(1 +j,1),(a+b, σ);



 (5.4)

Proof Consider the left side integral and using equation (5.4), we have

A≡

Z 1

0

za−1(1−z)b−1 j_pE_k,α,βγ,q (xzσ)dz

A≡ ∞

X

n=0

p(γ)(n+j)q,kxn pΓk(nα+β) (n+j)!

Z 1

0

zσn+a−1(1−z)b−1dz,

using definition of Beta function, we have

A≡ ∞

X

n=0

p(γ)(n+j)q,kxn pΓk(nα+β) (n+j)!

B(σn+a, b)

using equation (1.5),(1.6) and (1.24), we have

A≡ kp

jq−β kΓ(b)

Γ(γ_k) 3ψ3





(1,1),(γ_k +qj, q),(a, σ); xpq−αk

(β_k,α_k),(1 +j,1),(a+b, σ);



.

Hence.

Theorem 5.5 The Laplace transform of j-generalizedp-k Mittag-Leffler function, is given by,

Z ∞

0

za−1e−zs j_pE_k,α,βγ,q (xzσ)dz= kp jq−β

k s−a

Γ(γ_k) 3ψ2



 

(1,1),(γ_k +qj, q),(a, σ); xpq−αk

sσ

(β_k,α_k)(1 +j,1);





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Where k, p ∈ R+ _{− {}₀_}_; _{a, σ} _∈ _C_; _Re₍_α₎ _> ₀_{, Re}₍_β₎ _> ₀_{, Re}₍_γ₎ _> ₀_{, Re}₍_σ₎ _> _{0 and} _q _∈

(0,1)∪N, j∈N0,and | _sxσ |<1.

Proof Consider the right side integral and using equation(2.1), we have

A≡

Z ∞

0

za−1e−zs j_pE_k,α,βγ,q (xzσ)dz

A≡ ∞

X

n=0

xn (n+j)!

Z ∞

0

znσ+a−1e−zsdz,

using definition of gamma function, we have

A≡s−a

∞

X

n=0

p(γ)(n+j)q,k pΓk(nα+β) (n+j)!

Γ(σn+a)(x sσ)

n

using equation (1.5),(1.6) and (1.24), we have

A≡ kp

jq−β_k_s−a

Γ(γ_k) 3ψ2



 

(1,1),(γ_k+qj, q),(a, σ); xpq−αk

sσ

(β_k,α_k)(1 +j,1);



 .

Hence.

Theorem 5.6 The Mellin transform of j-generalized p-k Mittag-Leffler function, is given by,

Z ∞

0

ts−1j_pE_k,α,βγ,q (−wt)dt= kp

qj−β_k_Γ₍_s₎_Γ₍₁₋_s₎_Γ₍γ

k+qj−qs) Γ(β_k− αs_k)Γ(γ_k)Γ(1 +j−s) (

pαk−q

w )

s _(5.6)

Where k, p ∈ R+ − {0}; a, σ ∈ C; Re(α) > 0, Re(β) > 0, Re(γ) > 0, Re(s) > 0 and q ∈

(0,1)∪N, j∈N0.

Proof Putting z = -wt in equation (5.1), we have

j pE

γ,q

k,α,β(−wt) =

kpjq−βk

2πiΓ(γ_k)

Z

L

Γ(s)Γ(1−s)Γ(γ_k +jq−qs) Γ(β_k −αs_k )Γ(1 +j−s) (−wtp

q−α k)−sds.

j pE

γ,q

k,α,β(−wt) =

kpjq−βk

2πiΓ(γ_k)

Z

L

f∗(s)(t)−sds. (5.7)

where

f∗(s) = Γ(s)Γ(1−s)Γ( γ

k +jq−qs) Γ(γ_k)Γ(β_k− αs_k)Γ(1 +j−s)(−wp

q−α k)−s

using equation (1.28),(1.29) and (5.7), which immediately leads to (3.9).

Particular cases:Putting some particular values of j, p, q, k, α, β, γ we obtain all the results given by [1], [3],[8],[9],[10],[11],[12],[13]and [14].

References

[1] A. K. Shukla and J.C. Prajapati. On the generalization of Mittag-Leffler function and its properties. Journal of Mathematical Analysis and Applications,336 (2007) 797-811.

[2] A. Wiman. Uber den fundamental Satz in der Theories der Funktionen Eα(z), Acta Math. 29 (1905) 191-201.

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[4] G. Mittag-Leffler. Sur la nouvellefonction Eα(z) C.RAcad. Sci. Paris 137(1903) 554-558. [5] H. Kilbas, H. Srivastava, J. Trujillo, Theory and Application of Fractional Differential Equa-tions, Elsevier, 2006.

[6] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, John Wiley and Sons / Horword, New York/ Chichester,1984.

[7] I.N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill, New Delhi,1979.

[8]Kuldeep Singh Gehlot, The Generalized K- Mittag-Leffler function. Int. J. Contemp. Math. Sciences, Vol. 7 (2012) No. 45, 2213-2219.

[9] Kuldeep Singh Gehlot, Two Parameter Gamma Function and it’s Properties, arXiv:1701.01052v1[math.CA] 3 Jan 2017.

[10] Kuldeep Singh Gehlot, The p-k Mittag-Liffler function, Palestine Journal of Mathematics, Vol. 7(2)(2018), 628-632.

[11] Kuldeep Singh Gehlot, Fractional Integral and Diff. of p-k Mittag-Leffler function, under publication.

[12] Kuldeep Singh Gehlot, Recurrence relation and Integral representation of p-k Mittag-Leffler function, under publication.

[13] kuldeep Singh Gehlot, CR Choudhary and Anita Punia, Integral Transform of p-k Mittag -Leffler function, JETIR September 2018, Volume 5, Issue 9, 722-730.

[14] Luciano Leonardo Luque, On a Generalized Mittag-Leffler Function, International Journal of Mathematical Analysis, Vol. 13, 2019, no. 5, 223 - 234.

[15] Rafael Diaz and Eddy Pariguan. On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Mathematicas, Vol. 15 No. 2 (2007) 179-192.

[16] S.G. Samok, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives,Theory and Applications. Gordon and Breach, New York, 1993.