Some Integral Transforms of the Generalized k-Mittag-Leffler Function

Article

SOME INTEGRAL TRANSFORMS OF THE GENERALIZED

k-MITTAG-LEFFLER FUNCTION

FENG QI

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region 028043, China; Department of Mathematics, College of

Science, Tianjin Polytechnic University, Tianjin City 300387, China

KOTTAKKARAN SOOPPY NISAR

Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al dawaser, Riyadh region 11991, Saudi Arabia

Abstract. In the paper, the authors generalize the notion “k-Mittag-Leffler function”, estab-lish some integral transforms of the generalizedk-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized

k-Wright function.

1. _{Preliminaries}

Throughout this paper, let _C, _R, _R+₀, _R+_,

Z−0, and N denote respectively the sets of com-plex numbers, real numbers, non-negative numbers, positive numbers, non-positive integers, and positive integers.

The Pochhammer symbol (λ)ν can be defined forλ, ν∈Cby (λ)ν=

Γ(λ+ν) Γ(λ) , where

Γ(z) = lim n→∞

n!nz

Qn

k=0(z+k)

, z∈C\Z−0

is called the classical gamma function and its reciprocal _Γ1 is analytic on the whole complex plane C. See [14, Chapter 5], [16, Section 1], and [24, Section 1.1]. In particular, whenν ∈ {0} ∪N, the quantity

(λ)n= (

1, ν= 0

λ(λ+ 1)· · ·(λ+n−1), n∈N

is called the rising factorial. See [17] and closely-related references therein.

E-mail addresses:[email protected], [email protected], [email protected], [email protected], [email protected].

2010Mathematics Subject Classification. Primary 33E12; Secondary 33C20, 44A20, 44A30, 65R10.

Key words and phrases. integral transform; generalizedk-Mittag-Leffler function; generalized Wright function; generalizedk-Wright function; gamma function.

This paper was typeset using_AMS-LA_TEX.

Thek-Pochhammer symbol (λ)n,k was defined in [2] forλ, ν∈Candk∈Rby

(λ)ν,k=

Γk(λ+νk) Γk(λ)

, (1.1)

where

Γk(z) =kz/k−1Γ _z

k

(1.2)

is called thek-gamma function. In particular,

(λ)n,k = (

1, n= 0;

λ(λ+k)· · ·(λ+ (n−1)k), n∈_N.

In 1903, Mittag-Leffler, a Swedish mathematician, introduced and investigated in [12, 13] the so-called Mittag-Leffler function

Eα(z) =

∞

X

n=0 zn Γ(αn+ 1)

forz∈Cand α∈R+0. In 1905, Wiman [25] generalizedEα(z) as

Eα,β(z) =

∞

X

n=0 zn Γ(αn+β),

wherez∈C,α, β∈C, and<(α),<(β)>0. In 1971, Prabhakar [15] introduced the function

E_α,βγ (z) = ∞

X

n=0

(γ)n Γ(αn+β)

zn

n!

forz ∈ C, α, β, γ ∈C, and <(α),<(β),<(γ)>0. In 2012, Dorrego and Cerutti [3] introduced thek-Mittag-Leffler function

E_k,α,βγ (z) = ∞

X

n=0

(γ)n,k Γk(αn+β)

zn

n!,

where α, β, γ ∈ _C, k ∈ _R, <(α),<(β),<(γ)> 0, and (γ)n,k is thek-Pochhammer symbol. In 2012, a generalization of thek-Mittag-Leffler function

GE_k,α,βγ,q (z) = ∞

X

n=0

(γ)nq,k Γk(αn+β)

zn

n!

for α, β, γ ∈ C, k ∈ R, <(α),<(β),<(γ) > 0, and q ∈ (0,1)∪N was introduced and studied in [6]. For more information on generalizations of the Mittag-Leffler function, please refer to the papers [1, 9, 19, 20, 21] and closely-related references therein.

In this paper, we consider a more general generalization

E_k,α,β,δγ,τ (z) = ∞

X

n=0

(γ)nτ,k Γk(αn+β)

zn (δ)n

, (1.3)

where α, β, γ, δ, τ ∈ C, k ∈ R, <(α),<(β) > 0, and δ 6= 0,−1,−2, . . .. It is clear that E_k,α,β,γ,τ ₁(z) =GE_k,α,βγ,τ (z) andE_k,α,β,γ,1 ₁(z) =E_k,α,βγ (z).

It is well known [14, 18] that the generalized hypergeometric function can be defined by

pFq[(αp); (βq);z] =

∞

X

n=0

Πp_j=1(αj)n

Πq_j=1(βj)n

zn

for|z|<1 and p≤qwith p=q+ 1 and that the generalized Wright hypergeometric function

pΨq(z) is given by the series

pΨq(z) =pΨq[(ai, αi)1,p; (bj, βj)1,q;z] = Qq

j=1Γ(βj) Qp

i=1Γ(αi)

∞

X

k=0

Qp

i=1Γ(ai+αik) Qq

j=1Γ(bj+βjk)

zk

k! (1.4)

forai, bj∈Candαi, βj ∈Rwith 1≤i≤pand 1≤j≤q. Asymptotic behavior of the function

pΨq(z) for large values of argument of z ∈ C were studied in [5, 26, 27] under the condition Pq

j=1βj− Pp

i=1αi>−1. If puttingα1=· · ·=αp=β1=· · ·=βq = 1 in (1.4), then

pΨq(z) =pΨq[(a1,1), . . . ,(ap,1); (b1,1), . . . ,(bq,1);z]

= Qq

j=1Γ(bj) Qp

i=1Γ(ai)

pFq[a1, . . . , ap;b1, . . . , bq;z].

The generalizedk-Wright function was introduced in [7] as

pΨkq(z) =pΨkq[(ai, αi)1,p; (bj, βj)1,q;z] =

∞

X

n=0

Qp

i=1Γk(ai+αin) Qq

j=1Γk(bj+βjn) zn

n!

fork∈R+,z∈C,αi, βj∈R\ {0}, andai+αin, bj+βjn∈C\kZ− with 1≤pand 1≤j≤q. The Euler transform of a functionf(z) is defined by

B{f(z);α, β}= Z 1

0

zα−1(1−z)β−1f(z) dz (1.5)

forα, β∈Cand<(α),<(β)>0. The Laplace transform of a functionf(t) is defined as

F(s) =L{f(t);s}= Z ∞

0

e−stf(t) dt, <(s)>0.

The Fourier transform of a functionu=u(t)∈S(_R) is defined by

b

u==[u](w) = Z

R

u(t)eiwtdt,

whereS(_R) denotes the Shwartzian space of rapidly decreasing test functions on_R. The fractional Fourier transform of orderαfor 0≤α≤1 was defined in [10] by

ˆ

uα(w) ==α[u](w) = Z

R

eiw1/αtu(t) dt (1.6)

foru∈Φ(_R), where

Φ(R) ={ϕ∈S(R) : ˆϕ∈V(R)}

denotes the Lizorkin space of functions and

V(R) =v∈S(R) :v(n)(0) = 0, n= 0,1,2, . . . .

Whenα= 1, the quantity (1.6) reduces to the Fourier transform; whenw >0, the transform (1.6) reduces to the fractional Fourier transform.

2. _{Main results}

Now we are in a position to state and prove our main results.

Theorem 1. Ifk∈R,α, β, γ, a, b, σ∈C,<(α),<(β)>0,δ6= 0,−1,−2, . . ., andq >0, then

Z 1

0

za−1(1−z)b−1E_k,α,β,δγ,τ (xzσ) dz

= k

1−β/k_Γ(b)Γ(δ)

Γ(γ_k) 3Ψ3

γ k, τ

,(a, σ),(1,1); β k, α k

,(a+b, σ),(δ,1);kτ−α/kx

. (2.1)

Proof. Denote the left-hand side of the equation (2.1) by I1. By definition of the generalized k-Mittag-Leffler function and (1.5), we have

I1=

Z 1

0

za−1(1−z)b−1Eγ,τ_k,α,β,δ(xzσ) dz= Z 1

0

za−1(1−z)b−1 ∞

X

n=0

(γ)nτ,k Γk(αn+β)

(xzσ₎n (δ)n

dz.

By interchanging the order of the integration and summation, we obtain

I1= ∞

X

n=0

(γ)nτ,k Γk(αn+β)

xn (δ)n

Z 1

0

za+σn−1(1−z)b−1dz= ∞

X

n=0

(γ)nτ,k Γk(αn+β)

xn (δ)n

Γ(a+σn)Γ(b) Γ(a+b+σn).

From (1.1) and (1.2), we acquire

I1=k

1−β/k_Γ(b)Γ(δ)

Γ(γ_k)

∞

X

n=0

Γ(γ_k +nτ)Γ(a+σn)Γ(δ)

Γ(β_k +α_kn)Γ(a+b+σn)Γ(δ+n) knτ kαn/k.

In view of (1.4), we arrive at the desired result.

Remark 2.1. Takingδ= 1 in Theorem 1 gives [23, Eq. (24)] which reads that

Z 1

0

za−1(1−z)b−1E_k,α,βγ,τ (xzσ) dz=k

1−β/k_Γ(b)

Γ(γ_k) 2Ψ2 _γ

k, τ

,(a, σ); _β

k, α k

,(a+b, σ);kτ−α/kx

.

Settingδ= 1 andτ=q >0 in Theorem 1 leads to [23, Eq. (25)] which states that

Z 1

0

za−1(1−z)b−1E_k,α,βγ,q (xzσ) dz=k

1−β/k_Γ(b)

Γ(γ_k) 2Ψ2 _γ

k, q

,(a, σ); _β

k, α k

,(a+b, σ);kq−α/kx

.

Further lettingk= 1 in the above equation derives [23, Eq. (26)] which formulates that

Z 1

0

za−1(1−z)b−1E_α,βγ,q(xzσ) dz= Γ(b)

Γ(γ)2Ψ2[(γ, q),(a, σ); (β, α),(a+b, σ);x].

Theorem 2. If k ∈ R, α, β, γ, a, σ ∈ C, <(α),<(β),<(s) > 0, τ ∈ C, _sxσ

< 1, and δ 6= 0,−1,−2, . . ., then

Z ∞

0

za−1e−szEγ,τ_k,α,β,δ(xzσ) dz= k

1−β/k_Γ(δ)

sa_Γ(γ k)

3Ψ2

_γ

k, τ

,(a, σ),(1,1); _β

k, α k

,(δ,1);xk τ−α/k

sσ

.

(2.2)

Proof. Denote the left-hand side of (2.2) byI2. Applying definition of the generalizedk -Mittag-Leffler function results in

I2=

Z ∞

0

za−1e−szE_k,α,βγ,τ (xzσ) dz= Z ∞

0

za−1e−sz ∞

X

n=0

(γ)nτ,k Γk(αn+β)

(xzσ₎n (δ)n

Interchanging the order of the integration and summation leads to

I2= ∞

X

n=0

(γ)nτ,k Γk(αn+β)

xn (δ)n

Z ∞

0

za+σn−1e−szdz.

In view of definition of the Laplace transform, we have

I2= ∞

X

n=0

(γ)nτ,k Γk(αn+β)

xn (δ)n

Γ(σn+a) sσn+a .

Utilizing (1.1) and (1.2) derives the required result.

Remark 2.2. If settingδ= 1 in Theorem 2, then we deduce [23, Eq. (27)] which formulates that

Z ∞

0

za−1e−szE_k,α,βγ,τ (xzσ) dz= k

1−β/k_s−a

Γ(γ_k) 2Ψ1 _γ

k, τ

,(a, σ); _β

k, α k

;xk

τ−α/k

sσ

.

If takingτ=q >0,k= 1, and δ= 1, then we acquire [23, Eq. (29)] which reads that

Z ∞

0

za−1e−szE_α,βγ,q(xzσ) dz= s −a

Γ(γ)2Ψ1

(γ, q),(a, σ); (β, α); x sσ

.

If takingk=q= 1 and δ= 1 in the above equation reduces to

Z ∞

0

za−1e−szE_α,βγ (xzσ) dz= s −a

Γ(γ)2Ψ1

(γ,1),(a, σ); (β, α); x sσ

which is the main result in [22].

Recall that Z ∞

0

tv−1e−t/2Wλ,µ(t) dt=

Γ(1₂+µ+v)Γ(1₂−µ+v)

Γ(1−λ+v) , <(v±µ)>− 1

2, (2.3)

where the Whittaker function

Wλ,µ(t) =

Γ(−2µ)

Γ(1₂−µ−λ)Mλ,µ(t) +

Γ(2µ)

Γ(1₂+µ−λ)Mλ,−µ(t)

and

Mλ,µ(t) =zµ+1/2e−t/21F1

₁

2 +µ+v; 2µ+ 1;t

are given in [4, 11].

Theorem 3. If k∈R,α, β, γ, δ, τ, η ∈C,<(α),<(β),<(ρ)>0, δ6= 0,−1,−2, . . ., and <(ρ± µ)>−1

2, then

Z ∞

0

tρ−1e−pt/2Wλ,µ(pt)Ek,α,β,δγ,τ (wt

η_{) dt=} k1−

β/k_p−ρ_Γ(δ)

Γ(γ_k)

×3Ψ3

γ k, τ

,

1

2±µ+ρ, η

,(1,1);

β k,

α k

,(1−λ+ρ, η),(δ,1);wk τ−α/k

pη

.

Proof. Lettingpt=v, interchanging the integration and summation, and using the formula for the Whittaker transform (2.3) yields

Z ∞

0

tρ−1e−pt/2Wλ,µ(pt)E γ,τ k,α,β,δ(wt

η_{) dt}

= Z ∞

0

e−v/2

_v

p

ρ−1

Wλ,µ(v)

∞

X

n=0

(γ)nτ,k Γk(αn+β)

wn (δ)n

_v

p

δn₁

= ∞

X

n=0

(γ)nτ,k Γk(αn+β)

wn (δ)n

Z ∞

0

e−v/2

_v

p

ρ−1_v

p

δn

Wλ,µ(v) 1 pdv

= 1 pρ

∞

X

n=0

(γ)nτ,k Γk(αn+β)(δ)n

_w

pδ

nZ ∞

0

e−v/2vδn+ρ−1Wλ,µ(v) dv

= 1 pρ

∞

X

n=0

(γ)nτ,kΓ(1₂+µ+δn+ρ)Γ(1₂−µ+δn+ρ) Γk(αn+β)(δ)nΓ(1−λ+δn+ρ)

w pδ

n

.

In view of (1.1) and (1.2), we find the desired result.

Remark 2.3. Takingδ= 1 in Theorem 3 gives [23, Eq. (30)] which reads that

Z 1

0

tρ−1e−pt/2Wλ,µ(pt)E γ,τ k,α,β(wt

η_{) dt}

= k

1−β/k_p−ρ

Γ(γ_k) 2Ψ2 _γ

k, τ

,

₁

2 ±µ+ρ, η ; _β k, α k

,(1−λ+ρ, η);wk τ−α/k

pη

.

Settingτ =q >0,k= 1, andδ= 1 results in [23, Eq. (32)] which states that

Z 1

0

tρ−1e−pt/2Wλ,µ(pt)E_α,βγ,q(wtη) dt

= 1

pρ_Γ(γ)2Ψ2

(γ, τ),

₁

2±µ+ρ, η

; (β, α),(1−λ+ρ, η); w pη

.

Lettingq= 1 in the above equation derives a result given in [22].

Theorem 4. Ifk∈R,α, β, γ, a, b, σ∈C,<(α),<(β)>0,τ ∈C, andδ= 06 ,−1,−2, . . ., then

=σ[E_k,α,β,δγ,τ (t)](w) =

k1−β/k Γ(γ_k)

∞

X

n=0

(−1)nn!k

(τ−β/k)n_Γ(γ k +nτ)i

n−1_w−(n+1)/σ

(δ)nΓ(αkn+ β k)

.

Proof. Using definitions of the generalizedk-Mittag-Leffler function and the fractional Fourier transform and interchanging the integration and summation give

=σ[E_k,α,β,δγ,τ (t)](w) = Z 1

0

exp(iw1/σt)E_k,α,β,δγ,τ (t) dt= ∞

X

n=0

(γ)nτ,k Γk(αn+β)(δ)n

Z

R

exp(iw1/σt)tndt.

Lettingiw1/σ_{t=−η reduces to}

=σ[E γ,τ

k,α,β,δ(t)](w) =

∞

X

n=0

(γ)nτ,k Γk(αn+β)(δ)n

Z 0

−∞

exp(−η)

_−η

iw1/σ

n_−dη

iw1/σ

= ∞

X

n=0

(γ)nτ,k

Γk(αn+β)(δ)nin+1w(n+1)/σ(−1)n Z ∞

0

e−ηηndη

= ∞

X

n=0

(γ)nτ,ki−n−1w−(n+1)/σ(−1)nn! Γk(αn+β)(δ)n

Further using the formulas (1.1) and (1.2) arrives at the required result.

Remark 2.4. If takingδ= 1 in Theorem 4, then the equation

=σ[E γ,τ

k,α,β(t)](w) =

k1−β/k Γ(γ_k)

∞

X

n=0 (−1)nk

(τ−β/k)n_Γ(γ k +nτ)i

n−1_w−(n+1)/σ

Γ(α kn+

β k)

in [23, Eq. (33)] follows readily.

If settingδ= 1 and τ=qin Theorem 4, then the equation

=σ[Ek,α,βγ,τ (t)](w) =

k1−β/k Γ(γ_k)

∞

X

n=0 (−1)nk

(q−β/k)n_Γ(γ k +nq)i

n−1_w−(n+1)/σ

Γ(α_kn+β_k)

in [23, Eq. (34)] can be derived immediately.

If lettingδ=k= 1 andτ=qin Theorem 4, then

=σ[Eγ,qα,β(t)](w) = 1 Γ(γ)

∞

X

n=0

(−1)nΓ(γ+nτ)i

n−1_w−(n+1)/σ

Γ(αn+β)

in [23, Eq. (34)] can be deduced straightforwardly.

Remark 2.5. In [8, Section 3], the quantity ik _{for k ∈}

N was computed generally by three approaches.

3. Concluding remark

Some integral transforms of the generalizedk-Mittag-Leffler function are established and the results are expressed in terms of the generalized Wright function. By taking δ = 1 and using the formulas (1.1) and (1.2), we express Theorems 1 to 3 in terms of the generalizedk-Wright function as follows.

It is noted that, using the appropriate formulas mentioned in Section 1, one can easily express the Euler integral in terms of thek-Wright function as

Z 1

0

za−1(1−z)b−1E_k,α,βγ,τ (xzσ) dz= Γ(b)k b

Γk(γ)

2Ψk2[(γ, τ k),(ak, σk); (β, α),((a+b)k, σk);x].

By applying suitable formula for thek-gamma function, Theorem 2 can be expressed in terms of thek-Wright function as

Z 1

0

za−1e−szE_k,α,βγ,τ (xzσ) dz= k 2−γ/k

(sk)a_Γ k(γ)

2Ψk1

(γ, τ k),(ak, σk); (β, α); x (ks)σ

.

Theorem 3 can be expressed in terms of thek-Wright function as

Z 1

0

tρ−1e−pt/2Wλ,µ(pt)E γ,τ k,α,β(wt

η_{) dt}

=p

−ρ_k1−ρ−λ

Γk(γ)

2Ψk2

(γ, τ k),

₁

2 ±µ+ρ

k, ηk

; (β, α),((1−λ+ρ)k, ηk);xk

τ−η−α/k

pη

.

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