CUBO A Mathematical Journal
Vol.21, No¯ 01, (49–60). April 2019

http: // dx. doi. org/ 10. 4067/ S0719-06462019000100049

Certain integral Transforms of the generalized
Lommel-Wright function

S. Haq

Department of Applied Mathematics, Faculty of Engineering and Technology,

Aligarh Muslim University, Aligarh-202002, UP, India

sirajulhaq007@gmail.com

K.S. Nisar

Department of Mathematics, College of Arts and Science,

Prince Sattam bin Abdulaziz University, Wadi Aldawaser, Riyadh region 11991, Saudi Arabia

ksnisar1@gmail.com

A.H. Khan

Department of Applied Mathematics, Faculty of Engineering and Technology,

Aligarh Muslim University, Aligarh-202002, UP, India

ahkhan.amu@gmail.com

D.L. Suthar

Department of Mathematics, Wollo University,

Ethiopia

dlsuthar@gmail.com

ABSTRACT

The aim of this article is to establish some integral transforms of the generalized

Lommel-Wright functions, which are expressed in terms of Wright Hypergeometric

function. Some integrals involving trigonometric, generalized Bessel and Struve func-

tions are also indicated as special cases of our main results.

http://dx.doi.org/10.4067/S0719-06462019000100049


50 S. Haq, K.S. Nisar, A.H. Khan and D.L. Suthar CUBO
21, 1 (2019)

RESUMEN

El objetivo de este art́ıculo es establecer algunas transformadas integrales de las fun-

ciones generalizadas de Lommel-Wright, que se expresan en términos de la función

hipergeométrica de Wright. Algunas integrales que involucran funciones trigonométricas,

de Bessel generalizadas y de Struve también se obtienen como casos especiales de nue-

stros resultados principales.

Keywords and Phrases: Gamma function, generalized Wright hypergeometric function pψq ,

generalized Lommel-Wright functions J
µ m
ν,λ (z), Integral Transforms.

2010 AMS Mathematics Subject Classification: 33B20, 33B15, 65R10, 33C20.



CUBO
21, 1 (2019)

Certain integral Transforms of the generalized Lommel-Wright . . . 51

1 Introduction

The k-Pochhammer symbol (λ)ν,k is defined (for ν,λ ∈ C;k ∈ R) by [4]

(λ)ν,k =
Γk(λ + ν k)

Γk(λ)
(λ ∈ C/0) (1.1)

and the k-gamma function has the relation

Γk(z) = k
z/k−1Γ(z/k), (1.2)

is such that Γk(z) → Γ(z) if k → 1 .

The Wright hypergeometric function defined by the series [21]

pψq









(α1,A1), ...,(αp,Ap);

(β1,B1), ...,(βq,Bq)

z









=

∞∑

k=0

p∏

j=1

Γ(αj + Ajk)z
k

q∏

j=1

Γ(βj + Bjk)k!

, (1.3)

where the coefficients A1, ...,Ap and B1, ...,Bq are positive real numbers such that

1 +

q∑

j=1

Bj −

p∑

j=1

Aj ≥ 0. (1.4)

can be slightly generalized (1.3) as given below.

pψq









(α1,1), ...,(αp,1);

(β1,1), ...,(βq,1);

z









=

p∏

j=1

Γ(αj)

q∏

j=1

Γ(βj)
pFq









α1, ..,αp;

β1, ...,βq;

z









, (1.5)

where pFq is the generalized hypergeometric function defined by [19, 21]

pFq









α1, ...,αp;

β1, ...,βq

z









=

∞∑

k=0

(α1)n, ...,(αp)nz
n

(β1)n, ...,(βq)nn!
= pFq(α1, ...,αp;β1, ...,βq;z), (1.6)

where (λ)n is the well known Pochhammer symbol [21].

The generalization of (λ)n is given as

(λ)n = λ(λ + 1)(λ + 2), ...,(λ + n − 1)) ,n > 0 (1.7)

(λ)n =

n∏

m=1

(λ + m − 1), (λ)0 = 1, λ 6= 0



52 S. Haq, K.S. Nisar, A.H. Khan and D.L. Suthar CUBO
21, 1 (2019)

(λ)n =
Γ(λ + n)

Γ(λ)

Generalized Bessel, Lommel, Struve and Lommel-Wright function have originated from concrete

problems in Mechanics, Physics, Engineering and Astronomy.

The series representation of the generalized Lommel Wright function as [8];

J
µ,m
ν,λ (z) =

∞∑

k=0

(−1)kΓ(k + 1)( z
2
)2k+ν+2λ

Γ(λ + k + 1)mΓ(ν + kµ + λ + 1)k!
, (1.8)

(z ∈ N/(−∞,0] m ∈ N, ν,λ ∈ C,µ > 0).

Also, we have the following relations of generalized Lommel Wright functions with trigonometric

functions and the generalized Bessel function and Struve function:

J1,11/2,0(z) =
√

(
2

πz
) sin(z) (1.9)

J1,1−1/2,0(z) =
√

(
2

πz
) cos(z) (1.10)

J
µ,1
ν,λ(z) = J

µ
ν,λ(z) (1.11)

J1,1ν,1/2(z) = Hν(z) (1.12)

Further, we recall the following results [5].

∫
∞

0

tu−1 exp(−t/2)Wλ,µ(t)dt =
Γ(1/2 + µ + u)Γ(1/2 − µ + u)

Γ(1 − λ + u)
, (1.13)

(Re(u ± µ) > −1/2),

where the Whittaker function Wλ,µ(t) is given in[5, 11].

Wλ,µ(t) =
Γ(−2µ)

Γ(1/2 − µ − λ)
Mλ,µ(t) +

Γ(2µ)

Γ(1/2 + µ − λ)
Mλ,−µ(t)

where Mλ,µ(t) is defined as

Mλ,µ(t) = z
1/2+µ exp(−t/2) 1F1

(

1/2 + µ + u;2µ + 1;t

)

Definition 1.1. Euler Transform:

Let ρ,σ ∈ C and Re(ρ),Re(σ) > 0, then the Euler transform of the function f(z) is defined by

B(f(z);ρ,σ) =

∫1

0

zρ−1(1 − z)σ−1f(z)dz (1.14)



CUBO
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Certain integral Transforms of the generalized Lommel-Wright . . . 53

Definition 1.2. Laplace Transform:

The Laplace transform of the function f(t) is defined as

F(δ) = L(f(t);δ) =

∫
∞

0

exp(−tδ)f(t)dt, Re(δ) > 0 (1.15)

Definition 1.3. Fourier Transform:

The following integral gives the Fourier transform

u = Im[u](w) =

∫

R

u(t) exp(iwt)dt, (1.16)

where u = u(t) be a function of the space S(R) Shwartzian space of the function that decay rapidly

at ∞ together with all derivatives.

Definition 1.4. The Fractional Fourier Transform (FFT):

Let u be the function belonging to φ(R), the Lizorkin space of function, where

φ(R) = {φ ∈ S(R)} : Im[φ] ∈ V(R)
and V(R) is the set of functions defined by

V(R) = {v ∈ S(R)} : Vu0 = 0,n = 0,1,2, ...
then FFT of order α, 0 ≤ α ≤ 1 is given by

Uα(w) = Imα(w) =

∫

R

exp(i wα t)u(t)dt (1.17)

particularly, if α = 1 (1.17) reduces to FT and for w > 0 (1.17) reduces to FFT given by Luchko

et al [10].

The aim of this paper is to obtain the Euler, Laplace, Whittaker and Fractional Fourier transforms

of Lommel-Wright function.

Various generalizations, integrals, transforms and fractional calculus of special functions have been

investigated by many researchers (see, for details, [1, 2, 6, 7, 9, 12, 13, 14, 15, 16, 17, 18, 20]).

In this sequel, here, we aim at establishing certain new generalized integral formula involving the

generalized Lommel-Wright function J
µ,m
ν,λ (z) interesting integral formulas which are derived as

special cases.

2 Main Results

This section deals with some integral formulas involving Lommel-Wright function.



54 S. Haq, K.S. Nisar, A.H. Khan and D.L. Suthar CUBO
21, 1 (2019)

Theorem 2.1. For t ∈ N/(−∞,0] m ∈ N, ν,λ ∈ C and µ > 0 , the following integral formula
holds true

∫1

0

tα−1(1 − t)β−1J
µ,m
ν,λ (x t

σ)dt =

(

x

2

)ν+2λ

Γ(β)

× 2ψm+2

[

(1,1),(α + νσ + 2λσ,2σ);

(λ + 1,1), ...,(λ + 1,1),(ν + λ + 1,µ),(α + β + νσ + 2λσ,2σ);
−
x2

4

]

. (2.1)

Proof. On using (1.8) in the integrand of (2.1) and then interchanging the order of integral sign

and summation which is verified by uniform convergence of the involved series under the given

conditions we get

∫1

0

tα−1(1 − t)β−1J
µ,m
ν,λ (x t

σ)dt

=

(

x

2

)ν+2λ ∞∑

k=0

Γ(k + 1)(−x2/4)k

Γ(λ + k + 1)mΓ(ν + kµ + λ + 1)k!

×
∫1

0

tα+σ(2k+ν+2λ)−1(1 − t)β−1dt. (2.2)

Now using (1.14) in the above equation we get

∫1

0

tα−1(1 − t)β−1J
µ,m
ν,λ (x t

σ)dt = Γ(β)

(

x

2

)ν+2λ

×
∞∑

k=0

Γ(k + 1)Γ(α + νσ + 2λσ + 2kσ)(−x
2

4
)k

Γ(λ + k + 1)mΓ(α + β + νσ + 2λσ + 2kσ)Γ(ν + kµ + λ + 1)k!
. (2.3)

Finally, using (1.3) in the above equation, we get our assertion (2.1). This completes the proof of

Theorem 2.1.

Theorem 2.2. For t ∈ N/(−∞,0] m ∈ N, ν,λ ∈ C and µ > 0 , the following integral formula
holds true

∫
∞

0

tα−1 exp(−tδ)J
µ,m
ν,λ (x t

σ
)dt =

(

x

2 δ−α

)ν+2λ

(δ)−α

× 2ψm+1

[

(1,1),(α + νσ + 2λσ,2σ);

(λ + 1,1), ...,(λ + 1,1),(ν + λ + 1,µ);
−

x2

4 δ2σ

]

. (2.4)

Proof. On using (1.8) in the integrand of (2.4) and then interchanging the order of integral sign

and summation which is verified by uniform convergence of the involved series under the given



CUBO
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Certain integral Transforms of the generalized Lommel-Wright . . . 55

conditions we get
∫
∞

0

tα−1 exp(−δ t)J
µ,m
ν,λ (x t

σ
)dt

=

(

x

2

)ν+2λ ∞∑

k=0

Γ(k + 1)(−x2/4)k

Γ(λ + k + 1)mΓ(ν + kµ + λ + 1)k!

×
∫
∞

0

tα+σ(2k+ν+2λ)−1 exp(−δ t)dt. (2.5)

Now using (1.15) in the above equation we get
∫
∞

0

tα−1 exp(−δ t)J
µ,m
ν,λ (x t

σ)dt = (δ)−α
(

x

2δσ

)ν+2λ

×
∞∑

k=0

Γ(k + 1)Γ(α + νσ + 2λσ + 2kσ)( −x
2

4 δ2σ
)k

Γ(λ + k + 1)mΓ(ν + kµ + λ + 1)k!
. (2.6)

Finally, using (1.3) in the above equation, we get our assertion (2.6). This completes the proof of

Theorem 2.2.

Theorem 2.3. For t ∈ N/(−∞,0] m ∈ N, ν,λ ∈ C and µ > 0 , the following integral formula
holds true

∫
∞

0

tη−1 exp(−p t)/2 Wλ,µ(p t)J
µ,m
ν,λ (w t

δ)dt =

(

w

pδ

)ν+2λ

× 3ψm+2

[

(1,1),(1/2 + µ + η + δ ν + 2δλ,2δ),(1/2 − µ + η + δ ν + 2δλ,2δ);

(λ + 1,1), ...,(λ + 1,1),(ν + λ + 1,µ),(1 − λ + η + νδ + 2δλ,2δ);
−

w2

4 p2δ

]

.

(2.7)

Proof. On using (1.8) in the integrand of (2.7) and then interchanging the order of integral sign

and summation which is verified by uniform convergence of the involved series under the given

conditions we get
∫
∞

0

(u/p)η−1 exp(−u/2)Wλ,µ(u)J
µ,m
ν,λ (w (u/p)

δ)du

=

(

w

pδ

)ν+2λ ∞∑

k=0

Γ(k + 1)(−w2/4 p2δ)k

Γ(λ + k + 1)mΓ(ν + kµ + λ + 1)k!

×
∫
∞

0

uη+δ(2k+ν+2λ)−1 exp(−u/2)Wλ,µ(u)du. (2.8)

Now using (1.13) in the above equation we get

∫
∞

0

tη−1 exp(−p t)/2 Wλ,µ(p t)J
µ,m
ν,λ (w t

δ)dt =

(

w

pδ

)ν+2λ

×
∞∑

k=0

Γ(k + 1)Γ(1/2 + µ + η + 2kδ + δν + 2δλ)Γ(1/2 − µ + η + 2kδ + δν + 2δλ)( −w
2

4 p2δ
)k

Γ(λ + k + 1)mΓ(ν + kµ + λ + 1)Γ(1 − λ + η + 2kδ + δν + 2δλ)k!
. (2.9)



56 S. Haq, K.S. Nisar, A.H. Khan and D.L. Suthar CUBO
21, 1 (2019)

Finally, using (1.3) in the above equation, we get our assertion (2.9). This completes the proof of

Theorem 2.3.

3 Special Cases

In this section, we get some integral formulas involving trigonometric function and generalized

Lommel-Wright function as follows:

Corollary 3.1. If we take m = 1,µ = 1,λ = 0 and ν = 1/2 in (2.1) and then by using (1.9), we

derive the following integral formula:

∫1

0

tα−σ/2−1(1 − t)(β−1) sin(x tσ)dt

=
√
π

(

x

2

)

Γ(β) 1ψ2









(α + σ/2,2σ);

(3/2,1),(α + β + σ/2,2σ);

−
x2

4









(3.1)

Corollary 3.2. If we take m = 1,µ = 1,λ = 0 and ν = 1/2 in (2.4) and then by using (1.9), we

derive the following integral formula:
∫
∞

0

tα−σ/2−1 exp(−δ t) sin(x tσ)dt

= δ−α
√

π

δσ

(

x

2

)

Γ(β) 1ψ1









(α + σ/2,2σ);

(3/2,1);

−
x2

4 δ2σ









(3.2)

Corollary 3.3. Further if we take m = 1,µ = 1,λ = 0 and ν = 1/2 in (2.7) and then by using

(1.9), we derive the following integral formula:
∫
∞

0

tη−δ/2−1 exp(−pt/2)Wλ,µ(p t) sin(w t
δ)dt

= w

√

π

2 pδ
2ψ2









(η + δ/2 + 3/2,2δ)(η + δ/2 − 1/2,2δ), ;

(3/2,1),(η + δ/2 + 1,2δ);

−
w2

4 p2δ









(3.3)

Corollary 3.4. If we take m = 1,µ = 1,λ = 0 and ν = −1/2 in (2.1) and then by using (1.10),

we derive the following integral formula:

∫1

0

tα−σ/2−1(1 − t)(β−1) cos(x tσ)dt

=
√
πΓ(β) 1ψ2









(α − σ/2,2σ);

(1/2,1),(α + β − σ/2,2σ);

−
x2

4









(3.4)



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Certain integral Transforms of the generalized Lommel-Wright . . . 57

Corollary 3.5. If we take m = 1,µ = 1,λ = 0 and ν = −1/2 in (2.4) and then by using (1.10),

we derive the following integral formula:
∫
∞

0

tα−σ/2−1 exp(−δ t) cos(x tσ)dt

= δ(σ−α)
√
π 1ψ1









(α − σ/2,2σ);

(1/2,1);

−
x2

4 δ2σ









(3.5)

Corollary 3.6. Further if we take m = 1,µ = 1,λ = 0 and ν = −1/2 in (2.7) and then by using

(1.10), we derive the following integral formula:
∫
∞

0

tη−δ/2−1 exp(−pt/2)Wλ,µ(p t) cos(w t
δ)dt

= w

√

π

2
2ψ2









(η − δ/2 + 3/2,2δ)(η − δ/2 − 1/2,2δ), ;

(1/2,1),(η − δ/2 + 1,2δ);

−
w2

4 p2δ









(3.6)

Corollary 3.7. If we take m = 1 in (2.1) and then by using (1.11), we derive the following integral

formula:

∫1

0

tα−1(1 − t)(β−1)J
µ
ν,λ(x t

σ
)dt =

(

x

2

)ν+2λ

Γ(β)

×2ψ3









(1,1),(α + νσ + 2λσ,2σ);

(λ + 1,1),(ν + λ + 1,µ),(α + β + νσ + 2λσ,2σ);

−
x2

4









(3.7)

Corollary 3.8. If we take m = 1 in (2.4) and then by using (1.11), we derive the following integral

formula:
∫
∞

0

tα−1 exp(−δ t)J
µ
ν,λ(x t

σ)dt

=

(

x

2

)ν+2λ

δ−α 2ψ2









(1,1),(α + νσ + 2λσ,2σ);

(λ + 1,1),(ν + λ + 1,µ);

−
x2

4 δ2σ









(3.8)

Corollary 3.9. Further if we take m = 1 in (2.7) and then by using (1.11), we derive the following

integral formula:

∫
∞

0

tη−1 exp(−pt/2)Wλ,µ(p t)J
µ
ν,λ(w t

δ
)dt =

(

w

pδ

)ν+2λ

×3ψ3









(1,1),(1/2 + µ + η + νδ + 2λδ,2δ),(1/2 − µ + η + νδ + 2λδ,2δ);

(λ + 1,1),(ν + λ + 1,µ),(1 − λ + η + δν + 2δλ,2δ);

−
w2

4 p2δ









(3.9)



58 S. Haq, K.S. Nisar, A.H. Khan and D.L. Suthar CUBO
21, 1 (2019)

Corollary 3.10. If we take µ = 1,m = 1 and λ = 1/2 in (2.1) and then by using (1.12), we derive

the following integral formula:

∫1

0

tα−1(1 − t)(β−1)Hν(x t
σ)dt =

(

x

2

)ν+1

Γ(β)

×2ψ3









(1,1),(α + νσ + σ,2σ);

(3/2,1),(ν + 3/2,1),(α + β + νσ + σ,2σ);

−
x2

4









(3.10)

Corollary 3.11. If we take µ = 1,m = 1 and λ = 1/2 in (2.4) and then by using (1.12), we derive

the following integral formula:

∫
∞

0

tα−1 exp(−δ t)Hν(x t
σ
)dt =

(

x

2 δσ

)ν+1

δ−α

×2ψ2









(1,1),(α + νσ + σ,2σ);

(3/2,1),(ν + 3/2,1);

−
x2

4 δ2σ









(3.11)

Corollary 3.12. Further if we take µ = 1,m = 1 and λ = 1/2 in (2.7) and then by using (1.12),

we derive the following integral formula:

∫
∞

0

tη−1 exp(−pt/2)Wλ,µ(p t)Hν(w t
δ)dt =

(

w

pδ

)ν+1

×3ψ3









(1,1),(η + νδ + δ + 3/2,2δ),(η + νδ + δ − 1/2,2δ);

(3/2,1),(ν + 3/2,1),(η + δν + δ + 1/2,2δ);

−
w2

4 p2δ









(3.12)



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Certain integral Transforms of the generalized Lommel-Wright . . . 59

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	Introduction
	Main Results
	Special Cases