Kottakkaran Sooppy Nisar,Waseem Ahmad Khan and Mohd Ghayasuddin

1Department of Mathematics,College of Arts and Science-Wadi Aldawaser,Prince Sattam bin Abdulaziz University,11991,Alkharj,Kingdom of Saudi Arabia

2Department of Mathematics,Faculty of Science,Integral University,Lucknow-226026,India

Abstract.The objective of this note is to provide some(potentially useful)integral transforms(for example,Euler,Laplace,Whittaker etc.)associated with the generalized k-Bessel function defined by Saiful and Nisar[3].We have also discussed some other transforms as special cases of our main results.

Key Words:Gamma function,k-Bessel function,generalized k-Bessel function,integral transforms.

1 Introduction

The Bessel function of fist kind has the power series representation of the form[4]:

Romero et al.[16]introduced the k-Bessel function of the first kind defined by the series

where k∈R;α,λ,γ,υ∈C;<(λ)>0 and<(υ)>0.

Very recently,Saiful and Nisar[3]gave a new generalization of k-Bessel function called the generalized k-Bessel function of the first kind defined for k∈R;σ,γ,υ,c,b∈C;<(σ)>0,<(υ)>0 as:

where the k-Pochhammer symbol(γ)n,kis defined by[1]:

and the k-gamma function has the relation

such that Γk(z)→Γ(z)if k→1.

The generalized hypergeometric function represented as follows[6]:

provided p≤q,p=q+1 and|z|<1 and(α)nis well known Pochhammer symbol(see[6]).The Fox-Wright generalizationpΨq(z)of hypergeometric functionpFqis given by(c.f.[7–9,15]):

where Aj>0(j=1,2,···,p);Bj>0(j=1,2,···,q)and

for suitably bounded value of|z|.

The generalized k-Wright function introduced in[10]as:For k∈R+;z∈C,αi,βj∈and

Also,we recall here the following definitions:

Definition 1.1.Euler Transform:Let α,β∈C and<(α),<(β)>0,then the Euler transform of the function f(t)is defined by

Definition 1.2.Laplace Transform:The Laplace transform of the function f(t)is defined as

Definition 1.3.K-Transform:The K-transform with p as a complex parameter defined by

Also,we need the following formula(see[13,page 78])

Definition 1.4.Whittaker Transform:The integral transform

where Wλ,μis the Whittaker function[12].

2 Main results

In this section,we give some integral transforms(for example,Euler,Laplace,Whittaker etc.)of generalized k-Bessel function given in(1.3).

where2Ψ3is the Wright hypergeometric function defined by(1.7).

Proof.In order to derive(2.1),we denote the L.H.S.of(2.1)by I1and then by using the definition of generalized k-Bessel function given in(1.3),we have

Interchanging the integration and summation with suitable convergence conditions,we get

which upon using the definition of Beta function gives

Now using(1.4)and(1.5),we get

In view of the definition of(1.7)we get the desired result.

Corollary 2.1.If we take k=1,in Theorem 2.1,then we have the following integral transform

Proof.Let I2denoted by the left hand side of Theorem 2.2.Applying(1.3)on the L.H.S.of(2.2),to get

Interchanging the integration and summation allow us to write,

In view of the definition of Laplace transform,we get

Now by applying(1.5)in the above equation,we have

which by using the definition(1.7),gives our desired result.

Corollary 2.2.For k=1 in Theorem 2.2,we have

Proof.Let I3denoted by the left hand side of Theorem 2.3.Applying(1.3)on the L.H.S.of(2.3),to get

Interchanging the integration and summation allow us to write,

Using the formula given in(1.12)in the above expression,we get

By applying(1.5),we have

In view of definition(1.7),we get the desired result.

Corollary 2.3.For k=1 in Theorem 2.3,we get the following interesting integral transform

In the following theorem,we derive the Whittaker transform of generalized k-Bessel function.Here,we recall the following results:

where the Whittaker function Wλ,μ(t)is given in[12](also see[14]),and

where Mλ,μ(t)is defined as

Theorem 2.4.If k∈R+,λ,μ,β,b,c,γ,λ∈C,0,min{<(δ),<(λ)}>0,<(ρ?λ0)>0,then

where Wλ0,μis the Whittaker function of second kind(see[1]).

Proof.Let pt=v,then

Interchanging the integration and summation allow us to write

Using the formula for Whittaker transform(2.4),we get

Applying(1.4)and(1.5),to get

In view of(1.7),we get the desired result.

Corollary 2.4.For k=1 in Theorem 2.4,we get

Theorem 2.5.If k∈R+,λ,μ,β,b,c,γ,λ∈C,and,then

Proof.Let pt=v,then

Interchanging the integration and summation allow us to write

On using(2.5),we get

By applying(1.4)and(1.5),we obtain

In view of(1.7),we get the desired result.

Corollary 2.5.For k=1 in Theorem 2.5,we get

Acknowledgements

The authors would like to thank reviewer’s valuable comments and suggestions.