ON THE(p,q)-MELLIN TRANSFORM AND ITS APPLICATIONS∗

Pankaj JAIN Chandrani BASU Vivek PANWAR

Department of Mathematics,South Asian University Akbar Bhawan,Chanakya Puri,New Delhi-110021,India

E-mail:pankaj.jain@sau.ac.in,pankajkrjain@hotmail.com;chandrani.basu@gmail.com;vivek.pan1992@gmail.com

Abstract In this paper,we introduce and study a(p,q)-Mellin transform and its corresponding convolution and inversion.In terms of applications of the(p,q)-Mellin transform,we solve some integral equations.Moreover,a(p,q)-analogue of the Titchmarsh theorem is also derived.

Key words q-Mellin transform;(p,q)-Mellin transform;inversion formula;convolution;integral equation

1 Introduction

Integral transforms play an important role in solving many differential and integral equations.Riemann[1] first recognized the Mellin transform in 1876 in his famous memoir on prime numbers.The explicit formulation was given by Cohen in 1894,and almost simultaneously,Mellin[2]gave an elaborate discussion of it,along with its inversion formula.

The Mellin transform and its inversion formula can be derived from the complex Fourier transform.More precisely,the Mellin transform of a suitable function f over(0,∞)is given by

The integral(1.1)is well de fined in a(possibly empty)maximal open vertical strip〈α,β〉,which is called a fundamental strip.The inversion formula for the Mellin transform is given by the following line integral:

The Mellin convolution product of two suitable functions f and g is de fined by

The Mellin transform and the corresponding convolution satisfy the following relations:

The development of quantum calculus,also called q-calculus or‘limitless’calculus,was started in the 1740s by Euler,and its progress continued under C.F.Gauss,who in 1812 invented the hypergeometric series and its contiguity relations[3].The study of quantum calculus or q-calculus has accelerated in the past two decades.It has been used in several fields in mathematical,physical and engineering sciences.Fitouhi et al.[4]introduced the concept of a q-Mellin transform and studied its applications in solving some integral equations.Later on,Brahim et al.[5]applied the q-Mellin transform to solving partial differential equations.The finite Mellin transform[6]and two dimensional Mellin transforms[7]have also been studied in the framework of quantum calculus.

The notion of q-calculus has further been generalized to post-quantum calculus,or(p,q)-calculus[8–10].In the recent past,(p,q)-calculus has been applied in several areas,such as approximation theory,computer aided geometric design,inequalities etc..For the relevant literature on these applications,one may refer to[11–18].The Laplace transform in the(p,q)-framework[9]has also been studied.

In this paper,we introduce and study some properties of the(p,q)-Mellin transform,as well as its inversion and convolution.Also,we appy the(p,q)-Mellin transform to solving some integral equations.Moreover,the Titchmarsh theorem[19]is proved in the framework of(p,q)-calculus.

2 Preliminaries

2.1 q-Calculus

Throughout this paper we shall take q∈(0,1).Here we shall give some basic notions and notations used in q-calculus.Let x∈C,and n∈N.The q-analogue of x and the q-factorial of n are de fined,respectively,by

and

Suppose that 0

provided that the series on the right converges absolutely.Also,

The improper q-integral is de fined by

provided,again,that the series on the right converges absolutely.For a systematic study of basic properties of q-calculus,one may refer to[3,20].

2.2 (p,q)-Calculus

In this section,we give a brief introduction of(p,q)-calculus.Throughout this paper we shall take 0

Suppose that 0

and the improper(p,q)-integral is de fined by

provided that all the series involved are absolutely convergent.

Note that when p=1,most of the notions in(p,q)-calculus reduce to the corresponding notions of q-calculus.For more on(p,q)-calculus,one may refer to[8–10].

Remark 2.1

Let us point out that one may be tempted to study(p,q)-calculus in terms of q-calculus by making a variable transformation and thereby be tempted to believe that the extra parameter p is redundant.This,however,is not the case.Let us see this through the following situation:in the classical integral theory and also in q-integration,the integral of a non-negative function de fined on an interval remains non-negative.However,this is not true in(p,q)-interation.Indeed,consider

2.3 q-Mellin Transform

De fine the set Rby

For a suitable function f de fined on R,its q-Mellin transform is de fined by

Remark 2.2

There exists a(possibly empty)maximal open vertical strip in which the integral(2.2)is well de fined.We denote it by〈α,β〉and call it a fundamental strip,or simply a strip.

We mention the following results from[4]:

Proposition A

Let f be a function de fined on Rand let u,v∈R with u>v.If

then M(f)(s)exists in the strip〈−u,−v〉.

Proposition B

Let f be de fined on R.Then M(f)is analytic on the strip〈α,β〉and we have,for all s∈〈α,β〉,

Next,we will mention some basic properties of the q-Mellin transform;see[4].

Proposition C

(a)For a∈Rand s∈〈α,β〉,

(b)For s∈〈−β,−α〉,

(c)For s∈〈1−β,1−α〉,

(d)For s∈〈α,β〉,

(e)For s∈〈α+1,β+1〉,

Moreover,for n∈N and s∈〈α+n,β+n〉,the following holds:

(f)For s∈〈α+1,β+1〉,

Moreover,for n∈N and s∈〈α+n,β+n〉,the following holds:

(g)For s∈〈α−1,β−1〉,

(h)Given ρ>0 and s∈〈ρα,ρβ〉,we have that

(i)Let{µ}be a sequence in R,let{λ}be a sequence in C,and let f be a suitable function.Then

provided that the sum converges.

Lemma D

([4]) For k∈Z,we have

where

The inversion formula for the q-Mellin transform is given by the following theorem:

Theorem E

([4]) Let f be a function de fined over Rand let c∈(α,β).Then,for all x∈R,

The q-Mellin convolution product of two functions f and g is de fined by

provided that the q-integral exists.

The q-Mellin convolution is a commutative operation.Moreover,the q-Mellin convolution equality holds.More precisely,the following is known:

Proposition F

([4]) If the q-Mellin convolution product of f and g exists,then the following hold:

(i)f∗g=g∗f;

(ii)M[f∗g]=M(f)M(g).

We also have that the following Parseval-type relations hold:

Proposition G

([4]) For suitable functions f and g,the following hold:

3 The(p,q)-Mellin Transform

De fine the set Rby

De finition 3.1

Let f be a function de fined on R.We de fine the(p,q)-Mellin transform of f by

Remark 3.2

(i)For a suitable function f,M(f)(s)becomes M(f)(s)as p→1.

(ii)There exists a(possibly empty)maximal open vertical strip in which the integral(3.1)is well de fined.We denote it by〈α,β〉and call it a fundamental strip,or simply a strip.

We shall be using the following result,which is also of independent interest,and which gives a relation between the q-integral and(p,q)-integral:

Proof

We have

and we are done.

By using Lemma 3.3,it can be proved that the q-Mellin transform and the(p,q)-Mellin transform are related.Indeed,the following can be proved:

We now prove

Proposition 3.5

Let f be de fined on R.Then M(f)is analytic on the strip〈α,β〉and for all s∈〈α,β〉,we have that

Proof

For every s∈〈α,β〉,we have that

The next theorem provides some of the basic properties of the(p,q)-Mellin transform.The proof can be obtained in view of the de finition of the(p,q)-Mellin transform,Lemmas 3.3 and 3.4,and Proposition C.

Theorem 3.6

(a)For a∈Rand s∈〈α,β〉,we have that

(b)For s∈〈−β,−α〉,we have that

(c)For s∈〈1−β,1−α〉,we have that

(d)For any a∈R,s∈〈α−a,β−a〉,we have that

(e)For s∈〈α,β〉,we have that

(f)For s∈〈α+1,β+1〉,we have that

(g)For s∈〈α+1,β+1〉,we have that

(h)For s∈〈α−1,β−1〉,we have that

(i)Given ρ>0 and s∈〈ρα,ρβ〉,we have that

(j)Let{µ}be a sequence in R,let{λ}be a sequence in C and let f be a suitable function.Then we have that

provided that the sum converges.

(k)For a,b∈R and s∈〈α,β〉∩〈α,β〉,we have that

Remark 3.7

(i) The expression(3.2)can be obtained for higher order derivatives as well.

(a)For second order derivative,it holds that for s∈〈α+2,β+2〉,we have

and therefore,proceeding as in(a)above,we get that

For general n∈N,we conjecture that for s∈〈α+n,β+n〉,the following holds:

(ii)Similarly,the expression(3.3)can be obtained for higher order derivatives as well.The following can be proved:

Again,for general n∈N,we conjecture that for s∈〈α+n,β+n〉,the following holds:

4 Inversion Formula and Convolution

Theorem 4.1

Let f be a function de fined over Rand let c∈(α,β).Then,for all x∈R

The above series converges uniformly with respect to s,so that we can change the order of integration and summation.Therefore,by using Lemma D with q replaced by q/p,and making a variable substitution,we get

and the assertion follows.

Next,we de fine the appropriate convolution for the(p,q)-Mellin transform.

De finition 4.2

The(p,q)-Mellin convolution product of two functions f and g is de fined by

provided that the(p,q)-integral exists.

We now prove some of the properties of the convolution de fined above.

Next,we prove the Parseval type relation for the(p,q)-Mellin transform.

Proposition 4.4

For suitable functions f and g,the following holds:

Proof

Let c∈R be such that c∈(α,β)and 1−c∈(α,β).Then

Proposition 4.5

For suitable functions f and g,the following holds:

Proof

By using Theorems 4.1 and 4.3(iii),we obtain that

Now the assertion follows by taking x=1 in the last equality and by applying the de finition of convolution.

5 Applications

In this section we will solve a(p,q)-integral equation with the help of the(p,q)-Mellin transform.We begin with the following lemma:

Now,applying the inversion formula,we have that

and it follows,in view of Propositin 4.4,that(5.5)is a solution of(5.4).

Moreover,if(5.6)is satis fied,then L=K,and we are done.

Along similar lines,by using Lemma 5.1(ii),we can immediately obtain the following:

Theorem 5.3

Let K and g be functions de fined on R.For a suitable real c,we put

then the integral equation has a solution

Finally,we prove a result which is the(p,q)-analogue of the Titchmarsh Theorem[19].

Proof

If we write

then(5.7)takes the form

Applying Lemma 5.1 to both of these equations,we have that

Changing s into 1−s in one of these equations and multiplying with the other,we get the desired result.

6 Conclusion

In this paper,we have introduced and studied the(p,q)-Mellin transform,which generalizes the known notion of q-Mellin transform.In this regard,the corresponding convolution has been de fined and the inversion formula has been derived.In terms of applications of the(p,q)-Mellin transform,we have solved some integral equations.Moreover,a(p,q)-analogue of the Titchmarsh theorem has also been derived.