# On Transformation Involving Basic Analogue of MultivariableH-Function

Authors: Dinesh Kumar; Frédéric Ayant; Jessada Tariboon
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, fractional orderq-integrals and q-derivatives involving a basic analogue of multivariable H-function have been obtained. We give an application concerning the basic analogue of multivariable H-function and q-extension of the Leibniz rule for the fractional q-derivative for a product of two basic functions. We also give the corollary concerning basic analogue of multivariable Meijer’s G-function as a particular case of the main result.


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## Body

## 1. Introduction and Preliminaries

Theq-calculus is not of recent appearance, it was introduced in the twenties of last century. In 1910, Jackson [1] introduced and developed q-calculus systematically. The fractional q-calculus is the expansion of ordinary fractional calculus in the q-theory. Recently, there was a significant work done by many authors in the area of q-calculus due to lots of applications in mathematics, statistics, and physics.Since special functions play significant roles in mathematical physics, it is persuaded to think that some deformation of the ordinary special functions based on theq-calculus can also play comparable roles in this area of research. Further, many authors have derived images of various q-special functions under fractional q-calculus operators; see, for example, [2–7], and may more. The q-fractional integrals and derivatives was firstly studied by Al-Salam [8] (see also, [9]). Many researchers have used these operators to evaluate fractional q-calculus formulas for various special function, general class of q-polynomials, basic analogue of Fox’s H-function, fractional q-calculus formulas for various special function, and etc. One may refer to the recent work [2–7, 10–14] on fractional q-calculus. Throughout this article, let ℤ,ℂ,ℝ,ℝ+, and N be the sets of integers, complex numbers, real numbers, positive real numbers, and positive integers, respectively, and let. ℕ0≔ℕ∪0.The objective of this article is to establish fractionalq-integral and q-derivative of Riemann-Liouville type involving a basic analogue of multivariable H-function. We also give an application of q-Leibniz formula.In theq-calculus theory, for a real parameter q∈ℝ+\1, we have a q-real number aqandq-shifted factorial (q-analogue of the Pochhammer symbol) as given by
(1)aq1−qa1−q,a;qn=∏i=1n−11−aqia∈ℝ,n∈ℕ∪∞.Theq-Factorial function is defined by
(2)a;qn=∏1,ifn=0n=1k=01−aqk,ifn∈ℕ.Its extension is(3)a;qn=a;q∞aqn;q∞n∈ℤ,which can be elaborated to n=α∈ℂ, given by
(4)a;qα=a;q∞aqn;q∞α∈ℂ;q<1,where the principal value of qα is taken.In terms of theq-gamma function, (2) can be written as
(5)a;qn=Γqa+n1−qnΓqaa∈ℝ\0,−1,−2,,where the q-gamma function [15] is given by
(6)Γqa=q;q∞1−q1−aqa;q∞=q;qa−11−qa−1q<1;a∈ℝ\0,−1,−2,,obviously,
(7)Γqa+1=aqΓqaq!q<1.Theq-analogue of the familiar Riemann-Liouville fractional integral operator of a function fxis defined by (see Al-Salam [8])
(8)Iqμfx=1Γqμ∫x0x−tqμ−1ftdqtRμ>0,q<1,also q-analogue of the power function is defined as
(9)x−yυ=xυyx;qυ=xυ∏n=0∞1−y/xqn1−y/xqn+υ,x≠0.The basic integral is given by (see Gasper and Rahman [15])
(10)∫x0ftdqt=x1−q∑k=0∞qkfxqk.The equation (8) in conjunction with (10) yield the following series representation of the Riemann-Liouville fractional integral operator
(11)Iqμfx=xμ1−qΓqμ∑k=0∞qk1−qk+1μ−1fxqk.In particular, forfx=xλ−1, we have [4]
(12)Iqμxλ−1=ΓqλΓqλ+μxλ+μ−1Rλ+μ>0.


## 2. Basic Analogue of MultivariableH-Function

In this section, we introduce the basic analogue of multivariableH-function [16, 17], given by the following manner:
(13)Hz1,zr;q=Hp,q′:p1q1;⋯;pr,qr0,n:m1,n1;⋯;mr,nrz1⋅⋅⋅zr;qaj;αjr⋯,αjr1,p:cj′,γ′1,p1,⋯,cjr,γjr1,prbj;βj′,⋯,βjr1,q′:dj′,δ′1,q1,⋯,djr,δjr1,q1=12πωr∫L1⋯∫Lrπrϕs1,⋯,sr;q∏i=1rθisi;qx1s1⋯xrsrdqs1⋯dqsr,where ω=−1, and
(14)ϕs1,⋯,sr;q=∏j=1nGq1−αj+∑i=1rαjisi∏j=n+1pGqaj−∑irαjisi∏j=1q′Gq1−bj+∑i=1rβjisi,θis1;q=∏j=1miGqdji−δjisi∏j=1niGq1−cji+γjisi∏j=mi+1qiGq1−dji+δjisi∏j=ni+1piGqcji+γjisiGq1−sisinπsi,here, i=1,⋯,r and
(15)Gqa=∏n=0∞1−qa+n−1=1qa;q∞.The integersn,p,q,mi,ni,pi,qi are constrained by the inequalities 0≤n≤p,0≤q′,1≤mi≤qi and 0≤ni≤pi,i=1,⋯,r. The poles of integrand are assumed to be simple. The quantitiesajj=1,⋯,p;cjij=1,⋯,pi;bjj=1,⋯,q′;djij=1,⋯,qi,i=1,⋯,r are complex numbers and the following quantities αjij=1,⋯,p;γjij=1,⋯,pi;βjij=1,⋯,q′;δjij=1,⋯,qi,i=1,⋯,r are positive real numbers.The contourLi in the complex si-plane is of the Mellin-Barnes type which runs from −ω∞ to ω∞ with indentations, if necessary to make certain that all the poles of Gqdji+δjisij=1,⋯,mi are separated from those of, Gq1−cji+γjisii=1,⋯,niGq1−aj+∑i=1rαjisij=1,⋯,n. For large values of si, the integrals converge if Rslogzi−logsinπsi<0i=1,⋯,r.If the quantitiesαjij=1,⋯,p;γjij=1,⋯,pi;βjij=1,⋯,q′;δjij=1,⋯,qi=1fori=1,⋯,r, then the basic analogue of multivariable H-function reduces in basic analogue of multivariable Meijer’s G-function defined by Khadia and Goyal [18], we obtain
(16)Gz1,⋯,zr;q=Gp,q′:p1q1;⋯;pr,qr0,n:m1,n1;⋯;mr,nrz1⋅⋅⋅zr;qajp:cj′p1,⋯,cjrprbjq′:dj′q1,⋯,djrq1=12πωr∫L1⋯∫Lrπrυs1,⋯,sr;q∏i=1rυisi;qz1s1⋯zrsrdqs1⋯dqsr,where
(17)υs1,⋯,sr;q=∏j=1nGq1−αj+∑i=1rsi∏j=n+1pGqaj−∑i=1rsi∏j=1q′Gq1−bj+∑i=1rsi,υs1;q=∏j=1miGqdji−si∏j=1niGq1−cji+si∏j=mi+1qiGq1−dji+si∏j=ni+1piGq1−cji−siGq1−sisinπsi,where i=1,⋯,r; the integers n,p,q,mi,ni,pi,qi are constrained by the inequalities 0≤n≤p,0≤q′,1≤mi≤qi and 0≤ni≤pi,i=1,⋯,r. The poles of integrand are assumed to be simple. The quantities αjij=1,⋯,p;cjij=1,⋯,pi;bjj=1,⋯,q′;djij=1,⋯,qi;i=1,⋯,r are complex numbers.The contourLi in the complex si-plane is of the Mellin-Barnes type which runs from −ω∞ to ω∞ with indentations, if necessary to ensure that all the poles of Gqdji+δjisij=1,⋯,mi are separated from those of Gq1−cji+sii=1,⋯,niGq1−aj+∑i=1rsij=1,⋯,n. For large values of si, the integrals converge if Rslogzi−logsinπsi<0i=1,⋯,r.


## 3. Main Results

In this section, we establish two fractionalq-integral formulas about the basic analogue of multivariable H-function.Let(18)U=m1,n1;⋯;mr,nr;V=p1,q1;⋯;pr,qr;A=aj;αj′,⋯,αjr1,p;B=cj′,γ′1,p1,⋯,cjr,γjr1,pr;C=bj;βj′,⋯,βjr1,q′;D=dj′,δ′1,q1,⋯,djr,δjr1,qr.Theorem 1.
LetRμ>0,q<1, the Riemann-Liouville fractionalq-integral of a product of two basic functions exists, and we have(19)Iqμxλ−1Hp,q′:V0,n:Uz1xP1⋅⋅⋅z1xPr;qA:BC:D=1−qμxλ+μ−1Hp+1,q′+1:V0,n+1:Uz1xP1⋅⋅⋅z1xPr;q1−λ;p1,⋯,prA:BC1−λ−μ;p1,⋯,pr:D,where pi∈ℕ,Rslogzi−logsinπsi<0fori=1,⋯,r.Proof.
To prove the result (19), we consider the left hand side of equation (19) (say I) and take the definitions (8) and (13) into account, we have
(20)I=1Γqα∫x0x−yqα−1yλ−12πwr∫L1⋯∫Lrπrϕs1,⋯,sr;q∏i=1rθisi;q×z1sr⋯zrsryρ1s1+⋯ρrsrdqs1⋯dqsrdqy.
Interchanging the order of integrations which is permissible under the given conditions, we obtain(21)I=1Γqα2πwr∫L1⋯∫Lrπrϕs1,⋯,sr;q∏i=1rθisi;qz1sr⋯zrsr×∫x0x−yqα−1yρ1s1+⋯ρrsr+λ−1dqydqs1⋯dqsr.
The above equation writes(22)I=12πwr∫L1⋯∫Lrπrϕs,t;qθ1s;q⋯θrs;qz1s1⋯zrρr×Iqμxρ1s1+⋯+ρrsr+λ−1dqs1⋯dqsr.
Now using the result (12), then the equation (22) reduces as
(23)I=12πwr∫L1⋯∫Lrπrϕs,t;qθ1s;q⋯θrs;qz1s1⋯zrρr×Γqρ1s1+⋯ρrsr+λΓqρ1s1+⋯ρrsr+λ+μxρ1s1+⋯ρrsr+λ+μ−1.
Next, interpreting theq-Mellin-Barnes multiple integrals contour in terms of the basic analogue of multivariable H-function, then we get the desired result (19).
If we replaceμ by −μ in Theorem 1, and use the fractional q- derivative operator defined as
(24)Iq−μfx=Dx,qμfx=1Γq−μ∫0xx−tq−μ−1ftdqtRμ<0,and power function formula
(25)Dx,qμxλ−1=ΓqλΓqλ−μxλ−μ−1λ≠−1,−2,⋯,then we have the following result:Theorem 2.
LetRμ>0,q<1the Riemann Liouville fractionalq-derivative of a product of two basic functions exists, and given by(26)Dx,qμxλ−1Hp,q′:V0,n:Uz1xρ1⋅⋅⋅zrxρr;qA:BC:D=1−q−μxλ−μ−1Hp+1,q′+1:V0,n+1:Uz1xρ1⋅⋅⋅zrxρr;q1−λ;ρ1,⋯,ρrA:BC1−λ+μ;ρ1,⋯,ρr:D,whereρi∈ℕ,Rslogzi−logsinπsi<0fori=1,⋯,r.Proof.
The proof of result asserted by Theorem2 runs parallel to that of Theorem 1.
The details are, therefore, being omitted.


## 4. Leibniz’s Application

In this section, we give an application concerning the basic analogue of multivariableH-function and q-extension of the Leibniz rule for the fractional q-derivative for a product of two basic functions.We have theq-extension of the Leibniz rule for the fractional q-derivatives for a product of two basic functions in terms of a series involving the fractional q-derivatives of the function, in the following manner [9]:Lemma 3.
(27)Dx,qαWxYx=∑n=0∞−1nqnn+1/2q−μ;qnq;qnDx,qμ−nWxqnDx,qnYx,whereWxandYxare two regular functions.Theorem 4.
LetRμ<0,, then the Riemann-Liouville fractionalq-derivative of a product of two basic function exists and given by(28)Hp+1,q′+1:V0,n+1:Uz1xρ1⋅⋅⋅zrxρr;q1−λ;ρ1,⋯,ρr,A:BC,1−λ+μ;ρ1,⋯,ρr:D=∑n=0∞−1nqnλ+nn−1/2q−μ;qnq;qnqλ;qn−μHp+1,q′+1:V0,n+1:Uz1xρ1⋅⋅⋅zrxρr;q0;ρ1,⋯,ρrA:BCn;ρ1,⋯,ρr:D,where pi∈ℕ,Rslogzi−logsinπsi<0fori=1,⋯,r.Proof.
For applyingq-Leibniz rule, we let
(29)Wx=xλ−1andYxHp,q′:V0,n:Uz1xρ1⋅⋅⋅zrxρr;qA:BC:D.
By using the lemma3, we have
(30)Dx,qμxλ−1Hp,q′:V0,n:Uz1xρ1⋅⋅⋅zrxρr;qA:BC:D=∑n=0∞−1nqnn−1/2q−μ;qnq;qnDx,qμ−nxqnλ−1Dx,qnHz1xρ1,⋯,zrxρr;q.
Next, by settingλ=1 and using the Theorem 2, we arrive at
(31)Dx,qnHz1xρ1,⋯,zrxρr;q=1−q−μx−μ×Hp+1,q′+1:V0,n+1:Uz1xρ1⋅⋅⋅zrxρr;q0;ρ1,⋯,ρrA:BCn;ρ1,⋯,ρr:D,where ρi∈ℕ,Rslogzi−logsinπsi<0i=1,⋯,r.Now, by using (25) and (31) we obtain the desired result (28) after several algebraic manipulations.


## 5. Particular Case

In this section, the basic analogue of multivariableH-function reduces in basic analogue of multivariable Meijer’s G-function [18].Let(32)U=m1,n1;⋯;mr,nr;V=p1,q1;⋯;pr,qr;A1=aj1,p;B1=cj1,p1,⋯,cjr1,pr;C1bj1,q′;D1=dj1q1,⋯,djr1,qr.Corollary 5.
(33)Dx,qμtλ−1Gp+1,q′+1:V0,n+1:Uz1xρ1⋅⋅⋅zrxρr;qA1:B1C1:D1=∑n=0∞−1nqnλ+nn−1/2q−μ;qnq;qnqλ;qn−μGp+1,q′+1:V0,n+1:Uz1xρ1⋅⋅⋅zrxρr;q0;ρ1,⋯,ρrA1:B1C1,n;ρ1,⋯,ρr:D1,where pi∈ℕ,Rslogzi−logsinπsi<0fori=1,⋯,r.Remark 6.
If the basic analogue of multivariableH-function reduces in basic analogue of Srivastava-Daout function [19], then we obtain the results given by Purohit et al. [20].Remark 7.
If the basic analogue of multivariableH-function reduces in basic analogue ofH-function of two variables defined by Saxena et al. [21], we obtain the result due to Yadav et al. [7]. Further, if the basic analogue of multivariableH-function reduces in basic analogue ofH-function of one variable defined by Saxena et al. [22], then we can easily obtain the similar result.


## 6. Conclusion

In the present article, we have proposed the fractional orderq-integrals and q-derivatives involving a basic analogue of multivariable H-function. The significance of our derived results lies in their diverse generality. By specializing the various parameters as well as variables in the basic analogue of multivariable H-function, we can obtain a large number of results involving a remarkably wide range of useful basic functions (or product of such basic functions) of one and several variables. Hence, the derived formulas in this article are most general in character and may reaffirm to be useful in several interesting cases appearing in literature.



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