2000 Mathematics Subject Classification: 33D60, 26A33, 33C60The present paper envisages the applications of Riemann-Liouville fractional q-integral operator to a basic analogue of Fox H-function. Results involving the basic hypergeometric functions like Gq(.), Jv(x; q), Yv(x; q),Kv(x; q), Hv(x; q) and various other q-elementary functions associated with the Riemann-Liouville fractional q-integral operator have been deduced as special cases of the main result

Kalla, S.

Purohit, S.

Yadav, R.

Fractional Calculus and Applied Analysis

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ON THE RIEMANN-LIOUVILLE FRACTIONALq-INTEGRAL OPERATOR INVOLVING A BASICANALOGUE OF FOX H-FUNCTIONS. L. Kalla∗, R. K. Yadav∗∗ and S. D. Purohit∗∗AbstractThe present paper envisages the applications of Riemann-Liouville frac-tional q-integral operator to a basic analogue of Fox H-function. Resultsinvolving the basic hypergeometric functions like Gq(.), Jv(x; q), Yv(x; q),Kv(x; q),Hv(x; q) and various other q-elementary functions associated withthe Riemann-Liouville fractional q-integral operator have been deduced asspecial cases of the main result.2000 Mathematics Subject Classification: 33D60, 26A33, 33C60Key Words and Phrases: fractional q-integral, H-function, q-functions1. IntroductionAgarwal [1] introduced the q-analogue of the Riemann-Liouville frac-tional integral operator as follows:Iµq f(x) =1Γq(µ)x∫0(x− yq)µ−1 f(y)d(y; q), (1.1)where ‘µ‘ is an arbitrary order of integration such that Re(µ) > 0.314 S.L. Kalla, R.K. Yadav, S.D. PurohitFollowing Jackson [5], Al-Salam [2] and Agarwal [1], we have the basicintegration defined as:x∫0f(t)d(t; q) = x(1− q)∞∑k=0qkf(xqk). (1.2)In view of equation (1.2), (1.1) can be expressed as:Iµq f(x) =xµ(1− q)Γq(µ)∞∑k=0qk(1− qk+1)µ−1f(xqk), (1.3)where Re(µ) > 0.Further, for real or complex α and 0 < |q| < 1, the q-factorial is definedas:(α; q)n ≡ (qα; q)n ={1 ; n = 0(1− qα)(1− qα+1)...(1− qα+n−1) ; n = 1, 2, ...(1.4)or equivalently,(α; q)n =∞∏j=0(1− αqj)(1− αqn+j) =(α; q)∞(αqn; q)∞. (1.5)In terms of the basic analogue of the gamma function, we have(α; q)n =Γq(α+ n)(1− q)nΓq(α), n > 0 , (1.6)where the q-gamma function, cf. Gasper and Rahman [4], in various formis given byΓq(α) =(q; q)∞(qα; q)∞(1− q)α−1 =(1− q)α−1(1− q)α−1 =(q; q)α−1(1− q)α−1 , (1.7)(α 6= 0,−1,−2, ...).Indeedltq → 1−(qα; q)n(1− q)n = (α)n, (1.8)where(α)n = α(α+ 1)...(α+ n− 1). (1.9)ON THE RIEMANN-LIOUVILLE FRACTIONAL . . . 315The q-binomial series, cf. Gasper and Rahman [4], is given by1φ0 α ; q, x−; = (αx; q)∞(x; q)∞. (1.10)Saxena, Modi and Kalla [10], introduced the basic analogue of the Fox H-function in the following manner:Hm1,n1A,B[x; q∣∣∣∣ (a, α)(b, β)](1.11)=12pi i∫Cm1∏j=1G(qbj−βjs)n1∏j=1G(q1−αj+αjs)pixsB∏j=m1+1G(q1−bj+βjs)A∏j=n1+1G(qαj−αjs)G(q1−s) sinpisds,where 0 ≤ m1 ≤ B, 0 ≤ n1 ≤ A; α′is and β′js are all positive integers,the contour C is a line parallel to Re(ws), with indentations if necessary, insuch a manner that all poles of G(qbj−βjs ), 1 ≤ j ≤ m1 are to the right, andthose of G(q1−αj+αjs), 1 ≤ j ≤ n1, to the left of C. The integral convergesif Re[s log(x) − log sinpis] < 0 for large values of |s| on the contour, i.e.if |{arg(x) − w2w−11 log |x|}| < pi, where 0 < |q| < 1, log q = −w =(w1 + iw2), w, w1, w2 are definite quantities w1 and w2 being real.AlsoG(qα) ={ ∞∏n=0(1− qα+n)}−1=1(qα; q)∞. (1.12)If we set αj = βi = 1, 1 ≤ j ≤ A, 1 ≤ i ≤ B in (1.11), then it reduces to thebasic Meijer’s G-function, namelyHm1,n1A,B[x; q∣∣∣∣ (a, 1)(b, 1)]≡ Gm1n1A,B[x; q∣∣∣∣ a1, ..., aAb1, ..., bB]=12pi i∫Cm1∏j=1G(qbj−s)n1∏j=1G(q1−aj+s)pixsB∏j=m1+1G(q1−bj+s)A∏j=n1+1G(qαj−s)G(q1−s) sinpisds, (1.13)where 0 ≤ m1 ≤ B, 0 ≤ n1 ≤ A; and Re[s log(x)− log sinpis] < 0.316 S.L. Kalla, R.K. Yadav, S.D. PurohitFurther, if we set n1 = 0,m1 = B in (1.13), we get the basic analogueof MacRobert’s E-functionGB,0A,B[x; q∣∣∣∣ a1, ..., aAb1, ..., bB]= Eq[B; bj : A; aj : x]. (1.14)Saxena and Kumar [11], introduced the basic analogues of Jv(x), Yv(x),Kv(x), Hv(x) in terms of Gq(.) function as follows:Jv(x; q) = {G(q)}2G1,00,3[x2(1− q)24; q∣∣∣∣ −v2 ,−v2 , 1], (1.15)where Jv(x; q) denotes the q-analogue of Bessel function Jv(x).Yv(x; q) = {G(q)}2 G2,01,4[x2(1− q)24; q∣∣∣∣ −v−12v2 ,−v2 ,−v−12 , 1], (1.16)where Yv(x; q) denotes the q-analogue of the Bessel function Yv(x).Kv(x; q) = (1− q) G2,00,3[x2(1− q)24; q∣∣∣∣ −v2 ,−v2 , 1], (1.17)where Kv(x; q) denotes the basic analogue of the Bessel function of the thirdkind Kv(x).Hv(x; q) =(1− q2)1−αG3,11,4[x2(1− q)24; q∣∣∣∣ 1+α2v2 ,−v2 ,1+α2 , 1], (1.18)where Hv(x; q) is the basic analogue of Struve’s function Hv(x).In view of the definition (1.11), the following elementary basic (q-) func-tions are expressible in terms of the basic analogue of Meijer’s G-functionas:eq(−x) = G(q)G1,00,2[x(1− q); q∣∣∣∣ −0, 1](1.19)sinq(x) =√pi(1− q)−1/2{G(q)}2 G1,00,3[x2(1− q)24; q∣∣∣∣ z−12 , 0, 1](1.20)cosq(x) =√pi(1− q)−1/2{G(q)}2 G1,00,3[x2(1− q)24; q∣∣∣∣ −0, 12 , 1](1.21)sinhq(x) =√pii(1− q)−1/2{G(q)}2 G1,00,3[−x2(1− q)24; q∣∣∣∣ −12 , 0, 1](1.22)ON THE RIEMANN-LIOUVILLE FRACTIONAL . . . 317coshq(x) =√pi(1− q)−1/2{G(q)}2 G1,00,3[−x2(1− q)24; q∣∣∣∣ −0, 12 , 1]. (1.23)A detailed account of various functions expressible in terms of the MeijerG-function or Fox H-function can be found in research monographs due toMathai and Saxena [8], [9]. A systematic and unified development of anew generalized fractional calculus, closely related to special functions of arather general nature is given in [6], [7].In a recent paper, Yadav and Purohit [12] have investigated some appli-cations of Riemann-Liouville fractional q-integral operator to various basichypergeometric functions of one variable.The motive for the present paper is to evaluate the Riemann-Liouvillefractional basic integral operator involving basic analogue of the H-functionand various other basic hypergeometric functions. The results deduced arebelieved to find certain applications to the solutions of basic (q-) integralequations.2. Main resultsIn this section, we shall evaluate the following q-fractional integral ofthe Riemann-Liouville type involving a basic analogue of Fox’s H-functions:Iµq{Hm1,n1A,B[ρxλ; q∣∣∣∣ (a, α)(b, β)]}=xµ(1− q)µHm1,n1+1A+1,B+1[ρxλ; q∣∣∣∣ (0, λ), (a, α)(b, β), (−µ, λ)], λ ≥ 0xµ(1− q)µHm1+1,n1A+1,B+1[ρxλ; q∣∣∣∣ (a, α), (1− λ)(1 + µ,−λ), (b, β)], λ < 0(2.1)where 0 ≤ m1 ≤ B, 0 ≤ n1 ≤ A.P r o o f o f (2.1). To prove (2.1), we apply equations (1.3) and (1.11)to the left hand side to obtainxµ(1− q)µ∞∑k=0qk(qµ; q)k(q; q)k× 12pii∫Cm1∏j=1G(qbj−βjs)n1∏j=1G(q1−aj+αjs)pi(pxλqλk)sB∏j=m1+1G(q1−bj+βjs)A∏j=n1+1G(qaj−αjs)G(q1−s) sinpisds.318 S.L. Kalla, R.K. Yadav, S.D. PurohitInterchanging the order of summation and integration, which is valid underconditions given with (1.11), the above expression reduces toxµ(1− q)µ2pii∫Cm1∏j=1G(qbj−βjs)n1∏j=1G(q1−aj+αjs)pi(pxλ)sB∏j=m1+1G(q1−bj+βjs)A∏j=n1+1G(qaj−αjs)G(q1−s) sinpis×∞∑k=0qk(1+λs)(qµ; q)k(q; q)kds,on summing inner 1φ0(.) series with the help of (1.10), it yields after somesimplificationsxµ(1− q)µ2pii×∫Cm1∏j=1G(qbj−βjs)n1∏j=1G(q1−aj+αjs)G(q1+λs)pi(pxλ)sB∏j=m1+1G(q1−bj+βjs)G(q1+µ+λs)A∏j=n1+1G(qaj−αjs)G(q1−s) sinpisds,which on interpretation in view of (1.11), leads us to the right hand side ofthe result (2.1). The second part follows similarly.3. ApplicationsIn this section, we shall derive certain basic integrals of the Riemann-Liouville type, involving various basic functions expressible in terms of thebasic analogue of Fox’s H-function or Meijer’s G-function, as the applicationof the main result (2.1). These results are presented in tabular form, seeTable 1.Proofs of the results (3.1) and (3.2) follows directly from (2.1) withλ = ρ = 1 and on using the definitions (1.13)-(1.14) respectively.While if we assign m1 = 1, n1 = A = 0, B = 3, b1 = v/2, b2 = −v/2, b3 =1, λ = 2, ρ = (1−q)24 in equation (2.1), we obtainIµq{H1,00,3[x2(1− q)24; q∣∣∣∣ −(v2 , 1), (−v2 , 1), (1, 1)]}= xµ(1− q)µON THE RIEMANN-LIOUVILLE FRACTIONAL . . . 319×H1,11,4[x2(1− q)24; q∣∣∣∣ (0, 2),(v2 , 1), (−v2 , 1)(1, 1), (−µ, 2)], (3.12)in view of of the definitions (1.15), the left hand side of (3.12) reduces toIµq {Jv(x; q)} = xµ(1− q)µ{G(q)}2×H1,11,4[x2(1− q)24; q∣∣∣∣ (0, 2)(v2 , 1), (−v2 , 1)(1, 1), (−µ, 2)]. (3.13)This completes the proof of (3.3).The proof of (3.4)-(3.6) follows similarly. To prove (3.7), we take m1 =1, n1 = A = 0, B = 2, αi = βj = 1, b1 = 0, b2 = 1, λ = 1, ρ = (1 − q) in themain result (2.1) and then on making use of the definitions(1.13) and (1.19)we arrive at (3.7).The results (3.8)-(3.11) can be proved similarly by assigning particularvalues to the parameters m1, n1, A,B, λ and ρ, keeping in view the defini-tions (1.20)-(1.23).Finally, it is interesting to observe that in view of a limit formula forGq(.) function due to Saxena and Kumar [11], if we let q → 1−1, in (3.1),we obtain a known result mentioned in Erde´lyi [3] [eq. no. 96, table (13.1),p. 200].AcknowledgmentS. D. Purohit is grateful to CSIR, Govt. of INDIA, for providing aSenior Research Fellowship (No. 9/98(51) 2003. EMR.I) to enable him tocarry out the present investigations.320 S.L. Kalla, R.K. Yadav, S.D. PurohitTable 1ON THE RIEMANN-LIOUVILLE FRACTIONAL . . . 321Table 1, cont’d322 S.L. Kalla, R.K. Yadav, S.D. PurohitReferences1. R. P. Agarwal, Certain fractional q-integrals and q-derivatives. Proc.Camb. Phil. Soc. 66 (1969), 365-370.2. W. A. Al-Salam, Some fractional q-integrals and q-derivatives. Proc.Edin. Math. Soc. 15 (1966), 135-140.3. A. Erde´lyi (Editor), Tables of Integral Transforms. Vol. II, McGraw-Hill Book Co. Inc., New York (1954).4. G. Gasper, G. and M. Rahman, Basic Hypergeometric Series. Cam-bridge University Press, Cambridge (1990).5. F. H. Jackson, Basic integration. Quart. J. Math. (Oxford), 2 (1951),1-16.6. S. L. Kalla and V. S. Kiryakova, An H-function generalized fractionalcalculus based upon composition of Erde´lyi-Kober operators in Lp.Math. Japonica 35 (1990), 1151-1171.7. V. S. Kiryakova, Generalized Fractional Calculus and Applications.Longman Scientific & Tech., Essex (1994).8. A. Mathai and R. K. Saxena, Generalized Hypergeometric FunctionsWith Applications in Statistics and Physical Sciences. Springer-Verlag,Berlin (1973).9. Mathai, A. M and Saxena, R. K.: The H-function With Applicationin Statistics and Other Disciplines. John Wiley & Sons. Inc. NewYork. (1978).10. R. K. Saxena, G. C. Modi and S. L. Kalla, A basic analogue of Fox’sH-function. Rev. Tec. Ing., Univ. Zulia 6 (1983), 139-143.11. R. K. Saxena and R. Kumar, Recurrence relations for the basic ana-logue of the H-function. J. Nat. Acad. Math. 8 (1990), 48-54.12. R. K. Yadav and S. D. Purohit, Application of Riemann-Liouvillefractional q-integral operator to basic hypergeometric functions. Toappear in: Acta Ciencia Indica 30, No. 3 (2004).∗ Department of Mathematics and Computer ScienceP.O. Box-5969, Safat 13060, KUWAITe-mail: shyamkalla@yahoo.com Received: 24 April 2005∗∗ Department of Mathematics & StatisticsJ. N. Vyas University, Jodhpur-342001 (INDIA)e-mail: rkmdyadav@yahoo.co.in

On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function

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On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function

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