Mathematics Subject Classification: 33D60, 33D90, 26A33Fractional q-integral operators of generalized Weyl type, involving generalized basic hypergeometric functions and a basic analogue of Fox’s H-function have been investigated. A number of integrals involving various q-functions have been evaluated as applications of the main results

Kalla, S.

Purohit, S.

Yadav, R.

Fractional Calculus and Applied Analysis

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ON GENERALIZED WEYL FRACTIONAL q-INTEGRALOPERATOR INVOLVING GENERALIZED BASICHYPERGEOMETRIC FUNCTIONSR.K. Yadav 1, S.D. Purohit 2, S.L. Kalla 3AbstractFractional q-integral operators of generalized Weyl type, involving gen-eralized basic hypergeometric functions and a basic analogue of Fox’s H-function have been investigated. A number of integrals involving variousq-functions have been evaluated as applications of the main results.Mathematics Subject Classification: 33D60, 33D90, 26A33Key Words and Phrases: generalized Weyl fractional q-integral oper-ator, basic integration, generalized basic hypergeometric functions, basicanalogue of Fox’s H-function1. IntroductionAl-Salam [3] introduced the generalized Weyl fractional q-integral oper-ator in the following manner:Kη,µq {f(x)} =q−ηxηΓq(µ)∫ ∞x(y − x)µ−1 y−η−µf(yq1−µ)d(y; q), (1.1)where Re(µ) > 0, η is arbitrary and the basic integration, cf. Gasper andRahman [5], is defined as:∫ ∞xf(t)d(t; q) = x(1− q)∞∑k=1q−kf(xq−k). (1.2)In view of relation (1.2), operator (1.1) can be expressed as:130 R.K. Yadav, S.D. Purohit, S.L. KallaKη,µq {f(x)} = (1− q)µ∞∑k=0qkη(qµ; q)k(q; q)kf(xq−µ−k), (1.3)where Re(µ) > 0 and η being an arbitrary complex quantity.In the sequel we shall use the following notations and definitions:For real or complex a and |q| < 1, the q-shifted factorial is defined as:(a; q)n =1 ; if n = 0(1− a)(1− aq) · · · (1− aqn−1) ; if n ∈ N.(1.4)In terms of the q-gamma function, (1.4) can be expressed as(a; q)n =Γq(a+ n)(1− q)nΓq(a), n > 0, (1.5)where the q-gamma function (cf. Gasper and Rahman [5]), is given byΓq(a) =(q; q)∞(qa; q)∞(1− q)a−1 =(q; q)a−1(1− q)a−1 , (1.6)where a 6= 0,−1,−2, · · ·. Also,(x− y)ν = xν∞∏n=0[1− (y/x)qn1− (y/x)qν+n]= xν 1Φ0 q−ν ;q, yqν/x; .(1.7)The generalized basic hypergeometric series, cf. Gasper and Rahman [5], isgiven byrΦs a1, · · · , ar ;q, xb1, · · · , bs ;= ∞∑n=0(a1, · · · , ar; q)n(q, b1, · · · , bs; q)n xn{(−1)nqn(n−1)/2}(1+s−r),(1.8)where for convergence, we have |q| < 1 and |x| < 1 if r = s+1, and for anyx if r ≤ s. The abnormal type of generalized basic hypergeometric seriesrΦs(·) is defined asrΦs[a1, · · · , ar ; q, xb1, · · · , bs ; qλ]=∞∑n=0(a1, · · · , ar; q)n(q, b1, · · · , bs; q)n xnqλn(n+1)/2, (1.9)where λ > 0 and |q| < 1.The q-exponential series isON GENERALIZED WEYL FRACTIONAL q-INTEGRAL . . . 131eq(x) =∞∑n=0xn(q; q)n. (1.10)The q-binomial series is given by1Φ0 α ;q, x; = (αx; q)∞(x, q)∞. (1.11)The basic analogue of the Sine and Cosine functions aresinq(ax) =12i{eq(iax)− eq(−iax)} , (1.12)andcosq(ax) =12{eq(iax) + eq(−iax)} . (1.13)Similarly, we have the q-Laguerre polynomial:L(α)n (x; q) =(qα+1; q)n(q; q)n1Φ1 q−n ;q,−xqnqα+1 ; ; (1.14)the little q-Jacobi polynomial:P (α,β)n (x; q) =(qα+1; q)n(q; q)n2Φ1 q−n, qα+β+n+1 ;q, xqqα+1 ; ; (1.15)the Wall polynomial (or little q-Laguerre polynomial)Wn(x; b, q) = (−1)n(b; q)n qn(n+1)/2 2Φ1 q−n, 0 ;q, xb ; ; (1.16)and the Stieltjes-Wigert polynomial:sn(x; q) =n∑k=0(q−n; q)k(q; q)kqk(k+1)/2 (−x)k = 1Φ0[q−n ; q,−x; q1]. (1.17)Saxena, Modi and Kalla [9], introduced a basic analogue of the H-function in terms of the Mellin-Barnes type basic contour integral in thefollowing manner:Hm,nA,B[x; q∣∣∣∣ (a, α)(b, β)]132 R.K. Yadav, S.D. Purohit, S.L. Kalla=12pii∫Cm∏j=1G(qbj−βjs)n∏j=1G(q1−aj+αjs)pixsB∏j=m+1G(q1−bj+βjs)A∏j=n+1G(qaj−αjs) G(q1−s) sinpisds, (1.18)whereG(qα) =∞∏n=0{(1− qα+n)}−1 = 1(qα; q)∞, (1.19)and 0 ≤ m ≤ B; 0 ≤ n ≤ A; αj and βj are all positive integers. Thecontour C is a line parallel to Re(ωs) = 0, with indentations, if necessary,in such a manner that all the poles of G(qbj−βjs) ; 1 ≤ j ≤ m, are to itsright, and those of G(q1−aj−αjs) ; 1 ≤ j ≤ n, are to the left of C. The basicintegral converges if Re [s log(x)− log sinpix] < 0, for large values of |s| onthe contour C, that is if∣∣{arg(x)− ω2 ω−11 − log |x|}∣∣ < pi, where |q| < 1,log q = −ω = −(ω1 + iω2), ω, ω1 and ω2 are definite quantities. ω1 and ω2being real.For αj = βi = 1, j = 1, · · · , A; i = 1, · · · , B, the definition (1.18)reduces to the q-analogue of the Meijer G-function due to Saxena, Modiand Kalla [9], namely:Gm,nA,B[x; q∣∣∣∣ a1, · · · , aAb1, · · · , bB]=12pii∫Cm∏j=1G(qbj−s)n∏j=1G(q1−aj+s)pixsB∏j=m+1G(q1−bj+s)A∏j=n+1G(qaj−s) G(q1−s) sinpisds, (1.20)where 0 ≤ m ≤ B; 0 ≤ n ≤ A and Re [s log(x)− log sinpix] < 0.Further, if we set n = 0 and m = B in the equation (1.20), we get thebasic analogue of MacRobert’s E-function due to Agarwal [1], namelyGB,0A,B[x; q∣∣∣∣ a1, · · · , aAb1, · · · , bB]≡ Eq [B; bj : A; aj : x]=12pii∫CB∏j=1G(qbj−s)pixsA∏j=1G(qaj−s) G(q1−s) sinpisds, (1.21)where Re [s log(x)− log sinpix] < 0.ON GENERALIZED WEYL FRACTIONAL q-INTEGRAL . . . 133Saxena and Kumar [8], introduced the basic analogues of Jν(x), Yν(x),Kν(x), Hν(x) in terms of Hq(·) function as follows:Jν(x; q) = {G(q)}2H1,00,3 x2(1− q)24 ; q(ν2 , 1), (−ν2 , 1), (1, 1) , (1.22)where Jν(x; q) denotes the q-analogue of Bessel function of first kind Jν(x);Yν(x; q) = {G(q)}2H2,01,4(−ν − 12, 1)x2(1− q)24; q(ν2 , 1), (−ν2 , 1), (−ν−12 , 1), (1, 1) ,(1.23)where Yν(x; q) denotes the q-analogue of the Bessel function Yν(x);Kν(x; q) = (1− q)H2,00,3 x2(1− q)24 ; q(ν2 , 1), (−ν2 , 1), (1, 1) , (1.24)where Kν(x; q) denotes the basic analogue of the Bessel function of the thirdkind Kν(x);Hν(x; q)=(1−q2)1−αH3,11,4(1+α2, 1)x2(1−q)24; q(ν2 , 1), (−ν2 , 1), (1+α2 , 1), (1, 1) ,(1.25)where Hν(x; q) is the basic analogue of Struve’s function Hν(x).Following Saxena and Kumar [8], Mathai and Saxena [6], [7], we havethe following q-extensions of certain elementary functions in terms of a basicanalogue of the Meijer G-function as:eq(−x) = G(q)H1,00,2 x(1− q); q(0, 1), (1, 1) , (1.26)sinq(x) =√pi(1−q)−1/2{G(q)}2H1,00,3 x2(1−q)24 ; q(12 , 1), (0, 1), (1, 1) ;(1.27)134 R.K. Yadav, S.D. Purohit, S.L. Kallacosq(x)=√pi(1−q)−1/2{G(q)}2H1,00,3 x2(1− q)24 ; q(0, 1), (12 , 1), (1, 1);(1.28)sinhq(x)=√pii(1−q)−1/2{G(q)}2H1,00,3−x2(1−q)24 ; q(12 , 1), (0, 1), (1, 1);(1.29)coshq(x)=√pi(1−q)−1/2{G(q)}2H1,00,3−x2(1− q)24 ; q(0, 1), (12 , 1), (1, 1).(1.30)A detailed account of various classical special functions expressible interms of Meijer’s G-function or Fox’s H-function can be found in researchmonographs by Mathai and Saxena [6] and [7].The main motive of the present paper is to investigate the generalizedWeyl fractional q-integral operator involving basic hypergeometric functionsincluding the basic analogue of the H-function. Certain interesting specialcases have also been derived as the applications of the main results.2. Main resultsIn this section, we shall evaluate the following fractional q-integrals ofgeneralized Weyl type involving basic hypergeometric function rΦs(·) andbasic analogue of Fox’s H-function. The main results are presented in thefollowing theorems.Theorem 1. If Re(η − λ) > 0 and ρ is any complex number, then thegeneralized Weyl fractional q-integral of xλ-weighted basic hypergeometricfunction rΦs(·), is given byON GENERALIZED WEYL FRACTIONAL q-INTEGRAL . . . 135Kη,µqxλ rΦs a1, · · · , ar ;q, ρxb1, · · · , bs ; = xλq−µλΓq(η − λ)Γq(η − λ+ µ)r+1Φs+1 a1, · · · , ar, q1−η+λ−µ ;q, ρxb1, · · · , bs, q1−η+λ ; . (2.1)P r o o f. In view of relations (1.3) and (1.8), the left hand side of (2.1)becomes(1− q)µ∞∑k=0qηk(qµ; q)k(q; q)k(xq−µ−k)λ∞∑n=0(a1, · · · , ar; q)n(q, b1, · · · , bs; q)n×{(−1)nqn(n−1)/2}(1+s−r)(ρxq−µ−k)n,and, interchanging the order of summations, we further obtainxλ(1−q)µq−µλ∞∑n=0(a1, · · · , ar; q)n(q, b1, · · · , bs; q)n{(−1)nqn(n−1)/2}(1+s−r)(ρxq−µ)n×∞∑k=0(qµ; q)k(q; q)kqk(η−λ−n).On summing the inner 1Φ0(·) series with the help of the equation (1.11),the above expression reduces toxλ(1− q)µq−µλ∞∑n=0(a1, · · · , ar; q)n(q, b1, · · · , bs; q)n(qη−λ−n; q)µ×{(−1)nqn(n−1)/2}(1+s−r)(ρxq−µ)n,and by further simplification, the above expression yields to the right handside of (2.1).Theorem 2. Let Re(µ) > 0, λ ∈ I, ρ be any complex number, then thefollowing generalized Weyl fractional q-integral of Hq(.) function for λ ≥ 0and λ < 0 holds:Kη,µq{Hm,nA,B[ρxλ; q∣∣∣∣ (a, α)(b, β)]}136 R.K. Yadav, S.D. Purohit, S.L. Kalla= (1−q)µHm+1,nA+1,B+1[ρ(xq−µ)λ; q∣∣∣∣ (a, α), (µ+ η, λ)(η, λ), (b, β)], λ ≥ 0(2.2)= (1−q)µHm,n+1A+1,B+1[ρ(xq−µ)λ; q∣∣∣∣ (1− η,−λ), (a, α)(b, β), (1− µ− η,−λ)], λ < 0,(2.3)where 0 ≤ m ≤ B, 0 ≤ n ≤ A and Re [s log(x)− log sinpix] < 0.P r o o f. To prove the theorem, we consider the left hand side of (2.2)and use definitions (1.3) and (1.18) to obtain(1− q)µ2pii∞∑k=0qηk(qµ; q)k(q; q)k×∫Cm∏j=1G(qbj−βjs)n∏j=1G(q1−aj+αjs)pi(ρxλq−µλ−kλ)sB∏j=m+1G(q1−bj+βjs)A∏j=n+1G(qaj−αjs)G(q1−s) sinpisds.On interchanging the order of summation and integration, valid underthe conditions given with equation (1.18), the above expression reduces to(1− q)µ2pii∫Cm∏j=1G(qbj−βjs)n∏j=1G(q1−aj+αjs)pi(ρxλq−µλ)sB∏j=m+1G(q1−bj+βjs)A∏j=n+1G(qaj−αjs)G(q1−s) sinpis×∞∑k=0(qµ; q)k(q; q)kqk(η−λs)ds.On summing the inner 1Φ0(·) series, with the help of equation (1.11)and on using definition (1.19), the left hand side of (2.2) finally reduces to(1−q)µ2pii∫Cm∏j=1G(qbj−βjs)G(qη−λs)n∏j=1G(q1−aj+αjs)pi(ρxλq−µλ)sB∏j=m+1G(q1−bj+βjs)A∏j=n+1G(qaj−αjs)G(qµ+η−λs)G(q1−s) sinpisds.Interpreting the above expression in view of definition (1.18), we obtain theright hand side of (2.2). The second part of the theorem follows similarly.ON GENERALIZED WEYL FRACTIONAL q-INTEGRAL . . . 1373. Applications of the main resultsIn this section, we evaluate some basic integrals of generalized Weyltype, involving basic hypergeometric functions and various elementary basicfunctions expressible in terms of a basic analogue of Fox’s H-function, asapplications of the theorems from the previous section. These results arepresented in the table that follows.For the sake of brevity, we mention here the proofs of a few results givenin the table. For example the results (3.1) to (3.6) have been derived byassigning appropriate values to the parameters r, s and ρ in (2.1), Theorem1, keeping in view of the definitions given by equations (1.7)-(1.17).If we set r = s = 0 and ρ = 1 in (2.1), it reduces to (3.2). Further,on making use of the result (3.2), we can easily prove the results (3.7) and(3.8). The proof of the result (3.9) is similar to the result (2.1).The proofs of the results (3.11) and (3.12) follow directly from (2.2) withρ = λ = 1 and on using the definitions (1.20)-(1.21), respectively.While, if we assign m = 1, n = A = 0, B = 3, b1 = ν/2, b2 = −ν/2, b3 =1, λ = 2 and ρ = (1−q)24 in (2.1), we obtainKη,µq{H1,00,3[x2(1− q)24; q∣∣∣∣ −(ν2 , 1), (−ν2 , 1), (1, 1)]}= (1−q)µ×H2,01,4[x2(1− q)24q2µ; q∣∣∣∣ (µ+ η, 2)(η, 2), (ν2 , 1), (−ν2 , 1), (1, 1)].(3.22)In view of of definition (1.22), the above equation (3.22) reduces to theresult (3.13). The results (3.14)-(3.21) can be proved similarly by assigningparticular values to the parameters m,n,A,B, λ and ρ, keeping in minddefinitions (1.23)-(1.30), respectively.The results deduced in the present paper aim to contribute to the theoryof basic hypergeometric series and q-fractional calculus. They are expectedto find some applications to the solutions of fractional q-differ-integral equa-tions. We intend to take up this aspect in a next contribution.138 R.K. Yadav, S.D. Purohit, S.L. KallaON GENERALIZED WEYL FRACTIONAL q-INTEGRAL . . . 139140 R.K. Yadav, S.D. Purohit, S.L. KallaON GENERALIZED WEYL FRACTIONAL q-INTEGRAL . . . 1413.19 cosq(x)√pi(1− q)µ−1/2{G(q)}2H2,01,426664(µ+ η, 2)x2(1− q)24q2µ; q(η, 2), (0, 1), (12, 1), (1, 1)377753.20 sinhq(x)√pii(1− q)µ−1/2{G(q)}2H2,01,42664 (µ+ η, 2)−x2(1− q)24q2µ ; q(η, 2), ( 12, 1), (0, 1), (1, 1)37753.21 coshq(x)√pi(1− q)µ−1/2{G(q)}2H2,01,426664(µ+ η, 2)−x2(1− q)24q2µ; q(η, 2), (0, 1), (12, 1), (1, 1)37775References[1] N. Agarwal, A q-analogue of MacRobert’s generalizedE-function. Ganita11 (1960), 49-63.[2] R.P. Agarwal, Fractional q-derivatives and q-integrals and certain hyper-geometric transformations. Ganita 27 (1976), 25-32.[3] W.A. Al-Salam, Some fractional q-integrals and q-derivatives. Proc.Edin. Math. Soc. 15 (1966), 135-140.[4] A. Erde´lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables ofIntegral Transforms. Vol. II. MacGraw-Hill Book Co. Inc., New York(1954).[5] G. Gasper and M. Rahman, Basic Hypergeometric Series. CambridgeUniversity Press, Cambridge (1990).[6] A.M. Mathai and R.K. Saxena, Generalized Hypergeometric FunctionsWith Applications in Statistics and Physical Sciences. Springer-Verlag,Berlin (1973).[7] A.M. Mathai and R.K. Saxena, The H-function with Application inStatistics and Other Disciplines. John Wiley and Sons, New York(1978).142 R.K. Yadav, S.D. Purohit, S.L. Kalla[8] R.K. Saxena and R. Kumar, Recurrence relations for the basic analogueof the H-function. J. Nat. Acad. Math. 8 (1990), 48-54.[9] R.K. Saxena, G.C. Modi and S.L. Kalla, A basic analogue of Fox’s H-function. Rev. Tec. Ing. Univ. Zulia, 6 (1983), 139-143.[10] R.K. Saxena, R.K. Yadav, S.D. Purohit and S.L. Kalla, Kober fractionalq-integral operator of the basic analogue of the H-function. Rev. Tec.Ing. Univ. Zulia, 28(2) (2005), 154-158.[11] R.K. Yadav and S.D. Purohit, On applications of Weyl fractional q-integral operator to generalized basic hypergeometric functions. Kyung-pook Math. J., 46 (2006), 235-245.Received: December 12, 20071 Department of Mathematics and StatisticsJ. N. Vyas University, Jodhpur-342005, INDIAe-mail: rkmdyadav@yahoo.co.in2 Department of Basic-Sciences (Mathematics)College of Technology and EngineeringM.P. University of Agriculture and TechnologyUdaipur-313001, INDIAe-mail: sunil a purohit@yahoo.com3 Department of Mathematics and Computer ScienceKuwait UniversityP.O. Box 5969, Safat 13060, KUWAITe-mail: shyamkalla@yahoo.com

On Generalized Weyl Fractional q-Integral Operator Involving Generalized Basic Hypergeometric Functions

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On Generalized Weyl Fractional q-Integral Operator Involving Generalized Basic Hypergeometric Functions

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