ON GENERALIZED WEYL FRACTIONAL q-integral OPERATOR INVOLVING GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS. Abstract

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1 ON GENERALIZED WEYL FRACTIONAL q-integral OPERATOR INVOLVING GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS R.K. Yadav 1, S.D. Purohit, S.L. Kalla 3 Abstract Fractional 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. Mathematics Subject Classification: 33D60, 33D90, 6A33 Key Words and Phrases: generalized Weyl fractional q-integral operator, basic integration, generalized basic hypergeometric functions, basic analogue of Fox s H-function 1. Introduction Al-Salam [3] introduced the generalized Weyl fractional q-integral operator in the following manner: Kq η,µ {f(x)} = q η x η Γ q (µ) x (y x) µ 1 y η µ f(yq 1 µ )d(y), (1.1) where Re(µ) > 0, η is arbitrary and the basic integration, cf. Gasper and Rahman [5], is defined as: f(t)d(t) = x(1 q) q k f(xq k ). (1.) x k=1 In view of relation (1.), operator (1.1) can be expressed as:

2 130 R.K. Yadav, S.D. Purohit, S.L. Kalla K η,µ q {f(x)} = (1 q) µ k=0 q kη (qµ ) k (q) k f(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: 1 ; if n = 0 (a) n = (1.) (1 a)(1 aq) (1 aq n 1 ) ; if n N. In terms of the q-gamma function, (1.) can be expressed as (a) n = Γ q(a + n)(1 q) n, n > 0, (1.5) Γ q (a) where the q-gamma function (cf. Gasper and Rahman [5]), is given by (q) Γ q (a) = (q a ) (1 q) a 1 = (q) a 1, (1.6) (1 q) a 1 where a 0, 1,,. Also, [ ] 1 (y/x)q (x y) ν = x ν n q ν ; 1 (y/x)q ν+n = x ν 1Φ 0 q, yq ν /x. n=0 ; (1.7) The generalized basic hypergeometric series, cf. Gasper and Rahman [5], is given by rφ s a 1,, a r ; q, x b 1,, b s ; = n=0 (a 1,, a r ) n (q, b 1,, b s ) n x n { ( 1) n q n(n 1)/} (1+s r), (1.8) where for convergence, we have q < 1 and x < 1 if r = s + 1, and for any x if r s. The abnormal type of generalized basic hypergeometric series rφ s ( ) is defined as ] [ a1,, a r rφ s b 1,, b s, x λ where λ > 0 and q < 1. The q-exponential series is = n=0 (a 1,, a r ) n (q, b 1,, b s ) n x n q λn(n+1)/, (1.9)

3 ON GENERALIZED WEYL FRACTIONAL q-integral e q (x) = n=0 The q-binomial series is given by α ; 1Φ 0 q, x ; The basic analogue of the Sine and Cosine functions are and x n (q) n. (1.10) = (αx) (x, q). (1.11) sin q (ax) = 1 i {e q(iax) e q ( iax)}, (1.1) cos q (ax) = 1 {e q(iax) + e q ( iax)}. (1.13) Similarly, we have the q-laguerre polynomial: L (α) n (x) = (qα+1 ) n 1Φ 1 (q) n the little q-jacobi polynomial: P n (α,β) (x) = (qα+1 ) n Φ 1 (q) n q n ; q, xq n q α+1 ; q n, q α+β+n+1 ; q, xq q α+1 ; ; (1.1) ; (1.15) the Wall polynomial (or little q-laguerre polynomial) q n, 0 ; W n (x; b, q) = ( 1) n (b) n q n(n+1)/ Φ 1 q, x ; (1.16) b ; and the Stieltjes-Wigert polynomial: s n (x) = n k=0 (q n [ ) k q k(k+1)/ ( x) k q n, x = 1 Φ 0 (q) k 1 ]. (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 the following manner: [ ] H m,n A,B x (a, α) (b, β)

4 13 R.K. Yadav, S.D. Purohit, S.L. Kalla = 1 πi where C B j=m+1 m G(q b j β j s ) n G(q 1 a j+α j s )πx s G(q 1 b j+β j s ) G(q α ) = n=0 A j=n+1 ds, (1.18) G(q a j α j s ) G(q 1 s ) sin πs { (1 q α+n ) } 1 = 1 (q α ), (1.19) and 0 m B; 0 n A; α j and β j are all positive integers. The contour C is a line parallel to Re(ωs) = 0, with indentations, if necessary, in such a manner that all the poles of G(q b j β j s ) ; 1 j m, are to its right, and those of G(q 1 a j α j s ) ; 1 j n, are to the left of C. The basic integral converges if Re [s log(x) log sin πx] < 0, for large values of s on the contour C, that is if { arg(x) ω ω1 1 log x } < π, where q < 1, log q = ω = (ω 1 + iω ), ω, ω 1 and ω are definite quantities. ω 1 and ω being real. For α j = β i = 1, j = 1,, A; i = 1,, B, the definition (1.18) reduces to the q-analogue of the Meijer G-function due to Saxena, Modi and Kalla [9], namely: [ ] G m,n A,B x a 1,, a A b 1,, b B = 1 πi C B j=m+1 m G(q bj s ) n G(q 1 aj+s )πx s G(q 1 b j+s ) A j=n+1 ds, (1.0) G(q aj s ) G(q 1 s ) sin πs where 0 m B; 0 n A and Re [s log(x) log sin πx] < 0. Further, if we set n = 0 and m = B in the equation (1.0), we get the basic analogue of MacRobert s E-function due to Agarwal [1], namely [ ] G B,0 A,B x a 1,, a A E b 1,, b q [B; b j : A; a j : x] B B G(q bj s )πx s ds, (1.1) C A G(q aj s ) G(q 1 s ) sin πs = 1 πi where Re [s log(x) log sin πx] < 0.

5 ON GENERALIZED WEYL FRACTIONAL q-integral Saxena and Kumar [8], introduced the basic analogues of J ν (x), Y ν (x), K ν (x), H ν (x) in terms of H q ( ) function as follows: J ν (x) = {G(q)} H 1,0 0,3 Y ν (x) = {G(q)} H,0 1, x (1 q) ( ν ν, 1), (, 1), (1, 1), (1.) where J ν (x) denotes the q-analogue of Bessel function of first kind J ν (x); ( ν 1, 1) x (1 q) ( ν, ν ν 1, 1), (, 1), (, 1), (1, 1) (1.3) where Y ν (x) denotes the q-analogue of the Bessel function Y ν (x); K ν (x) = (1 q)h,0 0,3 x (1 q) ( ν ν, 1), (, 1), (1, 1) where K ν (x) denotes the basic analogue of the Bessel function of the third kind K ν (x);, (1.) ( ) 1 q 1 α ( 1+α H ν (x)= H 3,1, 1) 1, x (1 q), ( ν ν 1+α, 1), (, 1), (, 1), (1, 1) (1.5) where H ν (x) is the basic analogue of Struve s function H ν (x). Following Saxena and Kumar [8], Mathai and Saxena [6], [7], we have the following q-extensions of certain elementary functions in terms of a basic analogue of the Meijer G-function as: e q ( x) = G(q)H 1,0 0, sin q (x) = π(1 q) 1/ {G(q)} H 1,0 0,3 x(1 q) x (1 q) (0, 1), (1, 1), (1.6) ; ( 1, 1), (0, 1), (1, 1) (1.7)

6 13 R.K. Yadav, S.D. Purohit, S.L. Kalla cos q (x)= π(1 q) 1/ {G(q)} H 1,0 0,3 x (1 q) π sinh q (x)= i (1 q) 1/ {G(q)} H 1,0 0,3 x (1 q) (0, 1), ( 1, 1), (1, 1) ( 1 ; (1.8) ;, 1), (0, 1), (1, 1) cosh q (x)= π(1 q) 1/ {G(q)} H 1,0 0,3 x (1 q) (1.9). (0, 1), ( 1, 1), (1, 1) (1.30) A detailed account of various classical special functions expressible in terms of Meijer s G-function or Fox s H-function can be found in research monographs by Mathai and Saxena [6] and [7]. The main motive of the present paper is to investigate the generalized Weyl fractional q-integral operator involving basic hypergeometric functions including the basic analogue of the H-function. Certain interesting special cases have also been derived as the applications of the main results.. Main results In this section, we shall evaluate the following fractional q-integrals of generalized Weyl type involving basic hypergeometric function r Φ s ( ) and basic analogue of Fox s H-function. The main results are presented in the following theorems. Theorem 1. If Re(η λ) > 0 and ρ is any complex number, then the generalized Weyl fractional q-integral of x λ -weighted basic hypergeometric function r Φ s ( ), is given by

7 ON GENERALIZED WEYL FRACTIONAL q-integral K η,µ q xλ rφ s a 1,, a r ; q, ρx b 1,, b s ; r+1φ s+1 = xλ q µλ Γ q (η λ) Γ q (η λ + µ) a 1,, a r, q 1 η+λ µ ; q, ρx b 1,, b s, q 1 η+λ ;. (.1) P r o o f. In view of relations (1.3) and (1.8), the left hand side of (.1) becomes (1 q) µ q ηk (qµ ) k (xq µ k ) λ (a 1,, a r ) n (q) k (q, b 1,, b s ) n k=0 n=0 { ( 1) n q n(n 1)/} (1+s r) (ρxq µ k ) n, and, interchanging the order of summations, we further obtain x λ (1 q) µ q µλ n=0 (a 1,, a r ) { n ( 1) n q n(n 1)/} (1+s r) (ρxq µ ) n (q, b 1,, b s ) n k=0 (q µ ) k (q) k q k(η λ n). On summing the inner 1 Φ 0 ( ) series with the help of the equation (1.11), the above expression reduces to x λ (1 q) µ q µλ (a 1,, a r ) n (q, b n=0 1,, b s ) n (q η λ n ) µ { ( 1) n q n(n 1)/} (1+s r) (ρxq µ ) n, and by further simplification, the above expression yields to the right hand side of (.1). Theorem. Let Re(µ) > 0, λ I, ρ be any complex number, then the following generalized Weyl fractional q-integral of H q (.) function for λ 0 and λ < 0 holds: { [ Kq η,µ H m,n A,B ρx λ (a, α) (b, β) ]}

8 136 R.K. Yadav, S.D. Purohit, S.L. Kalla [ = (1 q) µ H m+1,n A+1,B+1 ρ(xq µ ) λ [ = (1 q) µ H m,n+1 A+1,B+1 ρ(xq µ ) λ (a, α), (µ + η, λ) (η, λ), (b, β) (1 η, λ), (a, α) (b, β), (1 µ η, λ) ], λ 0 where 0 m B, 0 n A and Re [s log(x) log sin πx] < 0. (.) ], λ < 0, (.3) P r o o f. To prove the theorem, we consider the left hand side of (.) and use definitions (1.3) and (1.18) to obtain (1 q) µ πi C q ηk (qµ ) k (q) k m G(q b j β j s ) k=0 B j=m+1 n G(q 1 a j+α j s )π(ρx λ q µλ kλ ) s G(q 1 b j+β j s ) A j=n+1 ds. G(q a j α j s )G(q 1 s ) sin πs On interchanging the order of summation and integration, valid under the conditions given with equation (1.18), the above expression reduces to m (1 q) µ G(q b j β j n s ) G(q 1 a j+α j s )π(ρx λ q µλ ) s πi C B A G(q 1 b j+β j s ) G(q a j α j s )G(q 1 s ) sin πs j=m+1 j=n+1 k=0 (q µ ) k (q) k q k(η λs) ds. On summing the inner 1 Φ 0 ( ) series, with the help of equation (1.11) and on using definition (1.19), the left hand side of (.) finally reduces to m (1 q) µ G(q b j β j s )G(q η λs ) n G(q 1 a j+α j s )π(ρx λ q µλ ) s ds. πi C B A G(q 1 b j+β j s ) G(q a j α j s )G(q µ+η λs )G(q 1 s ) sin πs j=m+1 j=n+1 Interpreting the above expression in view of definition (1.18), we obtain the right hand side of (.). The second part of the theorem follows similarly.

9 ON GENERALIZED WEYL FRACTIONAL q-integral Applications of the main results In this section, we evaluate some basic integrals of generalized Weyl type, involving basic hypergeometric functions and various elementary basic functions expressible in terms of a basic analogue of Fox s H-function, as applications of the theorems from the previous section. These results are presented in the table that follows. For the sake of brevity, we mention here the proofs of a few results given in the table. For example the results (3.1) to (3.6) have been derived by assigning appropriate values to the parameters r, s and ρ in (.1), Theorem 1, keeping in view of the definitions given by equations (1.7)-(1.17). If we set r = s = 0 and ρ = 1 in (.1), it reduces to (3.). Further, on making use of the result (3.), we can easily prove the results (3.7) and (3.8). The proof of the result (3.9) is similar to the result (.1). The proofs of the results (3.11) and (3.1) follow directly from (.) with ρ = λ = 1 and on using the definitions (1.0)-(1.1), respectively. While, if we assign m = 1, n = A = 0, B = 3, b 1 = ν/, b = ν/, b 3 = 1, λ = and ρ = (1 q) K η,µ q { H 1,0 0,3 in (.1), we obtain [ x (1 q) ]} ( ν, 1), ( ν, 1), (1, 1) = (1 q) µ [ H,0 x (1 q) 1, q µ (µ + η, ) (η, ), ( ν, 1), ( ν, 1), (1, 1) (3.) In view of of definition (1.), the above equation (3.) reduces to the result (3.13). The results (3.1)-(3.1) can be proved similarly by assigning particular values to the parameters m, n, A, B, λ and ρ, keeping in mind definitions (1.3)-(1.30), respectively. The results deduced in the present paper aim to contribute to the theory of basic hypergeometric series and q-fractional calculus. They are expected to find some applications to the solutions of fractional q-differ-integral equations. We intend to take up this aspect in a next contribution. ].

10 138 R.K. Yadav, S.D. Purohit, S.L. Kalla

11 ON GENERALIZED WEYL FRACTIONAL q-integral

12 10 R.K. Yadav, S.D. Purohit, S.L. Kalla

13 ON GENERALIZED WEYL FRACTIONAL q-integral cos q(x) π(1 q) µ 1/ {G(q)} H,0 1, 6 x (1 q) q µ (µ + η, ) (η, ), (0, 1), ( 1, 1), (1, 1) sinh q(x) 3.1 cosh q (x) π i (1 q)µ 1/ {G(q)} H,0 1, 6 x (1 q) q µ π(1 q) µ 1/ {G(q)} H,0 1, 6 x (1 q) q µ (µ + η, ) (η, ), ( 1, 1), (0, 1), (1, 1) (µ + η, ) (η, ), (0, 1), ( 1, 1), (1, 1) References [1] N. Agarwal, A q-analogue of MacRobert s generalized E-function. Ganita 11 (1960), [] R.P. Agarwal, Fractional q-derivatives and q-integrals and certain hypergeometric transformations. Ganita 7 (1976), 5-3. [3] W.A. Al-Salam, Some fractional q-integrals and q-derivatives. Proc. Edin. Math. Soc. 15 (1966), [] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of Integral Transforms. Vol. II. MacGraw-Hill Book Co. Inc., New York (195). [5] G. Gasper and M. Rahman, Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990). [6] A.M. Mathai and R.K. Saxena, Generalized Hypergeometric Functions With Applications in Statistics and Physical Sciences. Springer-Verlag, Berlin (1973). [7] A.M. Mathai and R.K. Saxena, The H-function with Application in Statistics and Other Disciplines. John Wiley and Sons, New York (1978).

14 1 R.K. Yadav, S.D. Purohit, S.L. Kalla [8] R.K. Saxena and R. Kumar, Recurrence relations for the basic analogue of the H-function. J. Nat. Acad. Math. 8 (1990), 8-5. [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), [10] R.K. Saxena, R.K. Yadav, S.D. Purohit and S.L. Kalla, Kober fractional q-integral operator of the basic analogue of the H-function. Rev. Tec. Ing. Univ. Zulia, 8() (005), [11] R.K. Yadav and S.D. Purohit, On applications of Weyl fractional q- integral operator to generalized basic hypergeometric functions. Kyungpook Math. J., 6 (006), Department of Mathematics and Statistics J. N. Vyas University, Jodhpur-3005, INDIA rkmdyadav@yahoo.co.in Department of Basic-Sciences (Mathematics) College of Technology and Engineering M.P. University of Agriculture and Technology Udaipur , INDIA sunil a purohit@yahoo.com 3 Department of Mathematics and Computer Science Kuwait University P.O. Box 5969, Safat 13060, KUWAIT shyamkalla@yahoo.com Received: December 1, 007