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Transformaciones de la funcion hipergeomotrica basica generalizada de dos variables.

On transformations involving generalized basic hypergeometric function of two variables *

1. Introduction

Recently in a couple of papers Yadav and Purohit [1,2] have used the q-Leibniz rule for the fractional q-derivatives of the product of various basic hypergeometric function full stop. This has resulted in deduction of several transformations and expansion formulae involving the basic hypergeometric functions. Earlier Denis [3] and Shukla [4] have used the q-Leibniz rule to derive certain transformations for basic hypergeometric functions.

Recently Yadav et al. [5] have investigated the fractional q-calculus operators involving the basic analogue of Fox's H-function and basic analogue of Meijer's G-function of two variables.

Motivated by the aforementioned work, we investigate the applications of the q-Leibniz rule to a product involving the basic analogue of Fox's H-function of two variables. This shall further be used to derive transformations involving the above mentioned functions.

The fractional q-differential operator of arbitrary order p, cf. Al-Salam [6], is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where Re([mu]) < 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where x and y are complex numbers with x [not equal to] 0.

The basic integration cf. Gasper and Rahman [7] is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

provided that the series converges.

By virtue of the equation (3), the equation (1) can be expressed as:

[D.sup.[mu].sub.x,q] f(x) [x.sup.-[mu]](1 - q)/[[GAMMA].sub.q](-[mu]) [[infinity].summation over (k=0)] [[q.sup.k][1 - [q.sup.k+l]].sub.-[mu]-1] f(x[q.sup.k]), (4)

where Re([mu]) < 0 and the q-gamma function cf. Gasper and Rahman [7], in various forms is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

with [alpha] [not equal to] 0, -1, -2, ... and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Further, for real or complex [alpha] and 0 < [absolute value of q] < 1, the q-shifted factorial is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

or

[(a; q).sub.n] = [([alpha];q).sub.[infinity]]/[([alpha][q.sup.n];q).sub.[infinity]]. (8)

In view of Agarwal [8], we have the q-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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where U(x) and V(x) are two regular functions.

Following Saxena, Modi and Kalla [9], the basic analogue of the H-function of two variables is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where 0 < [absolute value of q] < 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The coefficients [[gamma].sub.j] and [[gamma]'.sub.j], (1 [less than or equal to] j [less than or equal to] C); [[delta].sub.j] and [[delta]'.sub.j], (1 [less than or equal to] j [less than or equal to] D); [[alpha].sub.j] (1 [less than or equal to] j [less than or equal to] [P.sub.1]), [[alpha]'.sub.j] (1 [less than or equal to] j [less than or equal to] [P.sub.2]), [[beta].sub.j] (1 [less than or equal to] j [less than or equal to] [Q.sub.1]), [[beta]'.sub.j] (1 [less than or equal to] j [less than or equal to] [Q.sub.2]) are positive numbers, A, C, D, [P.sub.1], [P.sub.2], [Q.sub.1], [Q.sub.2], [M.sub.1], [M.sub.2], [N.sub.1] and [N.sub.2] are non negative integers, satisfying the inequalities 0 [less than or equal to] A [less than or equal to] C, 0 [less than or equal to] [M.sub.i] [less than or equal to] [Q.sub.i], 0 [less than or equal to] [N.sub.i] [less than or equal to] [P.sub.i], D > 0; [for all]i [lamber of] {1,2}.

The contours [C.sup.*.sub.1] and [C.sup.*.sub.2] are lines parallel to Re([w.sub.i]s) = 0 (i = 1,2), with indentations, if necessary, in such a manner that all the poles of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for j [member of] {1 ,..., [M.sub.1]} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for j [member of] {1 ,..., [M.sub.2]}, lie to the right and those of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for j[member of] {1 ,..., [N.sub.1]} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for j [member of] {1 ,..., [N.sub.2]} lie to the left of the contours. An empty product is interpreted as unity. The poles of the integrand are assumed to be simple. The integral converges if Re[s log([z.sub.1]) - log sin [pi] s] < 0 and Re[t log([z.sub.2]) - log sin [pi]t] < 0 for large values of [absolute value of s] and [absolute value of t] on the contours i.e. if [absolute value of {arg([z.sub.i]) - [w.sub.2][w.sup.-1.sub.1] log[absolute value of [z.sub.i]]}] < [pi] for i = 1,2. Where 0 < [absolute value of q] < 1 such that log q = -w = -([w.sub.1] + i[w.sub.2]), [w.sub.1] and [w.sub.2] being real numbers.

If we set [alpha] = [alpha]' = [beta] = [beta]' = [gamma] = [gamma]' = [delta] = [delta]' = 1 then the definition (10) reduces to a basic analogue of Meijer's G-function of two variables as under:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Further, we observe that for A = C = D = 0 the basic analogue of Fox's H-function of two variables given by (10) reduces to a product of two basic Fox's H-functions of one variables cf. Saxena, Modi and Kalla [9], as under:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where the basic analogue of Fox's H-function of one variable due to Saxena, Modi and Kalla [10], is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where 0 [less than or equal to] [M.sub.1], [less than or equal to] [Q.sub.1], 0 [less than or equal to] [N.sub.1] [less than or equal to] [P.sub.1], [[alpha].sub.j]'s and [[beta].sub.j]'s all positive integers. The contour C is a line parallel to Re([omega]s) = 0 with indentations if necessary, in such a manner that all the poles of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 1 [less than or equal to] j [less than or equal to] [M.sub.1] are to the right, and those of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], to the left of C. The integral converges if Re[s log(x) - log sin [pi]s] < 0 for large values of [absolute value of s] on the contour C i.e. if [absolute value of {arg(x) - [w.sub.2][w.sup.-1.sub.1] log [absolute value of x]}] < [pi], where 0 < [absolute value of q] < 1, log q = -w = -([w.sub.1] + i[w.sub.2]) , w, [w.sub.1], [w.sub.2] are definite quantities, [w.sub.1] and [w.sub.2] being real.

Further, if we set [[alpha].sub.i] = [[beta].sub.j] = 1, [for all]i and j in the equation (20), we obtain the basic analogue of Meijer's G-function, due to Saxena, Modi and Kalla [10], namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where 0 [less than or equal to] [M.sub.1] [less than or equal to] [Q.sub.1], 0 [less than or equal to] [N.sub.1] [less than or equal to] [P.sub.1] and Re[s log(x) - log sin [pi]s] < 0.

2. Transformations involving a basic analogue of Fox's H-function of two variables

In this section, we shall establish certain theorems involving some transformations associated with the basic analogue of the Fox's H-function and Meijer's G-function of two variables.

If Re([mu]) < 0, Re[s log([z.sub.1]) - log sin [pi]s] < 0, Re[t log([z.sub.2]) - log sin [pi]t] < 0,[rho] and [sigma] being any positive integers, then for [lambda] [not equal to] 0,-1,-2 ,..., the following theorems holds:

Theorem 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

Theorem 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Theorem 3

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Proof of (22): On taking, in the q-Leibniz rule (9) U (x) = [x.sup.[lambda]-1] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Following the recent communication of authors [5] we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where Re[s log([z.sub.1]) - log sin [pi]s] < 0 and Re[t log([z.sub.2]) - log sin [pi]t] < 0.

If we set [lambda] - 1 in the above result (25), we obtain the following transformation involving Fox's H-function of two variables:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

where Re[s log([z.sub.1]) - log sin [pi]s] < 0 and Re[t log([z.sub.2]) - log sin [pi]t] < 0.

On using the relation (26) and a fundamental result of fractional q-calculus, namely

[D.sup.[mu].sub.x,q] ([x.sup.[lambda]-1]) = [[GAMMA].sup.q]([lambda])/[[GAMMA].sub.q]([lambda] - [mu]) [x.sup.[lambda]-[mu]- 1], ([lambda] [not equal to] 0, - 1 , - 2, ...), (27)

we arrive at the Theorem 1 after some simplifications.

Proof of (23): On setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the q-Leibniz rule (9), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

In view of the known results due to Yadav, Purohit and Vyas [5], namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

by assigning A - C - D - 0, in the above equations (29J, (30) and using the result (19) we obtain the following fractional q-derivative formulae involving Fox's H-function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

On substituting (29), (31) and (32) in (28) we arrive at the Theorem 2 after some simplifications.

Proof of (24): The proof of Theorem 3 is similar to that of the Theorem 1: for sake of brevity we omit the proof.

3. Applications

The q-extension of the H-function of two variables defined by (10) in terms of the Mellin-Barnes type of basic contour integrals, possess the advantage that a number of q-special functions (including Fox's H-function of one variable) happen to be the particular cases of this function. The transformations deduced in the previous section can find many applications giving rise to the transformations for various q-special functions, which are special cases of the Fox's H-function.

For example, if we set [alpha] = [alpha]' = [beta] = [beta] = [gamma] = [gamma]' = [delta] = [delta]' = [rho] = [sigma] = 1 in the Theorem 1 and Theorem 2, we respectively obtain the following results involving Meijer's G-function of two variables:

(i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

On putting A = C = D = 0, [rho] = 1 and [z.sub.1] = 1 in the Theorem 3, we obtain a known result due to Yadav and Purohit [2, p.323, eq. (2.1)].

We conclude with an observation that the method used here can be employed to yield a variety of interesting results involving the expansions and transformations for the generalized basic hypergeometric functions of two variables.

Acknowledgement

The authors are thankful to the referees for their valuable comments, which have helped in improvement of the paper

References

[1.] Yadav, R.K. and Purohit, S.D.: Fractional q-derivatives and certain basic hypergeometric transformations. South-East Asian J. Math. & Math. Sc., 2(2) (2004), 37-46.

[2.] Yadav, R.K. and Purohit, S.D.: On fractional q-derivatives and transformations of the generalized basic hypergeometric function. J. Indian Acad. Math., 2 (2006), 321-326.

[3.] Denis, R.Y.: On certain special transformations of poly-basic hypergeometric functions. The Math. Student, 51(1-4) (1983), 121-125.

[4.] Shukla, H. L.: Certain results involving basic hypergeometric functions and fractional q-derivative. The Math. Student, 61(1-4) (1992), 107-112.

[5.] Yadav, R.K., Purohit, S.D. and Vyas, V.K.: On fractional q-calculus operators involving the basic hypergeometric function of two variables. Raj. Acad. Phy. Sci. 9(2) (2010), (In press).

[6.] Al-Salam, W.A.: Some fractional q-integral and q-derivatives. Proc. Edin. Math. Soc., 15 (1966), 135-140.

[7.] Gasper, G. and Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge, 1990.

[8.] Agarwal, R.P.: Fractional q-derivatives and q-integral and certain hypergeometric transformations. Ganita, 27 (1976), 25-32.

[9.] Saxena, R.K., Modi, G.C. and Kalla, S.L.: A basic analogue of H-function of two variables. Rev. Tec. Ing. Univ. Zulia, 10(2) (1987), 35-39.

[10.] Saxena, R.K., Modi, G.C. and Kalla, S.L.: A basic analogue of Fox's H-function. Rev. Tec. Ing. Univ. Zulia, 6 (1983), 139-143.

Recibido el 27 de Julio de 2009

En forma revisada el 12 de Abril de 2010

R.K. Yadav (1), S.D. Purohit (2), V.K. Vyas (1)

(1) Department of Mathematics and Statistics, J. N. Vyas University. Jodhpur- 342005, India.. rkmdyadav@gmail.com, vyas.vyay01@redffmail.com.

(2) Department of Basic Science (Mathematics), College of Technology and Engineering, M.P University of Agriculture and Technology. Udaipur, India.. sunil_a_purohit@yahoo.com

* 2010: Mathematics Subject Classification: Primary 33D60, secondary 26A33.
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Title Annotation:texto en ingles
Author:Yadav, R.K.; Purohit, S.D.; Vyas, V.K.
Publication:Revista Tecnica
Date:Aug 1, 2010
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