A study of composition formulas for the unified fractional integral operators *.Abstract In the present paper we derive three new and interesting composition formulas for a general class of fractional integral operators involving the product of a general class of polynomials, a general sequence of functions and a multivariable H-function. The operators of our study are quite general in nature and may be considered as extensions of a number of simpler fractional integral operators studied from time to time by several authors. An interesting case of the first composition formula has been given for the sake of illustration. The importance of our study lies in the fact that it unifies and extends a number of corresponding results lying scattered Scattered Used for listed equity securities. Unconcentrated buy or sell interest. in the literature. In addition, we can also evaluate several double finite integrals with the help of our composition formulas. The results obtained by Buschman [2], Erdelyi [3], Gupta and Soni [7], Goyal and Jain [5], Goyal, Jain and Gaur Gaur, ruined city, India Gaur (gour), ruined city, West Bengal state, India. Known also as Lakhnauti, the city was an ancient Hindu capital of Bengal. It was captured (c. [6] follow as simple special cases of our composition formulas. Keywords and Phrases: Unified fractional integral operators, Composition formulas, General sequence of functions, General class of polynomials, Multivariable H-function. 1. Introduction A general sequence of functions A general sequence of functions introduced by Agarwal and Chaubey [1, p. 1155] will be defined by the following Rodrigues type formula (see also Srivastava and Manocha [14, p. 447, Problem 16]) [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .] (1.1) where the differential operator differential operator In mathematics, any combination of derivatives applied to a function. It takes the form of a polynomial of derivatives, such as D2xx − D2xy ∙ D [T.sup.n.sub.[lambda],k] is defined as [T.sup.n.sub.[lambda],k] = [[[x.sup.k] ([lambda] + x [D.sub.x])].sup.n], [D.sub.x] = d/dx (1.2) [{[K.sub.n]}.sup.[infinity].sub.n=0] is a sequence of constants and [omega] (x) is independent of n and differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. an arbitrary number of times. If we set [omega] (x) = exp exp abbr. 1. exponent 2. exponential (- [sx.sup.r]) in (1.1), then the general sequence of functions can be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.3) Salim obtained the following series formula [11, p.169, eqn.(8) ] for the general sequence of functions: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.4) and the infinite series infinite series In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. on the right hand side of (1.4) is absolutely convergent absolutely convergent adj. Of, relating to, or characterized by absolute convergence. absolutely convergent Relating to or characterized by absolute convergence. . It may be noted that series formula obtained by Salim contained some omission in the power of x. On setting s = 0 and expanding [([cx.sup.q] + d).sup.n[delta]-v] in a series the equation (1.4) takes the following form: [R.sup.([alpha],[beta]).sub.n][x;a,b,c,d;p,q;[gamma],[delta];1] = [summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (v,u,l,e,h])[theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ](v,u,l,e,h) [x.sup.kn+q(v+h)+pl] (1.5) where [summation over (v,u,l,e,h)] = [n.summation over (v=0)][v.summation over (u=0)] [n.summation over (l=0)][[infinity].summation over (h=0)] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.6) A general class of polynomials Srivastava [12, p.1,eqn.(1)] introduced the general class of polynomials defined by [S.sup.U.sub.V] [x] = [[V/U].summation over (R=0)] [(-V).sub.UR][A.sub.V,R]/R! [X.sup.R], V = 0,1,2, ... (1.7) where U is an arbitrary positive integer integer: see number; number theory and the coefficients AV,R (V, R [greater than or equal to] 0) are arbitrary constants (Math.) a quantity of function that is introduced into the solution of a problem, and to which any value or form may at will be given, so that the solution may be made to meet special requirements. , real or complex. By suitably specializing the coefficients [A.sub.V,R], [S.sup.U.sub.V] V can be reduced to a number of known polynomials as its special cases. These include among others, the Leguerre polynomials, the Jacobi polynomials In mathematics, Jacobi polynomials are a class of orthogonal polynomials. They are obtained from hypergeometric series in cases where the series is in fact finite: The multivariable H-function The multivariable H-function occurring in this paper was introduced by Srivastava and Panda [15, p.130, eqn.(1.1)]. This function will be defined and represented in the following contracted form [13, pp.251-252, eqns.(C.1 - C.3)]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.8) where [omega] [square root of -1]. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.10) Throughout this paper it is assumed that this function satisfies its appropriate conditions of existence and convergence [13, pp.252-253, eqns.(C.4 - C.6)]. Throughout this paper A will denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the class of functions f(t) for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.11) We shall represent such a class of functions as f(t) [member of] A Throughout the paper the figure occurring below the curly brackets curly bracket - brace at any place will indicate the total number of similar quantities/pairs covered by it. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would mean r zeros and so on. In the present paper we study the unified fractional integral operators. The operators of our study involve the product of general class of polynomials, a general sequence of functions and a multivariable H-function having general arguments. The operators will be defined and represented as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.12) provided that (i) min Re ([micro], v, [sigma]) [greater than or equal to] 0 (ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (iii) Re ([eta]+[zeta]) + 1 > 0 (iv) The quantities [v.sub.1], *, * [v.sub.r] are all positive (some of them may decrease to zero provided that the resulting operator has a meaning). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.13) provided that (i) min Re ([micro], v, [sigma]) [greater than or equal to] 0 (ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (iii) Re ([w.sub.2]) > 0 or Re ([w.sub.2]) = 0 and Re ([eta] - [w.sub.1]) > 0 (iv) The quantities [v.sub.1], *, * [v.sub.r] are all positive (some of them may decrease to zero provided that the resulting operator has a meaning). 2. Composition Formulas for the Fractional Integral Operators Result (1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1) where f (t) [member of] A, min Re[([eta]' + 1), ([eta] + [zeta] + 1)] > 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and the appropriate conditions for the existence of fractional integral operators involved are satisfied. Also [A.sup.*] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [B.sup.*] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [theta] (v, u, l, e, h) stands for the expression mentioned in the equation (1.6) and [theta]' (v', u', l', e', h') stands for the expression obtained from it by replacing all the parameters involved therein by the same parameters but having dashes in them. To prove the above result, we first express both the I-operators involved in its left hand side, in the integral form with the help of equation (1.12), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2) Now, we interchange the order of t- and [x.sub.1]- integrals (which is permissible under the conditions stated), we easily have after a little simplification. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4) To evaluate [DELTA] we first replace both the general class of polynomials, general sequence of functions occurring in it in terms of their respective series with the help of equations (1.7) and (1.5) respectively and the multivariable H-functions in terms of their well-known Mellin-Barnes contour contour or contour line, line on a topographic map connecting points of equal elevation above or below mean sea level. It is thus a kind of isopleth, or line of equal quantity. integrals and interchange the order of summation and integration in the result thus obtained (which is permissible under the conditions stated), we arrive at the following expression after a little simplification: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5) To evaluate the [x.sub.1]-integral in right hand side of the above equation (2.5), we substitute [x.sup.s.sub.2] - [x.sup.s.sub.1]/[x.sup.s.sub.2] - [t.sup.5]. Now on expressing the [sub.2][F.sub.1] thus obtained in terms of their contour integral, reinterpreting the result thus obtained in terms of the H-function of (2r+1) variables and substituting the value of [DELTA] thus obtained in equation (2.3), we easily arrive at the desired result (2.1) after a little simplification. Result 2 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6) where f (t)[member of] A, the composite operator defined by left hand side of (2.6) exists and the following conditions are satisfied: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [C.sup.*] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [D.sup.*] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [theta] (v, u, l, e, h) stands for the expression mentioned in the equation (1.6) and [theta]' (v', u', l', e', h') stands for the expression obtained from it by replacing all the parameters involved therein by the same parameters but having dashes in them. Result 3 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7) where f(t) [member of] A, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and the appropriate conditions for the existence of fractional integral operators involved are satisfied. here [A.sup.**] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [B.sup.**] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [C.sup.**] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [D.sup.**] stands for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [theta] (v, u, l, e, h) stands for the expression mentioned in the equation (1.6) and [theta]' (v', u', [theta]', e' ,h') stands for the expression obtained from it by replacing all the parameters involved therein by the same parameters but having dashes in them. The proof of results (2.6) and (2.7) can be developed proceeding on the lines similar to the result (2.1). On making the obvious parametric interchanges and taking the help of Euler's transformation formula [4, p.64, eqn.(23)], we can easily verify the commutativity com·mu·ta·tive adj. 1. Relating to, involving, or characterized by substitution, interchange, or exchange. 2. Independent of order. of the fractional integral operators involved in the composition formulas (2.1), (2.6) and (2.7). 3. Special Cases Since our composition formulas involve the general sequence of functions, general class of polynomials and the multivariable H-functions, we can obtain from our main results a large number of other composition formulas involving simpler functions and polynomials which are special cases of the functions occurring as kernels in our main results. Now, we cite some known results that follow as special cases of our composition formulas given by equations (2.1), (2.6) and (2.7). Thus, if in these composition formulas, we take a = a' = c = c' = 1, b = b' = d = d' = n = n' = 0, then both the general sequence of functions reduce to unity, further we take s = 1, [sigma] = [sigma]' = 0, then we get the known results given by Gupta and Soni [7, pp.342, eqn.(9); pp.343-344, eqn.(10); pp.345-346, eqn.(15)] after a little simplification. Again if we take in the above results V = V' = 0 (the polynomials [S.sup.U.sub.V] and [S.sup.U'.sub.V'] will reduce to [A.sub.0,0] and [A.sup.'.sub.0,0] respectively and can be taken to be unity without loss of generality Without loss of generality (abbreviated to WLOG or WOLOG and less commonly stated as without any loss of generality) is a frequently used expression in mathematics. ) and reduce both the multivariable H-functions to the exponential functions exponential function In mathematics, a function in which a constant base is raised to a variable power. Exponential functions are used to model changes in population size, in the spread of diseases, and in the growth of investments. and tend them to zero, we arrive at the results given by Erdelyi [3, p.166, eqns. (6.1), (6.2); p.167, eqn.(6.3)]. If in our composition formulas (2.1), (2.6) and (2.7), we take a = a' = c = c' = 1, s = 1, d = d' = b = b'= n = n' = 0 and further reduce both the multivariable H-functions to the generalized hypergeometric functions In mathematics, a hypergeometric function can be:
Now, we evaluate a double finite integral with the help of our main result (2.1). Thus, if we put f (t) = 1, reduce [R.sup.([alpha],[beta]).sub.n] to [S.sup.([alpha],[beta],[tau]).sub.n] [9, p. 71] (see also Saigo, Goyal and Saxena [10, pp. 43-45]), [R.sup.([alpha]',[beta]').sub.n'] to Jacobi polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a [P.sup.([beta]',[alpha]').sub.n'] [8, p. 257], the multivariable H-functions involved in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to unity and involved in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [sub.P][[PSI].sub.Q] [13, p. 19, eqn. (2.6.11) ] and further take s = 1, we arrive at the following finite double integral after a little simplification: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1) where the appropriate conditions easily obtainable from the result (2.1) hold good. Acknowledgement The authors are thankful to Prof. K. C. Gupta for his valuable suggestions, to Prof. H. M. Srivastava for his constant encouragement and to the Director, M. N. I. T., Jaipur, for the facilities provided during the preparation of this paper. Received January 15, 2004, Accepted April 1, 2004. References [1] B. D. Agrawal and J. P. Chaubey, Certain derivation derivation, in grammar: see inflection. of generating relations for generalized polynomials, Indian J. Pure Appl. Math. 11 (1980), 1155-1157; ibid. 11 (1981), 357-359. [2] R. G. Buschman, Fractional integration, Math. Japon. 9 (1964), 99-106. [3] A. Erdelyi, Fractional integrals of generalized functions Not to be confused with generic function. In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. , Fractional Calculus Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of , 1975, 151-170. [4] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, 1953. [5] S. P. Goyal and R. M. Jain, Fractional integral operators and the generalized hypergeometric functions, Indian J. Pure Appl. Math. 18 (1987), 251-259. [6] S. P. Goyal, R. M. Jain and N. Gaur, Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials, Indian J. Pure Appl. Math. 22 (1991), 403-411. [7] K. C. Gupta and R. C. Soni, On composition of some general fractional integral operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 339-349. [8] E. D. Rainville, Special Functions In mathematics, special functions are particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. , Chelsea Publishing Company, Bronx, New York, 1971. [9] S. K. Raizada, A Study of Unified Representation of Special Functions of Mathematical Physics mathematical physics Branch of mathematical analysis that emphasizes tools and techniques of particular use to physicists and engineers. It focuses on vector spaces, matrix algebra, differential equations (especially for boundary value problems), integral equations, integral and Their Use in Statistical and Boundary Value Problems In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. , Ph.D. thesis, Bundelkhand University University Bundelkhand University is located in the town of Jhansi, in Uttar Pradesh in India. The university came into existance on August 26, 1975 and therefore, essentially belongs to the younger generation of the Indian Universities. , Jhansi, India, 1991. [10] M. Saigo, S. P. Goyal and S. Saxena, A theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. relating a generalized Weyl fractional integral, Laplace and Varma transforms with applications, J. Fract. Calc. 13 (1988), 43-56. [11] T. O. Salim, A series formula of a generalized class of polynomials associated with Laplace transform Laplace transform In mathematics, an integral transform useful in solving differential equations. The Laplace transform of a function is found by integrating the product of that function and the exponential function e−pt and fractional integral operators, J. Rajasthan Acad. Phys. Sci. 1 (2002), 167-176. [12] H. M. Srivastava, A contour integral involving Fox's H-function, Indian J. Math. 14 (1972), 1-6. [13] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi New Delhi (dĕl`ē), city (1991 pop. 294,149), capital of India and of Delhi state, N central India, on the right bank of the Yamuna River. and Madras Madras. 1 State and former province, India: see Tamil Nadu. 2 City, India: see Chennai. , 1982. [14] H. M. Srivastava and H. L. Manocha, A Treatise A scholarly legal publication containing all the law relating to a particular area, such as Criminal Law or Land-Use Control. Lawyers commonly use treatises in order to review the law and update their knowledge of pertinent case decisions and statutes. on Generating Functions, Halsted Press (John Wiley John Wiley may refer to:
[15] H. M. Srivastava and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine Angew. Math. 283/284 (1976), 265-274. [16] H. M. Srivastava and N. P. Singh, The integration of certain products of the multivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo (Ser. 2) 32 (1983), 157-187. Rashmi Jain ([dagger]) and Arti Sharma ([double dagger double dagger n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ]) Department of Mathematics, Malaviya National Institute of Technology  This article or section is written like an .Please help [ rewrite this article] from a neutral point of view. Mark blatant advertising for , using . , Jaipur 302017, Rajasthan, India * 2000 Mathematics Subject Classificatio n. 26A33, 33C60, 33C70. ([dagger]) E-mail.rashmi_mrec@rediffmail.com ([double dagger]) E-mail.arti_mnit@yahoo.co.in |
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