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Operational Images and Relations of Two and Three Variable Hypergeometric Series
Abstract
Based upon the classical derivative and integral operators we introduce a new symbolic operational images for hypergeometric functions of two and three variables. By means of these symbolic operational images a number of operational relations among the hypergeometric functions of two and three variables are then found. Other closely-related results are also considered.
... In 1982 Exton [8] published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as 2 2 () m n p a and 2 () m n p a , and introduced a set of 20 distinct triple hypergeometric functions 1 X to 20 X and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions 11 F , 1 0 F , a Humbert function 2 and a Humbert function 2 in their kernels. It is not out of place to mention here that Exton's functions 1 X to 20 X have been studied a lot until today; see, for example, the works [10,[3][4][5][6][11][12][13]. ...
... x D denote the derivative operator and the inverse of the derivative, respectively (see, for example, [14], [1] and [2]). ...
we aim in this work at establishing interesting operational connections between new quadruple hypergeometric series defined in [9] and certain class of triple series involving of Exton’s functions to , Srivastava's functions , Lauricella's functions and the general triple hypergeometric series . Some particular cases and consequences of our main results are also considered.
... In 1982 Exton [8] published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as 1 X to 20 X and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions 11 F , 01 F , a Humbert function 2 and a Humbert function 2 in their kernels. It is not out of place to mention here that Exton's functions 1 X to 20 X have been studied a lot until today; see, for example, the works [10,[3][4][5][6][11][12][13]. ...
... x D denote the derivative operator and the inverse of the derivative respectively ( see ,for example, [14], [1] and [2]). ...
we aim in this work at establishing interesting operational connections between new quadruple hyper-geometric series) 30 , , 1 () 4 ( i X i defined in [9] and certain class of triple series involving of Exton's functions 1
... x D denote the derivative operator and the inverse of the derivative, respectively (see, for example, [14], [1] and [2]). ...
We aim in this work at establishing interesting operational connections between new quadruple hypergeo-metric series) 30 , , 1 () 4 ( = i X i defined in [9] and certain class of triple series involving of Exton's functions 1 X to 20
... The subject of operational calculus has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering [4,11,12]. One of the most interesting developments on the use of operational calculus is the finding of symbolic representations for hypergeometric series and polynomials that play an important role in the investigation of various useful properties of the hypergeometric series and polynomials (for example [1,2,[5][6][7][8][9][10]). Recently, Bin-Saad et al. [3] have introduced five new quadruple hypergeometric functions whose names are ( ) , , , ; , , , ; , , , , , ; , , , ; , , , , , ; , , , ; , , , , , , , ( 0 , , , 3 2 1 3 2 2 2 1 3 1 1 2 1 3 2 1 1 [3]. ...
We aim in this work to establish new operational representations for the hypergeometric functions of four variables X^(4)_6, X^(4)_7, X^(4)_8, X^(4)_9, X^(4)_10. By means of these operational representations, a number of generation functions involving these hypergeometric functions are then found.
... In recent years, in one hand, various extensions of the hypergeometric matrix functions have been presented and investigated (see, e.g., [1,2,12,14,15,17,18,20] and the references cited therein). On the other hand, we may mention (see, e.g., [4,5,6,9,10,13,16]) who have contributed to the study of the operational formulae of some special functions. Indeed, the use of operational formulae, currently exploited in the theory of algebraic decomposition of exponential operators, may significantly simplify the study of hypergeometric matrix functions and the discovery of new relations, hardly achievable by conventional means. ...
... Operational representations and relations involving one and more variables hypergeometric series have been given considerable in the literature, see for example, Chen and Srivastava [5], Goyal, Jain and Gaur ( [7], [8]) Kalla [11], Kalla and Saxena ([12] and [13]), Kant and Koul [14], Chyan and Srivastava [5]. The present sequel to these earlier papers is motivated largely by the aforementioned work of Bin-Saad and Maisoon [2] in which a number of operational relations among the hypergeometric functions of two and three variables are found. The aim of this paper is to introduce some N-fractional operators involving certain hypergeometric functions. ...
The subject of fractional calculus has gained importance and popularity during the past three decades. Based upon
the N-fractional calculus we introduce a new N-fractional operators involving hyper-geometric function. By means of these N-
fractional operators a number of operational relations among the hyper-geometric functions of two, three, four and several
variables are then found. Other closely-related results are also considered.
Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented.
Based upon the classical derivative and integral operators we introduce a new operator which allows the derivation of new symbolic operational images for hypergeometric functions. By means of these symbolic operational images a number of decomposition formulas involving quadruple series are then found. Other closely-related results are also considered.
Recently [ Math. Zeitschr. 108(1969), 231-234] the authors have defined operators of fractional integration involving Gauss hypergeometric function. In the present paper we establish certain inversion formulae, formal properties and table of operators.
In this paper we introduce two operators of fractional integration which are the extension of Saxena's operators [4, p. 286] defined recently in this journal. The operators introduced here also include, as particular cases, the operators given earlier by Kober [3] and Erd61yi [-13. Three theorems on these operators are established. The first two theorems give expressions for the Mellin transform of these operators and the third theorem deals with the fractional integration by parts. It is expected that the solution of certain dual integral equations involving special functions of Mathematical Physics can be obtained easily by the application of these operators. Incidentally we also obtain the sum of a generalized hypergeometric series 3F2 with unit argument, which generalizes Saalschiitz's theorem [5, p. 49-1. 2. Definitions We define the fractional integration operators by means of the following integral equations: J [f(x)] =J [e, fl, 7; m, u, r/, a; f(x)]
Here we obtain certain generalised fractional integrals of products of the H-function of several variables and certain general classes of polynomials of several variables.
This paper is a continuation of part I [ibid. 22, No. 5, 403-411 (1991; Zbl 0747.44002)]. In the present paper, it is shown how these operators can be identified with elements of the algebra of functions having the Mellin convolution as the product. Inversion formulas and the relations of our operators with the generalized Hankel transforms are also established.
Many earlier works on the subject of fractional calculus contain interesting accounts of the theory and applications of fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, etc.). The main object of this paper is to examine rather systematically (and extensively) some of the most recent contributions on the applications of fractional calculus operators involving power functions and in finding the sums of several interesting families of infinite series. Various other classes of infinite sums found in the mathematical literature by these (or other) means, and their validity or hitherto unnoticed connections with some known results, are also considered.