Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus
Maged Gumaan Bin-Saad
Department of Mathematics, Aden University, Aden, Kohrmaksar, Yemen
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To cite this article:
Maged Gumaan Bin-Saad. Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus. Mathematics Letters. Vol. 2, No. 6, 2016, pp. 47-57. doi: 10.11648/j.ml.20160206.12
Received: August 27, 2016; Accepted: November 19, 2016; Published: December 20, 2016
Abstract: 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.
Keywords: N-fractional Calculus Operators, Horn’s Functions, Appell Functions, Saran Functions, Quadruple Functions, Hyper-Geometric of Several Variables
1. Introduction
The subject of fractional calculus is one of the most intensively developing areas of mathematical analysis, mainly due to its fields of application range from biology through physics and electrochemistry to economics, probability theory, special functions and statistics (see [16] and [17]). Indeed, on behalf of the nature of their definitions the fractional derivatives and integrals provide an excellent instrument for the modeling of memory and hereditary properties of various materials and processes. Half-order derivatives and integrals prove to be more useful for the formulation of certain electrochemical problems than the classical methods [19]. Appell (see [1]; also [21, p. 22-23]) defined the four hyper-geometric functions of two variables which he denoted by . Other hyper-geometric functions of two variables has been defined by Horn [6]. Seven of them he denoted by (see e.g. [21, p. 24] and [22, p. 56-57]).
Appell’s and Horn’s functions are all generalization of Gaussian hypergeometric function
(1)
where
In 1893, Lauricella [15] further generalized the four Appell’s functions to functions of n-variables respectively. After a gap of long time, Saran [18] initiated a systematic study of ten series of three variables with the notations In 1982, Exton [8] has studied 20 triple hyper-geometric series. He denoted four of them by Srivastava and Karlsson [21] provide an impressive tabulation and a wealth of information on the construction of the set of all 205 distinct triple Gaussian hypergeometric series. In their work symbols and references are given for those series that have been introduced previously and they denoted the new series by the symbol (cf. [21, p. 273]):
For instance, the series has the definition
,
and it would be presented in the form
Sharma and Parihar [20] introduced 83 complete hyper-geometric series of four variables . It is remarkable that out of these 83 series, the following 19 series had already been appeared in the literature due to Exton (see [9]) in the different notations:
For the purpose of this work, we recall here some definitions.
Definition 1. (By K. Nishimoto (see [16] and [17])
Let
be a curve along the cut joining two points z and ,
be a curves along the cut joining two points z and ,
be a domain surrounded by , be a domain surrounded by
Moreover, let f = f(z) be analytic (regular) function in D and it has no branch point inside C and on C, and
(2)
where
then is the fractional differ-integration of arbitrary order v (derivatives of order v for v>0, and integrals of order –v for v<0), with respect to z, of the function f, if ||<.
Furthermore, let be Nishimoto’s operator defined by [Refer to (2)]
(3)
with
Lemma 1. We have
(4)
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. Based upon these operators, we aim here to derive operational relations among the above said hypergeometric functions of two, three, four and multiple variables. The structure of this paper is the following: In section 2, we establish N-fractional operators. Section 3 deals with same applications of these operators and the presentation of operational relations between functions of two, three and four variables. Section 4 aims at introducing multivariable generalization of the N-fractional operators in section 2 and establishing some operational relations among hypergeoemtric functions of several variables.
2. N-fractional Calculus Operators
By using definitions (2) and (3), we introduce three kinds of N-fractional operators in the following definitions:
Definition 2. Let be a N-fractional operator defined by
=
=
(5)
where and .
Definition 3. Let be a N-fractional operator defined by
(6)
where {} and .
Definition 4. Let be a N-fractional operator defined by
(7)
where {} and .
In view of lemma 1 and the binomial theorem
the operators , and, can be written in the forms:
(8)
(9)
and
=
(10)
respectively, where throughout this work
(11)
and
(12)
On replacing α, β and γ by α – γ+1, 2- γ and β- γ+1 respectively, relation (8) reduces to a known result due to Nishimoto [9, p. 78 (3)]. Further, if , relation (10) yields
(13)
Whereas, if relation (10) yields
(14)
where is Appell’s function of two variables [12, p. 23 (3)].
3. Applications
By choosing a suitable hyper-geometric function the operators (5), (6) and (7) cab be applied to deduce relationships involving a fairly variety of hyper-geometric functions of one and more variables.
i. Relations among two and three variables functions
By making use of the N-fractional operator (5), we establish the following relationships between hyper-geometric series of two and three variables.
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
PROOF. To establish relation (15), we first write Appell’s function in it’s series form (see e.g. [12, p. 23 (3)]):
Interchange the order of summation and differ-integration, which is permissible due to the absolute convergence, we get
Now, making use of (5), the identity and the definition of Saran’s function [18, p. 42 (5)], with a little simplification, the desired result is obtained. The proof of relations (16) to (43) would run as above.
ii. Relations among two and four variables functions
Now, we establish N-fractional relationships between hyper-geometric functions of two and four variables. These relations can be proved on the same lines as adopted in the proof of the relations in the previous section, considering lemma 1 and the definitions of the hyper-geometric functions during the proof. Thus, using (6) and (7), we obtain the following formulas:
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
iii. Relations among three and four variables functions
In this section, we derive a number of relationships between hyper-geometric functions of three and four variables, which can be established on the same lines as in the proof of (15)
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
(73)
(74)
(75)
(76)
(77)
(78)
(79)
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(87)
4. Multivariable N-fractional Operator
Motivated by the results of the previous sections, we aim here at presenting a multivariable generalization of the N-fractional operator (5) as follows.
Definition 5. Let be a multivariable N-fractional operator defined by
=
(88)
where and , .
Clearly, for n =1, (88) reduces to (5). Moreover, in view of lemma 1 and the multinomial theorem [12, p. 329 (220)]:
(89)
the operators , can be written in the form
(90)
where is Lauricella function of n-variables (cf. [12, p. 33 (1)]) and throughout this work
(91)
Next, in [3] we have established hyper-geometric function in several variables and denoted it by , which provide multivariable generalization of a number of known hyper-geometric functions. The function is defined as below
(92)
By the general theory of convergence of multiple hypergeometric functions (see e.g [8, section 2.9]; also [21, Chapter 9]), it follows that the region of convergence for is Now, we consider only four interesting multivariable applications of the operator (88). Considering
On expressing in series form [22] and employing (89) and the Legendre duplication formula [22, p. 23 (24)]:
,
we find
,
which on using lemma 1 gives us the desired result:
(93)
For the other three Lauricella functions ,and [21, p. 33] the operator (88) [ in conjunction with (89) and lemma 1], similarly yields the following results:
(94)
(95)
(96)
where and are Exton’s generalized Horn’s functions [8, p. 97 (3.5)],
(97)
Finally, let us stress that the schema suggested in Sections 3 and 4 can be applied to find N-fractional relations for other generalized hypergeometric functions. In a forthcoming papers we will consider the problems of establishing N-fractional relations for other generalized hypergeometric functions by following the technique discussed in this paper.
References