Mathematics Letters
Volume 2, Issue 6, December 2016, Pages: 47-57

Relations Among Certain Generalized Hyper-Geometric Functions Suggested by N-fractional Calculus

Department of Mathematics, Aden University, Aden, Kohrmaksar, Yemen

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

Contents

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

1. Appell, P., Sur les séries hypérgeométriques de deux variables, et sur des équations différentielles linéaires aux déivées partielles. C. R. Acad. Sci. Paris 90, (1880), 296-298.
2. Maged G. Bin-Saad, Maisoon A. Hussein, Operational images and relations of two and three variable hypergeometric series, Volume 2, Issue 1., J. Prog. Res. Math. (JPRM), 2395-0218.
3. Bin-Saad, Maged G., A new multiple hyper-geometric function related to Lauricella’s  and  (communicated for publication).
4.  Chhaya Sharma and C. L. Parihar, Hypergeometric functions of four variables (I), J. Indian Acad. Math. Vol. 11, No.2, (1989), 121-133.
5. Chen, M. P., Srivastava, H. M. (1997). Fractional calculus operators and their applications involving power functions and summation of series. Appl. Math. Comput., 81, 283-304.
6. Goyal, S. P., Jain, R. M., Gaur, N.(1992). Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials II. Indian J. Pure Appl. Math.,23, 121-128.
7. Goyal, S. P., Jain, R. M., Gaur, N.(1991). Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials. Indian J. Pure Appl. Math., 22, 403-411.
8. Exton, H., Multiple hypergeometric functions and applications, Ellis Horwood Ltd., Chichester, New York, 1976.
9. Exton, H., Hypergeometric functions of three variables, J. Indian Acad. Math. Vol. 4,(1982), 111-119.
10. Horn, J., Hypergeometrische Funktionen zweier veranderlichen, Math. Ann. 105, (1931), 381-407.
11. Kalla, S. L. (1970). Integral operators of fractional integration. Mat. Notae 22, 89-93.
12. Kalla, S. L., Saxena, R. K.(1969). Integral operators involving hypergeometric functions. Math. Zeitschr.108231-234.
13. Kalla, S. L., Saxena, R. K.(1974). Integral operators involving hypergeometric functions II. Univ. Nac. Tucuman Rev. Ser. A 24, 31-36.
14. Kant, S., Koul, C. L.(1991). On fractional integral operators. J. Indian Math. Soc. (N. S) 56, 97-107.
15. Lauricella, G., Sulle funzioni ipergeometriche a piu variabili. Rend. Circ. Math. Palermo 7, (1898), 111-158.
16. Nishimoto, K., Fractional Calculus Vol. I, Descartes Press Co., Koriyuma, Japan,963(1984).
17. Nishimoto, K., Fractional Calculus Vol. II, Descartes Press Co., Koriyuma, Japan,963(1984).
18. Saran, S., Hypergeometric functions of three variables, Ganita 5, (1954), 77-91.
19. Samko, S. G. and A. A. Kilbas, and O. I. Marichev. Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993.
20. Sharma, C., Parihar C. L., Hypergeometric functions of four variables (I), Indian Acad. Math. 11 (1989) 121–133.
21. Srivastava, H. M. and Karlsson P. W., Multiple Gaussian hypergeometric series, Halsted Press, Bristone, London, New York, 1985.
22. Srivastava, H. M. and Manocha, H. L., A treatise on generating functions, Halsted Press, New York, Brisbane and Toronto, 1984.

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