A unified study of Fourier series involving the Aleph-function and the

Kampé de Fériet's function

Frédéric Ayant *Teacher in High School , France

E-mail : fredericayant@gmail.com

Dinesh KumarDepartment of Mathematics and Statistics

Jai Narain Vyas universityJODHPUR-342005, INDIA

E-mail address:dinesh_dino03@yahoo.com

Abstract : Recently Yashwant Singh et al [7] have studied Fourier series involving the I-function defined by V.P. Saxena [6]. Motivated by this work,we make an application of an integral involving sine function, exponential function, the product of Aleph-function of one variable and Kampé deFériet's function. We also evaluate a multiple integral involving the Aleph-function to make its application to derive a multiple exponential Fourierseries. Several particular cases are also given at the end.

2010 Mathematics Subject Classification: 33C05, 33C45, 33C60, 33C65.

Keywords : Fourier series , Aleph-function , Kampé de Fériet's function.

1. Introduction and notations

The Aleph- function , introduced by Südland [8] et al , however the notation and complete definition is presented here in the following manner in terms of the Mellin-Barnes type integral :

z

= (1.1)

for all different to and

= (1.2)

For convergence conditions and other details of Aleph-function , see Südland et al [8]. The Kampé de Fériethypergeometric function will represented as follows.

(1.3)

For further detail see Appell and Kampé de Fériet [1]. For brevity , we shall use the following notations.

Mishra [3] has evaluated the following integral :

= (1.4)

where denotes ; represents ; is a positive integer ; and .

We have the following results :

= ; = where if , else. (1.7)

= where = if , 1 if , else. (1.8)

2. Main results The integrals to be evaluate are :

a)

dx =

(2.1)

provided that , are positive integers. The numbers are positives.

Where with

b)

z =

=

(2.2)

provided that all the conditions of (2.1) are satisfied and , are positives integersfor . The numbers are positives.

Proof

To prove (2.1) , express the Aleph-function in the Mellin-Barnes integral with the help of (1.1) and the Kampé de Férietfunction in double serie with the help of (1.3). We change the order of integration and summation, wich is permissibleunder the conditions stated, now evaluate the x-integral with the help of (1.4) and reinterpreting the Mellin-Barnescontour integral in the form of Aleph-function, we get the desire result (2.1).

The integral (2.2) is obtained by the similar method.

3. Exponential Fourier series

Let

(3.1)

where is a continuous function and bounded variation with interval . Now , multiplied by both

sides in (3.1) and integrating it with respect x from to . Use the first relation of (1.7) and (2.1) , we get :

(3.2)

Use (3.1) and (3.2) , we obtain the following exponential Fourier serie

=

(3.3)

4. Cosine Fourier series

Let

(4.1)

Integrating it with respect x from to , we have :

(4.2)

Multiplying the both sides in (4.1) by and integrating it with respect x from to and use the equations (1.7) ,

(1.8) and (2.1), we obtain.

(4.3)

Use the equations (4.2) , (4.3) and (4.1), we obtain the following cosine Fourier serie

(4.4)

5. Sine Fourier series

Let

= (5.1)

Multiplying the both sides in (5.1) by and integrating it with respect x from to and use the equations (1.8) , and (2.1), we obtain.

(5.2)

Use the equations (5.1) and (5.2) , we get the following sine Fourier serie :

=

(5.3)

6. Multiple exponential Fourier series Consider a function continuous and bounded variations in the domain and

z =

(6.1)

We fix and multipling the both sides in (6.1) by and integrating with respect to from to , we obtain :

z

=

Use the first relation of (1.7) and (2.1) , from (6.2) , we get :

=

(6.3)

Using (6.1) and (6.3) , we obtain the multiple exponential Fourier serie.

z

=

(6.4)

7. Particular cases

The Aleph-function is a generalization of I-function and H-function , for more details , see D.Kumar et al [4 , 5]. Weobtain similar results with I-function and H-function of one variable , see Y.A. Singh et al [7].

Setting in (2.2) , we get the following integral :

z

=

(7.1)

with = ,

If in (7.1), we get :

z

=

(7 2)

Remark : We obtain the similar formulas with multivariable h-function , see R. C. Chandel [2]

8. Conclusion

The aleph-function, presented in this paper, is quite basic in nature. Therefore , on specializing the parameters of thisfunction, we may obtain various other special functions such as I-function ,Fox's H-function, Meijer's G-function,Wright's generalized Bessel function, Wright's generalized hypergeometric function, MacRobert's E-function,generalized hypergeometric function, Bessel function of first kind, modied Bessel function, Whittaker function,exponential function , binomial function etc. as its special cases, and therefore, various unified integral presentationscan be obtained as special cases of our results.

References [1] Appel P. and Kampé de Fériet J. Fonctions hypergéométriques et hyperspheriques ; Polynômes D'hermite ,Gauthier-Villars , Paris . 1926

[2] Chandel R.C.Singh , Agarwal R.D.and Kumar H. Fourier series involving the multivariable H- function ofSrivastava and Panda, Indian J. Pure Appl.Math., 23(5) , (1992), page 343-357.

[3] Mishra S. Integrals involving Legendre functions,generalized hypergeometric series and Fox’s H-function , andFourier-Legendre series for products of generalized hypergeometric functions, Indian J. Pure Appl.Math., 21(1990),page 805-812.

[4] Ram J. and Kumar D.; Generalized fractional integration of the Aleph-function, J. Raj. Acad. Phy. Sci., Vol. 10,No. 4, December (2011), page 373-382.

[5] Saxena R.K. and Kumar D. ; Generalized fractional calculus of the Aleph-function involving a general class ofpolynomials , Acta Mathematica Scientia , Volume 35, Issue 5, September 2015, page 1095–1110, (2015).

[6] Saxena V.P. Formal solution of certain new pair of dual integral equations involving H-function, Proc. Nat. Acad.Sci. India, A52, (1982), page 366-375.

[7] Singh Y. and Khan N.A. A unified study of Fourier series involving generalized hypergeometric function. Globaljournal of science frontier research:G.J.S.F.R. (F) vol 12(4) (2012) , page 44-55

[8] Südland N.; Baumann, B. and Nonnenmacher T.F. , Open problem : who knows about the Aleph-functions? Fract.Calc. Appl. Anal., 1(4) (1998), page 401-402.

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