Extended Struve Function

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume III, Issue III, March 2018|ISSN 2454-6194

Composition of Pathway Fractional Integral Operator on (p, q) -Extended Struve Function 1

Poonam Maheshwari and 2Harish Nagar

1

Research Scholar, School of Basic and Applied Science, Sangam University, Bhilwara, Rajasthan, India Associate Professor, School of Basic and Applied Science, Sangam University, Bhilwara, Rajasthan, India

2

Abstract: - In this paper we present Pathway Fractional Integral Operator involving (p, q) – extended Struve function

H v, p ,q ( z ), which is expressed in terms of of the

Hadamard product

(p, q) -extended Gauss Haypergeometric function and

the Fox-Wright function r s (z ) . The result obtained here is also reduced to the known result of R. K. Parmar and J. Choi as special case.

Key words: (p, q) -extended Struve function, Fox-Wright function, (p, q) -extended Gauss Haypergeometric function, Pathway integral operator, Hadamard product.

(see, e.g., [3, 11, 14-16, 17, 20]). In particular, Maŝireviĉ et al. [6] introduced and studied the (p, q) –extended Struve function H ν, p, q (z) of the first kind of order ν with Min {p, q} ≥ 0 and R (ν) > − 1 when p = q = 0 in the form: v 1

z 2   1 z 2n 2 n H v , p ,q ( z )    (  1 ) B ( n  1 , v  ; p , q ) 1  2 (2n  1)!   (v  ) n  0 2

Subject Classification: Primary 33B20, 33C20; Secondary 33B15, 33C05.

1.1 Pathway operators Let 𝑓 𝑥 ∈ 𝐿 𝑎, 𝑏 , 𝜌 ∈ 𝐶, 𝑅𝑒 𝜌 > 0, 𝑎 > 0 and let us take a pathway parameter 𝛼 < 1. Then the pathway fractional integration operator, as an extension of (1.1), is defined and represented as follows (see [20, p. 239]): 𝜌 ,𝛼,𝑎 𝑃0+

𝑡

𝜌

0

1−

𝑎 1−𝛼 𝜏 𝑡

𝜌 1−𝛼

1 B ( n  1 , v  ; p, q )  z2 n 2 H v, p,q ( z )  ( ) 3  3 n 1 4 v n 0 2  (v  ) ( ) B(1, v  ) n ! 2 2 2 z v 1

I. INTRODUCTION

𝑡 𝑎 1−𝛼

(1.3)

𝑓

𝑡 =



(1.4)

where B(x, y; p, q) is the ( p, q) -extended Beta function introduced by Choi et al. [12] Fractional Calculus of the ( p, q)-extended Struve Function

𝑓 𝜏 𝑑𝜏,

1

(1.1) Where 𝐿(𝑎, 𝑏)is the set of Lebesgue measurable functions defined on(𝑎, 𝑏). The pathway model is introduction by Mathai [1] and studied further by Mathai and Haubold [2], [3]. 1.2 Result Required

B ( x, y ; p , q )   t

x  1

(1  t )

y 1

e



p q  t 1 t

0

(min {R(x), R( y)} > 0; min{R( p), R(q)} ≥ 0)

(1.5)

The more generalized definitions of (1.5) are discussed in [11]. Clearly, the case p = 0 = q reduces immediately to the classical Struve function H ν (z) of the first kind (see, e.g., [8, p. 328, equation (2)])

The following formula is required (see [20, eq. (12)])

(1.2) Where 𝛼 < 1; 𝑅𝑒 𝜌 > 0; 𝑅𝑒 𝛽 > 0

H v ( z) 



 n 0

( 1) n ( (n 

z v  2 n 1 ) 2

3 3 ) (v  n  ) 2 2

Recently, many authors have investigated the (p, q) -variant (when p = q, the p-variant) associated with a set of related higher transcendental Haypergeometric type special functions

(1.6)

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International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume III, Issue III, March 2018|ISSN 2454-6194 Also, for our present investigation, we need the concept of Hadamard product (or convolution) of two analytic functions. It can help us in decomposing a newly emerged function into two known functions. If, in particular, one of the two power series defines an entire function, then the Hadamard product series defines an entire function, too. Indeed, let 

f ( z)   an z n n 0 

g ( z )   bn z n

( z  Rf )



( f  g )(z )   a n bn z n n 0

 ( g  f )(z ) ( z  R f )

Whose radius of convergence R is 1

1

1

1 1  lim sup( a n bn ) n  lim sup( a n ) n lim sup( bn ) n  n  n  R n  R f . Rg

In this paper, we aim to compositions of the pathway integral operator involving (p, q) -extended Struve function H ν, p, q (z). Also, we deduce those results, corresponding to the main identities, for the classical Riemann-Liouville involving the (p, q) - extended Struve function H ν, p, q (z). Further, we show that those compositions are expressed in terms of the Hadamard product (1.5) of the (p, q) -extended Gauss Haypergeometric function (see [2, p. 354, equation (7.1)])

F p , q (a , b ; c ; z ) 

n 0

B(b  n , c  b ; p , q) z B(b , c  b) n! n

( z  1; R(c)  R(b)  0) (1.8)

where B(a, b) is the familiar beta function (see, e.g., [7]; see also [9, Section 1.1]), and Fox-Wright function pΨq (z) ( p, q ∈ N0 ) (see, e.g., [5, 4]; see also [10]):

p



q



( 1 , A1 ) ,...( p , A p );    z (  1 , B1 ) ,...(  q , B q );   ( 1  A1 n)...(

 (  n 0

1

p

 B1 n)...(  p

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 Aj

q

) ( B j

B j

).

j 1

Here we present Pathway Integral Operator of (p, q) -extended

H v , p ,q ( z )

of the first kind of order v.

 , ,  , v ,   C be such that min {R 2 (p), R (q)}>0, R (v)>- , 3 Theorem 1: Let

P

( , ,a ) 0



{t  1 H v , p ,q (t )} ( x) 

 1,1 ; ( ) v 1 2 F  3   2 x2 3 p , q v  ;  (v  )  2 4 [a(1   )]2 2    (  v  1; 2) , (1  1   ; 0)     22    2 x2  3  ( 2 ;1) , (1      v  2 ; 2) ; 4 [a(1   )]2    x   v 1  [a(1   )]  v 1

R  R f . Rg .

  (a) n

j 1

1.3 Pathway Integral Operator of the (p, q) -extended Struve function

Struve function

(1.7)

and

p

z   : ( A j j 1

product is a power series defined by

j 1

Where the equality in the convergence condition holds true for

and

be two given power series whose radii of convergence are given by R f and Rg , respectively . Then their Hadamard

p

(1.9)

( z  Rg )

n 0

q

( A j  R  ( j  1,.. p) ; B j  R  ( j  1,...q); 1   B j   A j  0)

 A p n) z n  B p n) n !

   

(1.10) Where  denotes the Hadamard product in (1.7) and whose left-sided Haypergeometric fractional integral is assumed to be convergent. Proof Applying (1.2) to (1.4) and changing the order of integration and summation, which is valid under the given conditions here, we find

P   ( , 0

)



{t  1 H v , p , q (t )} ( x) =

1 B ( k  1 , v  ; p, q ) ( w)  w 2 k ( , )  v 1 2k 1 2 ( ) ( P0 t )( x) 3  1 4 v k 0 3 2 (v  ) ( ) k B(1, v  ) k ! 2 2 2 v 1



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International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume III, Issue III, March 2018|ISSN 2454-6194

1 (1) k B(k  1, v  ; p , q) x 2     v 1  1 [a(1   )] k 0 B(1, v  ) k ! 2  (  v  1  2k ) (1  )  w2 x 2 k 1    ( ) 3  4[a(1   )]2 (  k ) (    v  2  2k ) k ! 2 1    v 1



[6].

[7].

[8].

[9].

(1.11)

Expressing the last summation in (1.11) in terms of the Hadamard product (1.4) with the function (1.8) and (1.9), we obtain the right side of (1.10).

[10].

[11].

1.4 Special case (1) The result (1.10) for Riemann – Liouville fractional integral operator defined in [20] on setting 𝛼 = 0, 𝑎 = 1, then replacing 𝜂 by 𝜂 − 1. The result in (1.10) reduces to the result in [18, eq. (2.10), pg 550].

[12].

[13].

ACKNOWLEDGMENT

[14].

The authors would like to thanks referees for their valuable comments and suggestions which helped to improve the manuscript.

[15].

[16].

REFERENCES [1]. A. M. Mathai: A pathway to matrix- variate gamma and normal densities. Linear Algebra and its Applications 396, (2005), 317328. [2]. A. M. Mathai and H. J Houbold.: Pathway model, Superstatistics, T sallis statistics and a generalized measure of entropy. Physics A 375, (2007), 110-122. [3]. A. M. Mathai and H. J Houbol.: On generalized distributions and pathways. Physics Letters 372, (2008), 2109-2113. [4]. A. M. Mathai, R. K. Saxena and H. J. Haubold: The H-functions: Theory and Applications, Springer, New York, 2010. [5]. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland

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[20].

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