Integral Representation and Certain Properties of M – Series ... - Hikari

International Mathematical Forum, Vol. 8, 2013, no. 9, 415 - 426 HIKARI Ltd, www.m-hikari.com

Integral Representation and Certain Properties of M – Series Associated with Fractional Calculus Kuldeep Singh Gehlot Government Bangur P.G. College, Pali Pali-Marwar, Rajasthan, India-306401 [email protected] Copyright © 2013 Kuldeep Singh Gehlot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract This paper deals with the Integral Representations and Certain properties of MSeries. The main results are given in two sections namely 2 and 3. In section 2, we calculate four different Integral Representations of M-Series. In section 3, we calculate certain relation between M-Series and the Riemann- Liouville Fractional Integral and Derivatives are investigated. Also point out some particular cases. Keywords: M-Series, Mittag-Leffler function, Generalized Hypergeometric Function, Fractional Calculus Operators. Mathematics Subject Classification: 26A33, 33C20, 33E12.

1. Introduction The M-series, introduced by [4], as 

  , … . . ,          . 1    , . ,  ;  , . ,  ;      , … . . ,   
  1  

Here, α ∈ C, Reα $ 0 and &  , &  are Pochammer symbols. The series (1) is defined when none of the parameters & s, j  1,2, . . , q, is a negative integer or zero. If any numerator parameter a+ is a negative integer or zero, then the series terminates to a polynomial in z. This series is convergent for all z if p - q, it is convergent if p  q  1 and divergent , if p $ .  1 . When p  q  1 and |z|  1, the series can converge in some cases. Let 1  ∑& & 3 ∑& & . It can

Kuldeep Singh Gehlot

416

be shown that when p  q  1 the series is absolutely convergent for |z|  1if Reβ 4 0, conditionally convergent for z  31 if 0 - Reβ 4 1, and divergent for |z|  1 if 1 - Reβ.  Mittag- Leffler Function [1] is an entire function of order α defined by 

78    



 . 2 
  1

when there is no upper (p) and lower (q) parameter in equation (1), we have 
 . 3     78     
  1 

The Fractional Integral operators for ϑ $ 0 3, Definition 2.1, Page 33 are defined as, L JK MN 

S

1 MQ P  TQ , N $ 0, 4 O N 3 QRL

JRL MN  

 

MQ 1 P TQ , N $ 0. 5 O Q 3 NRL S

The Fractional Derivative for ϑ $ 0 3, Definition 2.2, Page 35 are defined as, LK T R\L] L [JK XK MN   Y Z M^ N  TN LK S 1 T M Q P  Y Z TQ, N $ 0, 6 N 3 Q\L] 1 3 \O] TN T XRL MN  Y Z TN

LK

JRR\L] MN



LK 

1 T  Y3 Z 1 3 \O] TN

P S

MQ TQ, N $ 0, 7 Q 3 N\L]

where ϑ means the maximal integer not extending ϑ and \ϑ] is the fractional part of ϑ.

2. Integral representation of M-series

Theorem 2.1 For & $ 0, d 3 & $ 0, ∀ f  1 Qg h iT i  1 Qg . and convergent condition of M-Series are satisfies, then   

1 l Rn R n R P1 3 Q m o Q o  Q TQ. 8     j j B& , d 3 &  & d

Proof : we know 
d 

q8Kd q8



, is Pochammer symbol, there for

Integral representation and certain properties

417

&     &  d  &  d 3 &    jj ,  ∏dd  ∏ d   &  d   d 3 &  & d d    d   ∏&&  &  d 3 &   d    jj ,  ∏dd  & d 3 &  &    d 3 &  ∏&

∏&&  

& d  v v st igs u 1 3 QR Q R TQ  vK , we have    ∏&&   d   P1 3 Qlm Rno R Q noKR TQ, 9  jj ∏dd  & d 3 &  & d 

from definition of M-series, equation (1), ∞
  … . .          ,   … . .    
  1 



using equation (9), we have    
 1 P1 3 QlmRno R Q no KR TQ, jj      
  1 x& , d 3 &  

 
    j j & d

 
    j j

Hence.

& d

& d





Q  1 P1 3 QlmRno R Q no R  TQ, x& , d 3 &   
  1  






1 lm Rno R no R  P1 3 Q Q   Q TQ, B& , d 3 &  

Corollary 2.1.1 For
 1, & $ 0, d 3 & $ 0, ∀ f  1 Qg h iT i  1 Qg . and convergent condition of M-Series are satisfies, then equation (8) reduces in the form y





   j j & d

1

x& , d 3 & 



P1 3 Qlm Rno R Q no R t z{ TQ. 10 

Remark : The formula (10) is a known relation, the integral representation of y   function ([2], Theorem 28, page 85). Lemma 2.1.1 For & $ 0, d 3 & $ 0, ∀ f  1 Qg h iT i  1 Qg . convergent condition of M-Series are satisfies, then

and

Kuldeep Singh Gehlot

418
     


1 l Rn R n R P1 3 Q | | Q | RR Q TQ, 11 B ,  3  

Proof : we know 
d 

 Γ8Kd

   ∏&}&    ∏&}&           ,  ∏dd  ∏d}d    ∏d}d         ∏&&  ∏&}&       3       ,  ∏dd  ∏d}d        3   ∏&&  ∏&}&       3       ,  ∏dd  ∏d}d    3     3       ∏&& 

Γ8

, is Pochammer symbol.

st igs u 1 3 QR Q R TQ  ∏&&  



∏&}& 

Γ Γ

, then we get 

1  P1 3 Ql| Rn| R Q n| KR TQ, 12    ∏dd  ∏d}d  B ,  3   ΓK



from definition of M-series, 
  , … . ,      ,         , … . ,    
  1   using equation (12), we have  
∏&}&    P1 3 Ql| Rn|R Q n| KR TQ      
  1 ∏d}d  x ,  3  
   


   

Hence.





∏&}&  Q  1  Rn R n R l | | | P1 3 Q Q   TQ, x  ,  3   ∏d}d  
  1 

 






1 l Rn R n R P1 3 Q | | Q | RR Q TQ. B ,  3   

Theorem 2.2 For 1 $ 0, and convergent conditions of M-series are satisfies, then 

z R ~ R  R ~ P t { Q K  , . ,  ;  , . ,  , 1  1; Q TQ. 13     1  1  ≡ Pt ∞



R

z {




Q K  , ,  ;  , . ,  , 1  1; Q TQ,

Proof : Consider ~

Integral representation and certain properties 



using (1), we have ≡Pt 

R

z { Q~





∏&& 

Q TQ, ∏dd  1  1 
  1

put {  , we obtain z



419



∏&&  



K~K ≡  P t R‚ K~ TQ, ∏dd  1  1 
  1 



using definition of gamma function  ∏&&  K~K   1  1  , ≡  ∏dd  1  1 
  1 

≡

≡

≡

≡

~K

~K ~K ~K

Hence.



1  1 

 

1  1 

 

1  1 

∏&&  

   1  1 , ∏dd  1  1 1  1
  1 ∏&& 

 1  1 , ∏dd  1  1 
  1 

∏&&  

 , ∏dd  
  1


 1  1    . 

Lemma 2.2.1 For there is no upper (p) and lower (q) parameter in equation (13), and
$ 0, 1 $ 0 then , there holds a Integral Representation formula for MittagLeffler function 
z 1 R ~ { P t Q  3; 1  1; QTQ. 14 78    ~K 1  1 

Theorem 2.3 For 1, ƒ ∈ „ , 1 $ ƒ $ 0 and convergent conditions of M-series are satisfies, then
KK  , . ,  , ƒ;  , . ,  , 1; 
 ~R…R  1 … P Y1 3 Q Z   † Q … ‡ TQ. 15 ƒ. xƒ, 1 3 ƒ

Proof: consider the integral,



Kuldeep Singh Gehlot

420 

 ~R…R

 ≡ P Y1 3 Q … Z 


    † Q … ‡ TQ,

using the definition of M-series, equation (1), we have  ∞  ~R…R   ∏&&  1  … … P Y1 3 Q Z Y Q Z TQ, ≡  ∏dd  
  1 







put Q  , we have  ∞  ∏&&  ƒ  ≡  P1 3 ~R…R  K…R T, ∏dd  
  1 | ˆ

using definition of Beta Function, ∞ ∏&&   ƒ 1 3 ƒƒ     ≡  , 1    ∏dd  
  1 

 ≡ ƒ.  1 3 ƒ  ∞



∏&& 

ƒ ƒ  ,  ∏dd  
  1 1  1  

∏&&  ƒ ƒ. 1 3 ƒƒ   ≡   , 1  ∏dd   
  1 1  
    ≡ ƒ. x ƒ, 1 3 ƒ K K  , . ,  , ƒ;  , . ,  , 1; . ∞

Hence.

Theorem 2.4 For ‰ ∈ „ , Št‰ $ 0 and convergent conditions of M-series are satisfies, then 

  R Rz{ PQ t  QTQ. 16 K ‹ , . ,  , ‰;  , . ,  ; Œ   ‰ 

Proof: Consider the integral, ∞
R Rz{ x ≡ PQ t  QTQ, 

using the definition of M-series, equation (1), we have   ∏&&    x≡  P t Rz{ QKR TQ, ∏dd  
  1 



using definition of Gamma Function, we have  ∏&&   ‰    x≡  , ∏dd  
  1 K 

Integral representation and certain properties

421

∏&&  ‰   ‰  [ ^ , x≡    ∏   
  1  d d 
Γρ  x ≡  K ‹ , . ,  , ‰;  , . ,  ; Œ. 

Hence.

3. Certain properties of M-series associated with Fractional Calculus

L Theorem 3.1 Let
$ 0, O $ 0, ƒ $ 0 iT  ∈ Š . Let JK be the left –sided operator of Riemann- Liouville fractional integral (given by equation (4)). Then there holds a formula,
N …KLR ƒ  L …R  Q‘   ŽJK Q N    ƒ  O


’ KK  , . ,  , ƒ;  , . ,  , ƒ  O; N . 17

Proof: By virtue of (1) and (4), we have
L …R  Q‘    ≡ ŽJK Q N ,  

∏&&   Q …KR 1    P TQ, ≡ O ∏dd  
  1 N 3 QRL 



S



put Q  N ,we have   ∏&&   N K…KLR 1  ≡   P  K…R 1 3 LR T, O ∏dd  
  1 



using the definition of Bata function,  ∏&&  N  ƒ    O N …KLR  ≡   , O ∏dd  
  1 ƒ  O   

 ∏&&  ƒ N  N …KLR ƒ  ≡   , ƒ  O ∏   ƒ  O 
  1  d d 
N …KLR ƒ  ≡ K K  , . ,  , ƒ;  , . ,  , ƒ  O; N . ƒ  O

Hence.

Kuldeep Singh Gehlot

422

Lemma 3.1.1 For there is no upper (p) and lower (q) parameter in equation (17), L and
$ 0, O $ 0, ƒ $ 0 iT  ∈ Š . Let JK be the left –sided operator of Riemann- Liouville fractional integral (given by equation (4)). Then there holds a formula,
N …KLR ƒ L  JK Q …R 78 Q N    ƒ; ƒ  O; N. 18 ƒ  O

L Lemma 3.1.2 For
 1, O $ 0, ƒ $ 0 iT  ∈ Š . Let JK be the left –sided operator of Riemann- Liouville fractional integral (given by equation (4)). Then equation (17) will reduce in the form, L ”Q …R y Q•– N “JK



N …KLR ƒ y  , . ,  , ƒ;  , . ,  , ƒ  O; N . 19 ƒ  O K K 

˜ Theorem 3.2 Let α $ 0, O $ 0, ƒ $ 0 iT  ∈ Š . Let IR be the right –sided operator of Riemann- Liouville fractional integral (given by equation (5)). Then there holds a formula,
L R…RL  Q R ‘ N ŽJR Q  


N R… ƒ R   K K  , . ,  , ƒ;  , . ,  , ƒ  O; N . 20 ƒ  O

Proof: By virtue of (1) and (5), we have
L R…RL  Q R ‘    ≡ ŽJR Q N ,  

∏&&  1   ≡   P Q R…RLR Q 3 NLR TQ, O ∏dd  
  1 



S





put Q  ‚ ,we have S



 ∏&&  N R  N R…  ≡   P  …KR 1 3  LR T, O ∏dd  
  1

using the definition of Bata function,



Integral representation and certain properties

423

∏&&  N R  ƒ  O N R…    , ≡ O ∏dd  
  1 ƒ  O   



 ∏&&  ƒ N R  N R… ƒ  ≡   , ƒ  O ∏dd  ƒ  O 
  1 
N R… ƒ R  ≡ K K  , . ,  , ƒ;  , . ,  , ƒ  O; N . ƒ  O

Hence.

Lemma 3.2.1 For there is no upper (p) and lower (q) parameter in equation (20), L and
$ 0, O $ 0, ƒ $ 0 iT  ∈ Š . Let JR be the right –sided operator of Riemann- Liouville fractional integral (given by equation (5)). Then there holds a formula,
N R… ƒ L R…RL R R    [JR ‹Q 78 Q Œ^ N    ƒ; ƒ  O; N . 21 ƒ  O

L Lemma 3.2.2 For
 1, O $ 0, ƒ $ 0 iT  ∈ Š . Let JR be the right –sided operator of Riemann- Liouville fractional integral (given by equation (5)). Then equation (20) will reduce in the form,

“JRL ”Q R…RL y Q R •– N 

N R… ƒ y  , . ,  , ƒ;  , . ,  , ƒ  O; N R . 22  ƒ  O K K 

L Theorem 3.3 Let
$ 0, ƒ $ O $ 0 iT  ∈ Š . Let XK be the left –sided operator of Riemann- Liouville fractional derivative (given by equation (6)). Then there holds a formula,
L …R ŽXK Q  Q‘ N


N …RLR ƒ  KK  , . ,  , ƒ;  , . ,  , ƒ 3 O; N . 23 ƒ 3 O

Proof: By virtue of (1) and (6), we have
L …R  Q‘    ≡ ŽXK Q N ,  

Kuldeep Singh Gehlot

424

∏&&  Q 1 T š  …R  šRLR ≡ Y Z PQ N 3 Q   TQ, ™ 3 O TN ∏dd  
  1 S







where ™  
  1, š S  ∏&&  1  T  ≡   Y Z P Q K…R N 3 QšRLR TQ, ™ 3 O ∏dd  
  1 TN 



put Q  N and using Beta function, we have š   ∏&&  T 1     Y Z N šK…KRLR x™ 3 O, ƒ   , ≡ ™ 3 O ∏dd  
  1 TN 
N …RLR ƒ ≡ KK  , . ,  , ƒ;  , . ,  , ƒ 3 O; N . ƒ 3 O Hence.

Lemma 3.3.1 For there is no upper (p) and lower (q) parameter in equation (23), L and
$ 0, ƒ $ O $ 0 iT  ∈ Š . Let XK be the left –sided operator of Riemann- Liouville fractional integral (given by equation (6)). Then there holds a formula,
N …RLR ƒ L  XK Q …R 78 Q N    ƒ; ƒ 3 O; N. 24 ƒ 3 O

L Lemma 3.3.2 For
 1, ƒ $ O $ 0 iT  ∈ Š . Let XK be the left –sided operator of Riemann- Liouville fractional integral (given by equation (6)). Then equation (23) will reduce in the form, L “XK ”Q …R y Q•– N 



N …RLR ƒ y  , . ,  , ƒ;  , . ,  , ƒ 3 O; N . 25 ƒ 3 O K K 

Theorem 3.4 Let ƒ $ O $ 0, s›Qœ ƒ 3 O  \O] $ 1 iT  ∈ Š. Let XRL be the right –sided operator of Riemann- Liouville fractional derivative (given by equation (7)). Then there holds a formula,
L LR…  Q R ‘ N ŽXR Q  

Integral representation and certain properties

425


N R… ƒ R  KK  , . ,  , ƒ;  , . ,  , ƒ 3 O; N . 26 ƒ 3 O

Proof: From (1) and (7) it follows that,
L LR…  Q R ‘    ≡ ŽXR Q N ,  

∏&&  Q R  1 T š Q LR…  ≡ Y3 Z P TQ,   KLRš     TN  ™3O Q3N ∏dd  
  1 

S





where ™  
  1, and put Q  ‚ and use Beta function, we have
N R… Γƒ R  ≡ K K  , . ,  , ƒ;  , . ,  , ƒ 3 O; N . Γƒ 3 O S

Hence.

Lemma 3.4.1 For there is no upper (p) and lower (q) parameter in equation (26), L and
$ 0, ƒ $ O $ 0iT  ∈ Š . Let XR be the rightt –sided operator of Riemann- Liouville fractional integral (given by equation (7)). Then there holds a formula,
N R… ƒ  L LR… R R [XR ‹Q 78 Q Œ^ N   ƒ; ƒ 3 O; N . 27 ƒ 3 O

L Lemma 3.4.2 For
 1, ƒ $ O $ 0 iT  ∈ Š . Let XR be the rightt –sided operator of Riemann- Liouville fractional integral (given by equation (7)). Then equation (26) will reduce in the form,

“XRL ”Q LR… y Q R •– N 

N R… ƒ y  , . ,  , ƒ;  , . ,  , ƒ 3 O; N R . 28 ƒ 3 O K K 

References

[1] Mittag- Leiiler , G., Sur la nouvelle function 78 N , C.R. Acad. Sic. Paris. (Ser. II), (1903), 137, 554-558. [2] Rainville, E.D., Special Function, The Macmillan Company, New York, 1963.

426

Kuldeep Singh Gehlot

[3] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives. Theory and Applications. Gordon and Breach, Yverdon et al. (1993). [4] Sharma, M. Fractional Integral and Fractional Differentiation of the M-Series, Fractional Calculus and Applied Analysis, Vol. 11, No. 2 (2008), 187-191.

Received: October, 2012