Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 343 https://internationalpubls.com The Fractional Calculus of Product of Special Functions Karuna Laddha1, Deepak Kumar Kabra2, Seema Kabra3 1,2,3Department of Mathematics, Sangam University 1karunaladdha9@gmail.com, 2dkabra20@gmail.com, 3kabraseema@rediffmail.com Article History: Received: 17-05-2024 Revised: 27-06-2024 Accepted: 12-07-2024 Abstract: Introduction: In this paper, we aim to establish a closed form for the Pathway fractional integral operator and Marichev-Saigo-Maeda fractional integral and differential operators involving the product of Special G function and Generalized Mittag – Leffler function. The obtained results are evaluated in terms of generalized Wright hyper geometric function. Keywords: Pathway fractional integral operator, Marichev-Saigo-Maeda operator generalized hyper geometric function , special G function , generalized Mittag- Leffler function. 1. Introduction Throughout this paper, R and C denote the sets of real and complex numbers, respectively. Also R + = (0, ) , N 0 = {0 , 1, …..} and Z − = {-1, -2, ……}. Pathway Fractional Integral Operator : In 2005 Mathai [1] presented the technique of evaluation and interpretation of special function and integral transform and its applications in statistics and physical sciences. It was further extended by Mathai and Hauhold [2,3], see also [14]. In 2009 Nair [4] derived a Pathway fractional integral operator as, Let 0,0)Re(,),,()(  aCbaLxf  and 0 and  is taken as pathway parameter such that .1 Then the pathway fractional integration operator is defined and represented as follows: )( ),,( 0 aP  + ( )x = dttf x a x a x )( )1( )1( )1( 0     − −       − (1) where (𝑎, 𝑏) is the set of Lebsgue measurable function defined on (𝑎, 𝑏). The pathway model is introduced by Mathai and studied further by Mathai and Hauboldm. 2. Objectives Result Required: The following result is required here           ++ −        − + − = + − + 1 1 1 1)( )1( )( 1),,( 0           a t tP a (2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 344 https://internationalpubls.com where ;1 ;0)Re(  0)Re(  . Marichev-Saigo-Maeda fractional Operators The following MSM integral operators are required here [15], see also [16] to obtain the required results Let C ,,,,, '' such that ( ) 0Re  (a) If ( ) ( ) ( )  −++− ''' Re,Re,0maxRe , then ( ) ( ) ( ) ( ) ( ) ( ) 1 ' ' '' '' 1,,,, 0 ''' ))(( −++−−− + ++−− ++−−−  ++−−+ ++− =      tttI (3) (b) If ( ) ( ) ( ) )Re(,Re,RemaxRe ''  +−−+−− , then ( ) ( ) ( ) ( ) ( ) ( )      −+−−− − +−++ +−+  +− +−++− = ''' '' '' ,,,, ))(( tttI (4) (c) If ( ) ( ) ( )  −−−−+− ''Re,Re,0maxRe , then ( ) ( ) ( ) ( ) ( ) ( ) 1 ' '' ' 1,,,, 0 ''' ))(( −+−+− + +−+ +−++  +−++− ++− =      tttD (5) (d) If ( ) ( ) ( ) )Re(,Re,RemaxRe '''  −+−+− , then ( ) ( ) ( ) ( ) ( ) ( )      −−+− − ++−−− ++−−  ++− ++−−+ = ''' ' ' '' '' ,,,, ))(( tttD (6) Special G function : The special  zaG ,,,  is defined by [5,6] as   ( ) ( ) ( )  = −− −+ = 0 1 ,, ! , n n n nn az zzaG      (7) Generalized Mittag- Leffler Function Gosta Mittag – Leffler the Swedish mathematician introduced the term Gosta Mittag – Leffler function i.e. , Mittag – Leffler function is defined [7] as ( ) ( ) ( )( )0; 1 )( 0  + =  =    RCd n d dE n n where is a gamma function , after this Wiman generalized the Mittag – Leffler function as follows, ( ) ( ) ( ) ( )( )0)min(;)( 0 ,  + =  =    RRCd n d dE n n there are number of ways in which Mittag- Leffler function Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 345 https://internationalpubls.com   ( ) ( ) !0 , m z m zE m m m  = + =    (8) where ( )( )0,,   RC Product of G function and Mittag Leffler function     ( ) ( ) ( ) ( ) ( ) !! , 0 0 1 ,,, m z mnn az zzEzaG m n m m n n   =  = −− +−+ =        let m = n = k then     ( ) ( ) ( ) ( ) ( )  = −− +−+ = 0 1 ,,, !! , k k k k k k z kkk az zzEzaG        )9( Fox – Wright Generalized Hypergeometric Function In 1933, E.M. Wright defined a more interesting generalized hypergeometric function of one variable[8] and further generalizations of the series qp F were given by Fox[9] and Wright [10,11,12]; ( ) ( ) ( ) ( ) ( )         = z BB AA z qq pp qpqp ,.,.....,..., ,,...,...., 11 11    = ( ) ( ) ( ) ( ) ( ) ( ) !........ ........ 0 2211 2211 n z nBnBnB nAnAnA n n qq pp   = +++ +++   (10) where the coefficients + RAA p........,,1 and + RBB q........,,1 such that 01 11 −+  == p i i q j j AB for suitably bounded values of z . qp  ,.....,,,,....,, 2121 are complex parameters. The Fox- Wright function is a special case of the Fox – H function as [13] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     −− −− −=         + qq ppp qp qq pp qp BB AA zHz BB AA ,1,.........,1 ,1,.........,1 ,.,.....,..., ,,...,...., 11 11,1 1, 11 11      (11) 3. Methods Theorem 1 Let 1 , Cba ,,,,,,  then for 0)Re(,)Re(,)Re(,)Re(,)Re(      zEzaGzP b    ,,, 1,, 0 , − + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 346 https://internationalpubls.com ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )           −      +−++ − − +−−+ −       − + = + + −−+ −−++ 1 1 631 1 11, 1 ,,,1,1,1,0,1,0,, 1,1,,1,,1 1 1 1              b az kb z )12( Proof : Let I be the left hand side of (12) and on applying (9) we get ( ) ( ) ( ) ( )       +−+ =   = −−++− + 0 11,, 0 !!k kk k kkb z kkkk a zPI    ( ) ( ) ( ) ( ) }{ !!0 11,, 0  = −−−+++ + +−+ = k kkb k kk zP kkkk a    ( ) ( ) ( ) ( ) ( ) ( )  1 0 1 111 1 1 11 !! −−++++  = −−+++  −      +−−++++ −        − +−−+++  +−+ =           kk k kk k kk z bkk kk kkkk a after using equation (10) we get the right hand side of (12). Corollary 1. The result of (12) can also be represented as Fox H function in the following manner.     zEzaGzP b    ,,, 1,, 0 , − + ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                 ++−− − −−−− +−+−− − − −       − + = + + −−+ −−++ kb az H b z 1, 1 1,1,1,0,1,1,1,1,,1,1,0 1,,1,1,1,1 )}1({ 1 1 1 1 )1( 3,1 7,3 1 1              (13) Corollary 2. On taking 0==  , the Generalized Mittag Leffler function reduces to the classical Mittag Leffler function and (12) becomes     zEzaGzP b   − + ,,, 1,, 0 ( ) ( ) ( )       +−+ =   = −−++− + 0 11,, 0 1!k kk k kb z kkk a zP    ( ) ( ) ( ) }{ 1!0 11,, 0  = −−−+++ + +−+ = k kkb k k zP kkk a    Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 347 https://internationalpubls.com ( ) ( ) ( ) ( ) ( )  1 0 1 111 1 1 11 1! −−++++  = −−+++  −      +−−++++ −        − +−−+++  +−+ =           kk k kk k k z bkk kk kkk a ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )           −      +−++ − − +−−+ −       − + = + + −−+ −−++ 1 1 421 1 11, 1 ,,1,1,0,, 1,1,,1 1 1 1              b az kb z Theorem 2 Let Cba ,,,,,,,,,,, ''  such that ( ) Cba  ,,,,,,,0Re  then for ( ) ( ) ( ) ( ) 0Re,Re,Re,Re      zEzaGzI    ,,, 1,,,, 0 , ,' − + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               +−−+++−−+−−+++−− +−−++− +−−+++−−− +−−+++−+−−+ = +−−+++−− 1 '' ' , 85 2 1,1,1,1 ,1,1,1,1,,,,,1,0,1,0 1,1 ,1,1'',1,1,1,,1,       azz ( )14 Proof : Let I be the left hand side of (14) and on applying (9) we get ( ) ( ) ( ) ( )       +−+ =   = −−++− + 0 11,,,, 0 !! ' k kk k kk z kkkk a zII    ( ) ( ) ( ) ( ) }{ !!0 11,,,, 0 '   = −−−+++ + +−+ = k kk k kk zI kkkk a    After using (3) we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 ' ' 0 ' '' ' 1 1 1'1 11 !! −−−+++++−−  = −−+++++−− −−+++++−−−  −−+++++−−−−++++ −−+++++−−−+++  +−+ =         kk k k kk z kk kk kkkk kkkk kkkk a after using equation (10) we get the right hand side of (14). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 348 https://internationalpubls.com Theorem 3 Let Cba ,,,,,,,,,,, ''  such that ( ) Cba  ,,,,,,,0Re  then for ( ) ( ) ( ) ( ) 0Re,Re,Re,Re      zEzaGzI    ,,, ,,,, , ,' − − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −−++−+−++−−++−+− −−+−−− −−++−+−+ −−++−+−+−−+++−− = +−++−++−− 1 '' ' 85 1 1,1,1,1 ,1,1,1,1,,,,,1,0,1,0 1,1 ,1,1',1,1,1,,1,       azz ( )15 Proof : Let I be the left hand side of (15) and on applying (9) we get ( ) ( ) ( ) ( )       +−+ =   = −−++− − 0 1,,,, !! ,' k kk k kk z kkkk a zII    ( ) ( ) ( ) ( ) }{ !!0 1(,,,, '   = ++−−−− − +−+ = k kk k kk zI kkkk a    After using (4) we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 '' ' 0 '' ' 1 1 11 11 !! ++−−−++−−  = ++−−−+−++ ++−−−+−+  ++−−−+−++−−− ++−−−++−−++−−−+  +−+ =         kk k k kk z kk kk kkkk kkkk kkkk a after using equation (10) we get the right hand side of (17). Theorem 4 Let Cba ,,,,,,,,,,, ''  such that ( ) Cba  ,,,,,,,0Re  then for ( ) ( ) ( ) ( ) 0Re,Re,Re,Re      zEzaGzD    ,,, 1,,,, 0 , ,' − + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               +−−++−++−−++−+ +−−++−− +−−++−++ +−−++−+−−+ = +−−++−+ 1 '' '' 85 1 1,1,1,1 ,1,1,1,1,,,,,1,0,1,0 1,1 ,1,1,1,1,1,,1,       azz ( )16 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 349 https://internationalpubls.com Proof : Let I be the left hand side of (16) and on applying (9) we get ( ) ( ) ( ) ( )       +−+ =   = −−++− + 0 11,,,, 0 !! ' k kk k kk z kkkk a zDI    ( ) ( ) ( ) ( ) }{ !!0 11,,,, 0 '   = −−−+++ + +−+ = k kk k kk zD kkkk a    After using (5) we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ' '' 0 ' 1 1 1'1 11 !! −−++++−+  = −−++++−+ −−++++−++  −−++++−+−−++++− −−++++−−−+++  +−+ =         kk k k kk z kk kk kkkk kkkk kkkk a after using equation (10) we get the right hand side of (16). Theorem 5 : Let Cba ,,,,,,,,,,, ''  such that ( ) Cba  ,,,,,,,0Re  then for ( ) ( ) ( ) ( ) 0Re,Re,Re,Re      zEzaGzD    ,,, ,,,, , ,' − − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               −−++−+−−−−−−++−++− −−++−− −−++−++−− −−++−++−−−−+++− = +−++−+−+ 1 ''' ' ' 85 1 1,1,1,1 ,1,1,1,1,,,,,1,0,1,0 1,1 ,1,1',1,1,1,,1,       azz ( )17 Proof : Let I be the left hand side of (17) and on applying (9) we get ( ) ( ) ( ) ( )       +−+ =   = −−++− − 0 1,,,, !! ,' k kk k kk z kkkk a zDI    ( ) ( ) ( ) ( ) }{ !!0 1(,,,, '   = ++−−−− − +−+ = k kk k kk zD kkkk a    After using (6) we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 350 https://internationalpubls.com ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 '' ' 0 '' ' ' 1 1 11 11 !! ++−−−+−+  = ++−−−++−−− ++−−−++−−  ++−−−++−++−−− ++−−−+−+++−−−+−  +−+ =         kk k k kk z kk kk kkkk kkkk kkkk a after using equation (10) we get the right hand side of (17). 4. 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