A Complete Solution to The Problem of Differentiation and Integration of Real Orders for the Gaussian Hypergeometric Function

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1 Jourl of Mtheti d Syte Siee 5 (5) doi:.765/59-59/5.. D DAVID PUBLISING A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui Mhei M. Beghorbl Motrel QC Cd. Reeived: Aril 5 / Aeted: My 8 5 / Publihed: Deeber 5 5. Abtrt: A olete olutio to the roble of fidig the th derivtive d the th ti-derivtive of eleetry d eil futio h bee give. It del with the roble of fidig forul for the th derivtive d the th ti-derivtive of eleetry d eil futio. We do ot liit to be iteger it be rel uber. I geerl the olutio i give through uified forul i ter of the Fo -futio d the Mieer G-futio whih i y e be ilified to le geerl futio. Thi i tur e the firt rel ue of thee two eil futio i the literture d how the eed of uh futio. I thi tl we would lie to reet the ide o the Gu hyergeoetri futio whih i well ow eil futio. Oe of the ey oit i thi wor i tht the roh doe ot deed o itegrtio tehiue. We dot the lil defiitio for geerlitio of differetitio d itegrtio.nely the th order of differetitio i foud ordig to the Rie-Liouville defiitio d f = ( t) f () t dt Γ( ) d where ( < < ) d =. The geerlied Cuhy -fold itegrl i doted for the th order of itegrtio f = ( t) f () t dt >. Γ. Itrodutio The otivtio of thi wor oe fro the re of yboli outtio well the re of lil d frtiol lulu. The ide i tht: Give futio f i vrible we or outer lgebr yte (CAS) fid forul for the th derivtive the th ti-derivtive or both? Thi ehe the ower of itegrtio d differetitio of CAS. I Mle the forul orreod to ivoig the od diff ( f $ ) for the th derivtive d it( f $ ) for the th ti-derivtive. The outtio of frtiol derivtive d frtiol itegrl ivolve evlutig oe tye of o eleetry itegrl. I ft itegrtio tehiue Correodig uthor: Mhei M. Beghorbl Motrel QC Cd. E-il: beghorbl@gil.o. do ot wor uully. Thi h led of thiig bout differet rohe to evlute the. A erie of er h bee etblihed [] [] [4] d reetly boo [] by the uthor of thi er to itrodue differet rohe to evlutig thi tye of itegrl. Thi wor i otiutio of thi erie.. Rie-Liouville Frtiol Derivtive Defiitio The ot widely ow defiitio of the frtiol derivtive i The Rie-Liouville defiitio (R-L FD) [8] [] []. It er reult of uifitio of the otio of iteger-order itegrtio d differetitio. The defiitio i give by d f = ( t) f () t dt Γ( ) d (

2 5 A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui where < < = d f( ) i futio with we igulrity over the itervl of itegrtio. If f( ) i otiuou over the itervl ( [ ] the by lettig oe get f ) ( ).. The th Derivtive of ( ) of Rel Order We re itereted i the frtiol derivtive (the th derivtive of rel order) of the futio f = ( ) () beue of it ltter ue. Subtitutig () i ( yield d y y dy Γ( α) d Γ ( + = ( ) Γ( + α ( ) = The bove forul give the th derivtive of rel order of the futio ( ).. The Geerlied Cuhy -fold Itegrl Defiitio The geerlied Cuhy -fold itegrl i geerlitio of the Cuhy -fold itegrl of iteger order f = ( t) f () t dt. Γ Relig the oditio o i the bove defiitio fro iteger to rel > doe ot ffet the overgee of the itegrl. f( ) i futio with we igulrity over the itervl of itegrtio.. The th Ati-Derivtive of ( ) of Rel Order Agi for our eed of the th ti-derivtive of the futio ( ) we give it i thi etio. It i the evlutio of the itegrl ( t) ( t ) dt Γ Clultig the itegrl reult i the deired forul Γ ( + ( ) Γ ( The Melli Trfor + () A tter of ft we eed the elli trfor of the Gu hyergeoetri futio it will be elied lter to fid the deired uified forul. So we itrodue the Melli trfor d it ivere through the followig two defiitio. For ore diuio of the Melli trfor; ee [9]. Defiitio. The Melli trfor [9] of lolly itegrble futio f( ) o ( ) i defied by [ ; ] = () = () M f F f d ( α < R () < where α d β re rel ott. The tri α < R () < β i ow the tri of lytiity. Defiitio. The ivere Melli trfor [9] i give by f ( + i ) [ ] M f = i d α β π ; < < i where α d β re the e oe i the lt defiitio. Uig the Melli trfor defiitio d evlutig the orreodig itegrl eliit ereio for the Melli trfor of the Gu hyergeoetri futio be foud F( b ; ; ) (4) ( ) ( ) ( b ) ( )( Γ Γ( b) Γ( ) Γ Γ Γ Γ G () = where Γ () i the g futio d i defied () t t e dt Re() (5) Γ = > (6) 5. The Gui The Gu hyergeoetri futio i eil futio d i olutio of the lier eod order

3 A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui differetil eutio d by b y d d + ( ) y = d + (( + + ) ) (7) Oe of the olutio of the bove differetil eutio the Gui hyergeoetri futio d i give i ter of ower erie ( b) F( b ; ; ) = (8) ()! 6. The -futio = The -futio i very geerl futio tht eoe the ot of eil futio iludig the Meier G-futio d the geerlied hyergeoetri futio; ee [5] [6]. Nottio ( A ( ( b B ( b B) ( ( ) ( ). ( b B) Defiitio. The -futio i defied by the Melli-Bre itegrl [9]. () = h() d πi C where h () i give by h () = for Γ( b ) ( ) B Γ + A = = Γ = + + Γ = + ( b B) ( A) (9) For the bove we reuire tht (i) A d B re oitive uber. (ii) d b re ole uber uh tht A ( b + ν) B ( λ νλ = ; = ; =. 5 Tht e the ole of Γ( b B ) for = d Γ( + A ) for = do ot oiide. (iii) The otour C erte the ole reultig fro Γ( b B ) ( ) fro thoe of Γ( + A ) ( ). 6. Eitee Coditio for The -futio For eitee oditio of the -futio we refer the reder to [6]. 6. Proertie of The -futio The followig re oe roertie of the -futio tht re ueful for our uroe (ee [6]). Proerty ( ( + α A α = ( b B) ( b + α B B) Proerty A A α α = α > B ( b ) B α ( b α ) Proerty ( ( b B) = ( b B) ( 7. Meier G-futio The G-futio i eil e of the -futio. A lrge uber of eil futio re eil e of thi futio. I thi etio we give oe defiitio of the futio without y roof. For detiled diuio of the G-futio we refer the reder to [7]. Nottio G G b b b G ( ) G Thee re the tdrd ottio ued i the literture. I the followig defiitio ety

4 5 A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui rodut i iterreted uity d. The Meier G-futio with the reter d b b i defied Melli-Bre tye itegrl follow [7]. Defiitio 4. G = g() d b b L πi g () = Γ( b ) Γ( + ) = = Γ( b + ) Γ( ) = + = + It i ler tht the Meier G-futio i eil e of the -futio d it i derived fro the ltter by lettig A d B eul oe. 7. The rth Derivtive d The rth Ati-Derivtive of The -futio The loure roerty of the -futio uder rel order of differetitio d itegrtio e it very owerful tool for fidig uified forul for the th derivtive d the th ti-derivtive of eleetry d eil futio. I other word oe ere rel order derivtive d itegrl of -futio i ter of ew -futio. Thi i very ie roerty of the -futio whih i ot oeed by other eil futio. The followig le illutrte the ide. Le. The forul ( ) = ( ) r ( ra ( r ( b rb B) ( () give (i) derivtive of rbitrry order if r > (ii) itegrl of rbitrry order if r < of the -futio ( ( b B) Proof: We give roof for rt (i). The roof for rt (ii) i iilr to rt(i) d oly oe eed to ue forul () for frtiol order itegrtio. Rellig the defiitio of the -futio (9) () = h() d πi C where h () i give by (9). Oe differetite both ide of the bove eutio rovided the bove itegrl overge uiforly for oe uig the forul give where r d Γ ( + = r d Γ ( r + ( r) r ( ( )) π i h = C ( h() = Γ + h () Γ ( r + r h () i give by eutio (9). Uig the ottio of the -futio the bove be writte ( A )( ( ) r r + = + + ( b B) ( r Proerty ( of the -futio ilifie the lt eutio to ore ot for ( ( )) ( r ) b rb B d ( ra A ) ( r = ( )( If C i te th ( the eitee oditio for the bove -futio i π = = = + = + rg( ) < A + B A B For Prt(ii) oe eed forul () Γ ( + + r Γ ( + r+ whih i derived fro the geerlied Cuhy forul for the -fold itegrl. The ret of the roof i etly iilr to rt (i). Forul () i very iortt for fidig iteger

5 A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui 5 d rbitrry order yboli derivtive d itegrl of both eleetry d eil futio log they re rereetble i ter of the -futio. 8. A Uified Forul for the th Derivtive d the th Ati-Derivtive of Gui of Rel Order Now we re i oitio to give olete olutio of the roble of yboli differetitio d itegrtio of the Gui hyergeoetri futio of rel order. Thi will be hieved by itroduig uified forul for the th derivtive d the th ti-derivtive where be y rel uber. The forul eoe the two geerlied defiitio of differetitio d itegrtio. Nely the Rie-Liouville defiitio for frtiol differetitio d the geerlied Cuhy -fold itegrl. Theore: Give the Gui hyergeoetri futio the the forul α e (l( α) + iπ) ( ) ( b ) ( ( )( )( iπ e ( α+ = d ( t) F ( b ;( αt+ ) dt if > Γ ( ) d ( t) F ( b ;( αt+ ) dt if < Γ ( where < α + β < α d b give () The origil futio if = (b) Derivtive of y order if > () Ati-Derivtive of y order if <. Proof: Rereetig the hyergeoetri futio i ter of Melli-Bre itegrl give Γ() F( b ; ;( α + ) = Γ Γ( b) ( Γ( ) Γ( b) Γ( ) ( α d π i + C Γ where C i uitble otour. Relig by i the lt eutio yield Γ() F( b ; ;( α + ) = Γ Γ ( b)πi C Γ ( + ) Γ ( b+ ) Γ( ) iπ e ( α + d Γ + with tig i oidertio the hge i the otour C. Uig the -futio ottio we get the -futio rereettio of our hyergeoetri futio Γ() F( b ; ;( α + ) = Γ Γ( b) ( ( b ( α + ( ( By eloitig roerty () of the -futio we ilify the bove -futio to Γ() F( b ; ;( α + ) = Γ Γ( b) b ( ) ( ) iπ e ( α+ β ) ( ) ( ) Alyig le ( for rel order derivtive d ti-derivtive to the bove futio give uified forul for the th derivtive d the th ti-derivtive of the Beel futio ( ) ( ) F( b ; ; α + β ) α ( α + = ( ) ( b ) ( iπ e ( α+ β ) ( )( )( ()

6 54 A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui A further ilifitio be de to our uified forul by elig to roerty ( ( F( ) ) b ; ; + α β = ( ) ( b ) iπ e α Γ() ( iπ e ( α+. Γ Γ( b) ( ) ( ) ( () By oiderig the rorite otour eitig oditio be obtied uh < ( α + <. Ele: A uified forul for the gui hyergeoetri futio b( α give by G F( ; ; + ) i + Γ() ( α Γ Γ( b) 4 44 b α β (4) Subtitutig = i the bove forul give the oe third ti derivtive of the hyergeoetri futio give i the bove ele ( t) F( b ; ;( α + ) dt Γ() Γ() = ( α Γ Γ( b) (5) G b + + α β A further ilifitio be de to the ltter Meier G futio i ter of the geerlied hyergeoetri futio ( t) F( b ; ;( α + ) dt Γ() 6 5 ( i) ( + Γ b = 48 α ( α+ 5 4 F + b+; +; 6 ( α + 4 (6) 9. Coluio The i hieveet i thi wor be uried i the followig tteet. A olete olutio to the roble of fidig the th derivtive d the th ti-derivtive of the Gui hyergeoetri futio of rel order i obtied. Thi be oidered big brethrough i the re of lil d frtiol lulu well the re of yboli outtio. The diffiulty h bee i the evlutio of o eleetry itegrl. The gol h bee rehed without elig to itegrtio tehiue. The Fo -futio h bee itrodued i tool to give olete olutio to the roble i oidertio. Aother iortt ft i thee uified forul give uifitio for the two lil defiitio of geerlied differetitio d itegrtio. I the re of yboli outtio thee forul ehe the ower of outer lgebr yte ie they re lug i forul. CAS uh Mle d Mtheti hve lredy the Meier G -futio ileeted. O theother hd the Fo -futio h ot bee ileeted yet. Referee [] Mhei Beghorbl Frtiol differetil eutio & yboli derivtive d itegrl Sholr Pre Gery 4. [] Mhei M. Beghorbl A uified forul for rbitrry order yboli derivtive d itegrl of the ower-eoetil l Itertiol Jourl of Pure d Alied Mtheti 7 (7) o [] Uified forul for iteger d frtiol order yboli derivtive d itegrl of the ower-ivere trigooetri l I Itertiol Jourl of Pure d Alied Mtheti 4 (7) o [4] A uified forul for the th derivtive d the th-ti derivtive of the ower-logrithi l Proeedig of the Itertiol Coferee of Coutig i Egieerig Siee d Ifortio Clifori Stte Uiverity Fullerto CliforiIEEE (9) -4. [5] Chrle Fo The G d -futio yetril fourier erel Tr. Aer. Mth. So. 98 ( [6] A. M. Mthi The -futio with litio i

7 A Colete Solutio to The Proble of Differetitio d Itegrtio of Rel Order for the Gui 55 ttiti d other diilie Joh Wiley d So 978. [7] A hdboo of geerlied eil futio for ttitil d hyil iee Oford Siee Publitio 99. [8] Keith B. Oldh d Jeroe Sier The frtiol lulu Adi Pre 974. [9] R. B. Pri d D.Kii Aytoti d Melli-Bre itegrl Cbridge. [] Igor Podluby Frtiol differetil eutio Adei Pre S Diego Clifori 999. [] S. E. So A. A. Kilb d O. I Mrihev Frtiol itegrl d derivtive theory d litio Gordo d Breh Siee Publiher 996.