International Journal of Mathematical Archive-5(1), 2014, Available online through ISSN

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1 Itertiol Jourl of Mthemticl Archive-5( Avilble olie through ISSN GENERALIZED FOURIER TRANSFORM FOR THE GENERATION OF COMPLE FRACTIONAL MOMENTS M. Gji F. Ghrri* Deprtmet of Sttistics Uiversity of Mohghegh Arbili Ir Deshghh Aveue Arbil. Deprtmet of Mthemtics Uiversity of Mohghegh Arbili Ir Deshghh Aveue Arbil. (Receive o: --3; Revise & Accepte o: --4 ABSTRACT Fourier trsform of frctiol orer usig the Mittg-Leffler-type fuctio E ( its comple type ws itrouce together with its iversio formul. The obtie trsform provie suitble geerliztio of the chrcteristic fuctio of rom vribles. It ws show tht comple frctiol momets which re comple momets of orer th of certi istributio re euivlet to Cput frctiol erivtio of geerlize chrcteristic fuctio (GCF i origi beig positive iteger <. The cse ws reuce to the comple momets. Filly fter itroucig frctiol fctoril momets of positive rom vrible we presete the reltioship betwee iteger momets frctiol momets (FMs frctiol fctoril momets (FFMs of positive rom vrible. Mthemtics Subject Clssifictio: MSC 6 MSC 58. Keywors: Frctiol clculus Mittg-Leffler-type fuctio Stirlig fuctio.. INTRODUCTION It is well ow tht the Fourier trsform of probbility esity fuctio is chrcteristic fuctio tht is ϕ ( t it it e e f ( Where the ottio. mes epecttio o the other h we hve: t φ ( t ( i (! This fuctio geertes comple momets of iteger orer s we hve: ( φ ( t i. (3 t u But i this wor we geerlize ( t φ i orer to obti comple o-iteger momets. Recetly frctiol momets of the type E[ ] hve bee itrouce [] showig tht such utities hve importt fetures: (i they re ect turl geerliztio of iteger momets s lie s frctiol ifferetil opertors geerlize the clssicl ifferetil clculus; (ii the iterestig poit is the reltioship betwee frctiol momets the frctiol specil fuctios. ( Correspoig uthor: F. Ghrri* Deprtmet of Mthemtics Uiversity of Mohghegh Arbili Ir Deshghh Aveue Arbil. E-mil: ftemeh. ghrri@yhoo.com Itertiol Jourl of Mthemticl Archive- 5( J. 4 93

2 M. Gji F. Ghrri* / Geerlize Fourier Trsform for the Geertio of Comple Frctiol Momets / IJMA- 5( J.-4. I this wor t first we efie Mittg-leffler-type fuctio Ε ( its comple type herefter clle the geerlize epoetil fuctio. This fuctio is prouct of Mittg-leffler fuctio power fuctio. Usig comple type of this fuctio we efie geerlize Fourier trsform. The obtie trsform provie suitble geerliztio of the chrcteristic fuctio of rom vribles; tht is usig the epecttio of comple geerlize epoetil fuctio we coul irectly obti the geerlize chrcteristic fuctio GCF of certi rom vrible. It ws show tht comple frctiol momets which re comple momets of orer th of certi istributio re euivlet to Cput frctiol erivtio of the GCF i origi beig positive iteger <. The cse ws reuce to the comple momets. I cotiue fter itroucig frctiol fctoril momets of positive rom vrible we presete the reltioship betwee iteger momets frctiol momets (FMs frctiol fctoril momets (FFMs of positive rom vrible. Our mi mes of Frctiol Clculus for this geerliztio were Reim-Liouville Cputo opertors frctiol Tylor series.. PRELIMINARIES I this sectio we briefly review the efiitios of frctiol itegrls frctiol erivtives the forml frctiol right Riem- Liouville Tylor series. Defiitio: Let f is fuctio efie o the itervl [b] is positive rel umber. The right Riem- Liouville frctiol itegrl is efie by: I f ( t f ( t t Γ lso the right Riem Liouville frctiol erivtive" is efie by: D f ( I f. Defiitio: Let [ ] C + the right Cputo frctiol erivtive ( D f is efie by: I f ( t f ( t t ( Γ t (6 the seuetil frctiol erivtives is give by: D D D... D C C C C Ktimes Defiitio: 3Let f be fuctio efie o the right eighborhoo of be ifiitely frctiolly- C ifferetible fuctio t tht is to sy ll ( D f (... Riem- Liouville Tylor series of fuctio is eist. The forml frctiol right (4 (5 f [ ( ] C ( D f. ( I (7 epilicity (. I Γ + C where D is the right Cput frctiol erivtive I is the right Riem- Liouville frctiol itegrl. The frctiol Tylor series of ifiitely frctiolly ifferetible fuctio is bse o fumetl theorem of Frctiol Clculus (see [6]. By fumetl theorem of frctiol clculus oe c sy tht the right Cput frctiol erivtive opertio the right Riem- Liouville frctiol itegrl opertio re i iverse to ech other. 4 IJMA. All Rights Reserve 94

3 M. Gji F. Ghrri* / Geerlize Fourier Trsform for the Geertio of Comple Frctiol Momets / IJMA- 5( J GENERALIZED FOURIER TRANSFORM The eplicit solutios to the eutio C ( D y λ y ( ; N λ R (8 i terms of this fuctio tht is y Ε ( λ. Seuetil frctiol erivtive of the fuctio gives C D y λ y. (9 i geerl cse ( C D Ε Ε ( I itio the geerlize epoetil fuctio stisfie ( λ ( λ ( λ E + y E E y ( ( λ ( λ λ E E E E Therefore ( E λ E λ E λ tht is ( Ε is the frctiol logue of Ep (. The frctiol Tylor series of this fuctio is s followig: Ε ( ( I becuse C ( D E (. It c be see tht { ( } s L Ε s [ ( ] Γ + ( (3 (4 where L is Lplce trsform. With substitutios the results (8 t (4 hve vli for the elemetry epoetil fuctio. We efie the geerlize epoetil fuctio ( Γ + ( Ε by the series below we hve the comple geerlize epoetil fuctio s followig: iπ Ε (( i.( i. e (6 Γ + Γ( + 4 IJMA. All Rights Reserve 95 (5

4 M. Gji F. Ghrri* / Geerlize Fourier Trsform for the Geertio of Comple Frctiol Momets / IJMA- 5( J.-4. lso we hve: iπ Ε (( i.( i. e. (7 Γ Γ( + ( + Now tht we hve geerliztio of the comple epoetil fuctio it shoul; of course be possible to costruct geerliztio of the Euler reltio tht beig Ε (( i cos ( + isi (. (8 From the rel prt of (6 we obti the eutio for the geerlize cosie fuctio cos ( ( Ε (( i + Ε (( i where by usig (6 (7 i recet eutio we c rewrite: π cos(.cos Γ( + So tht i the cse we hve: cos ( π.cos cos( Γ( + from the imgiry prt of (6 we obti the eutio for the geerlize sie fuctio si ( ( Ε i (( i Ε (( i where by usig (6 (7 i recet eutio we c rewrite: π si (.si Γ( + So tht i the specil cse we hve: si ( π.si si( Γ( + Also we hve Therefore we coclue tht the fuctio ( i E ( i ( T.. Ε is perioic with perio T efie s the solutio of the eutio Defiitio: 4 Let f : R C f.the geerlize Fourier trsform of the fuctio f is efie by itegrl ˆ f s E (( is f s C ( i( + y E ( i E ( iy E (9 for we hve the clssicl Fourier trsform ˆ is f s e f ( 4 IJMA. All Rights Reserve 96

5 M. Gji F. Ghrri* / Geerlize Fourier Trsform for the Geertio of Comple Frctiol Momets / IJMA- 5( J.-4. iverse Fourier trsform is s followig: f E ( ˆ is f( s s. T 4. THE GENERALIZED CHARACTERISTIC FUNCTION (GCF OF A RANDOM VARIABLE ~ Defiitio: 5 The geerlize chrcteristic fuctio of y rom vrible φ ( t is efie by: ϕ t E it ( where ( it Ε is the geerlize epoetil fuctio. I the specil cse we obti the oriry chrcteristic fuctio ( t Ep(it. φ Theorem: Suppose tht the frctiol geerlize chrcteristic fuctio of rom vrible is fiite i some ope itervl cotiig zero. The ll the comple frctiol momets eist t ϕ ( t ( i o Γ ( + tht is the comple frctiol momets re the coefficiets of the frctiol Mc-Louri series of ~ φ ( t ( ( the geerlize chrcteristic fuctio is ifiitely frctiolly ifferetible i tht ope itervl for <... ( i C D ( ϕ t ( C D ( ϕ ( t ( t t lso i the specil cse we obti: C ( ( D φ ( t φ ( t. ( i t t Proof: Sice the frctiol Mc-Louri series of ( i ( i Ε i Γ + it c be writte: ϕ t E (( it ( it Γ ( + i the other h by usig (9 we hve: C ~ C ( D ( t ( D ( E (( it Ε is t ( i Γ + C φ ( D ( E(( it ( i E (( it 5. THE FRACTIONAL FACTORIAL MOMENTS (FFMS. Stirlig fuctios of the first i S( c be efie vi their geertig fuctio ( ( S C N where ( ( ( Γ ( + (... + Γ + (3 4 IJMA. All Rights Reserve 97

6 M. Gji F. Ghrri* / Geerlize Fourier Trsform for the Geertio of Comple Frctiol Momets / IJMA- 5( J.-4. with the covetio S( δ (Kroecer s elt. The ltter gives turl possibility to efie Stirlig umber of frctiol orer S( with C N. I fct these Stirlig fuctios s oe my cll them which were itrouce by Butzer Huss Schmit [] my be efie vi the geertig fuctio ( < C. (4 Theorem: Suppose be rom vrible with support [ G (z is the probbility geertig fuctio fiite i some ope itervl cotiig the origi. The G (z is ifiitely frctiol ifferetible i tht ope lim z is fiite the (...( + eists is ifiite itervl if G z ( z (...( limg K + Z C where the ottio ( mes the right Cputo frctiol erivtive ( D f i the cse we hve: G K Z Γ( + ( + Γ ( z (...( lim + S ( where ( z P(. z. G o (5 Proof: It flows from the covergece of series (5 for z from Weierstrss theorem o uiformly coverget series of lytic fuctios tht G (z is lytic fuctio of z i z < with (5 s its power series epsio. Sice G (z is lytic it is ifiitely frctiolly ifferetible oe c write meigfully the frctiol Tylor epsio z ( G z G ( o Γ ( + lso ( G ( z z z ( ( z ( z G we hve z G ( z Γ( + z Γ( + ( Γ( + limg z Γ( + ( ( z ( if we hve K Γ( + limg z z Γ( + (. 5. THE RELATIONSHIP BETWEEN INTEGER MOMENTS WITH FMS AND FFMS I this sectio other theorem which brigs the reltioship betwee iteger momets FMs FFMs is presete for positive rom vrible. Theorem: 3 Suppose be rom vrible with support [. (i frctiol momets of eist if oly if iteger momets eist. 4 IJMA. All Rights Reserve 98

7 M. Gji F. Ghrri* / Geerlize Fourier Trsform for the Geertio of Comple Frctiol Momets / IJMA- 5( J.-4. (ii frctiol fctoril momets of eist if oly if iteger momets eist. Proof: (i let be o-iteger umber it c be writte: ( (! ( >! ( (!(! the we hve :!(! by tig epecttio of recet éutio we hve : c ( (ii By tig epecttio of epressio (4 we hve: ( S 6. REFERENCES (.. [] Butzer P. L. Huss M. Schmi M. Frctiol fuctios Stirlig umbers of frctiol orer Resoltte mth 6 ( [] Gji M. Eghbli N. Ghrri F. Some frctiol specil fuctios frctiol momets Ge. Mth. Notes (3-7. [3] Hilfer R. Applictios of frctiol clculus i physics Worl Scietific Sigpore ( [4] Kilbs A. A. Srivstv H. M. Trujillo J. J. Theory Applictio of frctiol Differetil Eutios El-sevier Amsterm (6. [5] Li M. Rogre J. Zhu T. Frctiol vector clculus frctiol specil fuctio rive:.889v [mth-ph] 7 J. [6] Li M. Rogre J. Zhu T. Series epsio i frctiol clculus frctiol ifferetil eutios riv: 9.489v [mth-ph] 3 Oct 9. [7] Olhm K. Spier J. The frctiol clculus Acemic Press New Yor (974. [8] Poluby I. frctiol ifferetil eutios Acemic Press S Diego (999. [9] Vsily E. Trsov Als of physics 33 ( Source of support: Nil Coflict of iterest: Noe Declre 4 IJMA. All Rights Reserve 99