Finding Formulas Involving Hypergeometric Functions by Evaluating and Comparing the Multipliers of the Laplacian on IR n

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1 Ieriol Jorl of Pril Differeil Eqios d Alicios,, Vol., No., 7-78 Ailble olie h://bs.scieb.com/ijde///3 Sciece d Edcio Pblishig DOI:.69/ijde---3 Fidig Formls Iolig Hergeomeric Fcios b Elig d Comrig he Mliliers of he Llci o I Mohmed Vll Old Mosh * Uiersie des Scieces, Techologie e de l Medecie(USTM Fcle des Scieces e Techiqes *Corresodig hor: khmes@sm.mr eceied Ocober, ; eised Ocober, ; Acceed Ocober 3, Absrc I his work we gie exc formls for some secrl mliliers of Llci o he Eclidi sce I. B comrig hese mliliers we fid old d ew formls iolig hergeomeric fcios. Kewords: hergeomeric fcio, he kerel, we kerel, resole kerel, llce kerel, hkel rsform, mlilier Cie This Aricle: Mohmed Vll Old Mosh, Fidig Formls Iolig Hergeomeric Fcios b Elig d Comrig he Mliliers of he Llci o I. Ieriol Jorl of Pril Differeil Eqios d Alicios, ol., o. (: doi:.69/ijde Irodcio Amog he clssicl eqios of mhemicl hsics he he eqio, he Schrodiger eqio, he Llce eqio i, he we eqio d he Helmolz eqio φ φ φ, where is he Llce oeror i I. These eqios re sdied wih d wiho iiil codiios b seerl hors sice log imes []. For exemle he he, he Schrodiger d he Llce eqios re ofe coled wih he iiil codiio (,x (x, he we eqio wih iiil codiios (,x (x, (,x (x d he Helmolz eqio wih bodrie codiio ifii clled Sommerfeld rdiio codio ([,6] d [7]. The im of his er is firs o gie exlici formls of he Schwrz kerels of he followig mliliers clled here reseciel he weighed he, Schrodiger, Llce, we, resole d geerlized resole kerels o ( H : e > (. i ( K : e > (. ( L : e > (.3 si ( W : > (. wih ( ω : cos > (.5 ( ( : Ι m > (.6 µ ( ( µ, : Ι m > (.7... x x x Noe h we c defie is he Llci o I. φ if he fcio φ is sch h F [ φ]( ξ is emered disribio ([],. 9 where F is he Forier rsform (see below for recise seme. ecll h he clssicl he, Schrodiger, Llce, we d resole Schwrz kerels re gie reseciel b ([],. 6 e 7-7 d ([6],. 59. / x H ( xx,, ' ( ex > / x (,, ' ( ex K x x i ( i Γ (,, ' ' / ( / (.8 (.9 L xx x x (. ( / W ( xx,, ' ( / ( / { x < } (. k >

2 73 Ieriol Jorl of Pril Differeil Eqios d Alicios ( / / W ( xx,, ' / / ( x (. k > ( / ω ( xx,, ' ( / ( / { x < } k > (.3 ( / / ω ( xx,, ' / / ( x (. k > ( / ( (,, ' i xx H ( x x / x ' Im > (.5 where ( H is he Hkel fcio of he firs kid. The ed of his secio is deoed o he relimiries o he Forier rsform o d formls elig he Hkel rsform of some fcios. The Forier rsform of fcio f L ( I ( ξ ( / ix. ξ is defied b he iegrl F f e f x dx. (.6 wih x. ξ x. ξ x. ξ... x. ξ is he ier rodc o. The Forier ierse rsform is gie b i. x F f / I ξ ξ e f ξ dξ (.7 ecll lso h he Forier rsform of rdil fcio f o is rdil d i c be wrie i erms of he Hkel rsform ([7],. 6 s / ( ξ ξ ( ξ / / F f f r J r r dr (.8 where (. J is he Bessel fcio of he firs kid d order. For more iformios o he Forier rsform he reder c cosls he book [7]. Proosiio.. The Schwrz kerel of he oeror φ is gie ls formll b ( φ,, ' ( ( ' φ / ' / K xx x x / J r x x r r dr (.9 The roof of his roosiio ses esseill he forml (.8 d i coseqece is lef o he reder. ecll he followig formls elig some Hkel rsform ([3]., 9, d 3. µ Γ µ x x e J x dx ( µ / Γ (. µ F, ; ( µ e > ;e > wih F (,c,z is he firs kid coee hergeomeric fcio. ( µ Γ ( µ x Γ z x e J x dx µ µ µ z F,, ; (. where F (,b,c;z is he Gss hergeomeric wih e >, e µ >. ρ x J x µ ( x k dx (( ρ / ( µ ( ρ / Γ ( µ Γ ( Γ Γ k F ( ρ /,( ρ / µ,, Γ µ 3ρ / ( µ ( ρ / Γ ( µ ( ρ µ ρ ρ µ k k F µ, µ ( ρ /, µ ( ρ /, for >, e < e ρ < e µ 7 / where ( ;, ( ( c (. F cz z (.3! (, ;, ( ( b ( c! F bcz z (. ( ;,, ( F bcz z (.5 b c! re he hergeomeric fcios.. Weighed He d Schrodiger Eolio Oerors o I I his secio we gie exc forml for he Schwrz iegrl kerel of he weighed he d Schrodiger eolio oerors e ( / d i / e o I. Theoreme.. For e > -, he Schwrz iegrl kerel of he weighed he eolio oeror e / o I is gie i erms of he firs order Kmmer coe hergeomeric fcio F (, c; z b Γ H xx (,, ' / ( Γ( / x F, ; (.

3 Ieriol Jorl of Pril Differeil Eqios d Alicios 7 Proof. B mkig se of he forml (.9 wih r φ r e r we c wrie ( / H xx,, ' x r / ( ' J x x r e r dr / (. sig he forml (. wih, µ /, d x we ge he forml (.. Corollr.. For e >-, he Schwrz iegrl kerel of he weighed Schrodiger eolio oeror i / e o I is gie i erms of he firs order Kmmer coe hergeomeric fcio F (, c; z b Γ K x x i (,, ' x F, ; i / Γ / 3. Weighed Poisso Oeror o I (.3 This secio is deoed o he comio of he weighed Poisso oeror / e. Theoreme 3.. For e >-, The Shwrz iegrl kerel of he weighed Poisso oeror / e o hergeomeric fcio F b L ( xx,, ' F I is gie i erms of he Gss Γ ( / ( Γ( / x, ;, wih F is he Gss hergeomeric fcio. ge r Proof. From he forml (.9 wih ( ( / L xx,, ' x / r (3. φ r e r we / J r x x ' r e dr. (3. d b he forml (. wih /, µ /, z x d oe c esil dedce he forml (3.. Proosiio 3.. (,, ' (,, ' H z x x d L x x re reseciel he weighed he d he weighed. Poisso kerels gie boe he we he (,, ' / z / L xx e z H (,, '. zxx dz (3.3 d ( 3 / x e F ( /, /, d (3. Γ ( x ( F, ;, Γ / Proof. We recll he for ml ([6],. 5 e / / e e d (3.5 B seig i (3.5 we c wrie for he Poisso semi-gro / / e e e d (3.6 d his gies he forml (3.3. To roe he forml (3. we se he forml (3.3, (. d (3... Weighed We Eolio Oeror o I I his secio we shll come exliciel he Schwrez iegrl kerel of he weighed we eolio si oerors / d cos / Theorem.. For > we he ( / ( / si 3/ e d i (. Here we shold oe h he iegrl i (. c be exeded oer coor srig, goig clockwise rod, d rerig bck o wiho cig he rel egie semi-xis. Proof. We sr b recllig he formls ([5],. 73 / si z z/ J z (. where J (. is he Bessel fcio of firs kid d of order gie b ([5],. 83 ( α / ( / z z J ( α z e d i (.3 roided h e α > d rg z. Moreoer, we he he followig forml: ( α / ( z / siαz α 3/ e d z i (. Pig α d relcig he rible z b he smbol (. we obi he forml (.. Theorme.. For > - he Schwrz iegrl kerel si of he weighed we oeror ( is gie b he followig formls

4 75 Ieriol Jorl of Pril Differeil Eqios d Alicios d W W ( x,, i ( /,, ' / 3/ ( xx,, ' e H x x d ( C x F, ;, < < where (,, ' (. C (.5 (.6 if x x ' < if x x ' H xx is he weighed he kerel gie i (( (( ( 3/ ( / Γ( / Γ / Γ / si 3 / d F is he clssicl Gss hergeomeric fcio. Proof. The forml (.5 is coseqece of (., o roe he forml (.6 se Γ W ( xx,, ' J(.7 iγ where ( / 3/ ( / / ( 3 / x J e F,, d (.8 he we he J J J J3 (.9 ( 3 / x J e F,, d (. γ ( 3 / x J e F,, d (. γ d ( 3 / x J3 e 3 F,, d (. γ where he hs γ, γ d γ3 re gie b i γ : z re ; є r <(boe he c -i γ : z re ; > r є (below he c -iθ γ3:z єe ; < θ < (rd he smll circle s є, we he i( 3 / i( 3 / J e I, J e I d J3. Addig he iegrls esbilishes he followig forml wih J isi( 3 / I (.3 ( 3 / x I e F,, d (. ecllig he forml [5],. eα>, eα >, ek, ez > e F c k d Γ z F z c α (.5 α z z k (,, α,,, wih α /, ( /, c /, z ( -/ d x k o ge I ( / Γ (( / x if x x ' < (.6 F, ;, if < < x x ' Combiig (.7, (.3 d (.6 we ge he forml (.5. Corollr.3. The Schwrz iegrl kerel for he weighed we eolio oeror / cos o he Eclidi sce c be wrie o he followig form ω ( x,, ( c x if x x ' < (.7 F, ;, if < < x x ' wih Γ c / Γ / si 3 /. (( (( ( / ( / Γ( / Proof. I iew of he Forml d ω ( xx,, ' W ( xx,, ' we c se he forml ([5], d. d z F ( bcz,,, z F (, bcz,, (.8 dz o obi he forml (.7 from he forml ( Weighed esole Oeror o I I his secio we gie exlici forml for he Schwrz iegrl kerel of he weighed resole /. oeror Theorem 5.. For Im > The Schwrz iegrl kerel for he weighed resole oeror is gie b

5 Ieriol Jorl of Pril Differeil Eqios d Alicios 76 (, xx, ' (( / ( ( / / ( Γ( / Γ Γ ( / F ( /, ( /, /, x ( ( ( / Γ / x / / Γ ( / F, ( /, /, x (5. where F is he hergeomeric series gie i (.5. Proof. Usig he forml (.9 wih φ r r r we ge ( ( J ( rx /, xx, ' x / r r dr. / (5. d b he forml (. wih /, µ, ρ /, x d k we ge he resl of he heorem. Proosiio 5.. Le (, xx, ' be he Schwrz kerel of he weighed resole oeror he we he he followig iegrl rereseios (,, ' (,, ' ; e < (5.3 xx e H xx d (,, ' (,, ' ; e < (5. xx i e K xx d (, xx, ' e W xx,, ' d; I (5.5 i i (, xx, ' e ω (,, ' ; xx d I (5.6 where H ( xx,, ', K ( xx,, ', W ( xx,, ', ω ( xx,, ' re reseciel he Schwrz iegrl kerel of he weighed he, Schrodiger, d we eolio oerors. Proof. We se reseciel he followig formls e d e > (5.7 i i e d e > (5.8 si x x e dx e > (5.9 x e cos xdx e > (5. Corollr 5.3. For e < we he he followig forml ( / x e F( /, /, d ( ( / Γ / F ( /, ( /, /, x ( (( / ( / Γ / Γ / x Γ Γ F, ( /, /, x ( / (5. Proof. This is coseqece of he roosiio (5. forml (5.3, (5. d (.. O c wrie he weighed resole i erms of he weighed we kerel. Corollr 5.. We he xx is ( e s ' x F, ;, s (( / ( ( / / ( Γ( / ds ( C ( Γ Γ F ( /, ( /, /, x ( / Γ( ( / x / ( Γ( / F, ( /, /, x / is ( x e s F, ;, ds x i s Γ (( / Γ( ( / ( / Γ / ( C ( / ( / F /, /, /, x ( / Γ / x Γ F, ( /, /, x / (5. (5.3 where C is s i he heorem.. Proof. The roof of his corollr c be see from he roosiio 5. (5.5, (5.6, (.6, (.7 d ( Weighed Geerlized esole Oerors o I

6 77 Ieriol Jorl of Pril Differeil Eqios d Alicios I his secio we geerlize some resls of he secio 5 b gie exlici exressio of he weighed geerlized resole kerels µ, µ /. Theorem 6.. For e < The Schwrz kerel of he weighed geerlized resole oeror is gie b µ, (, xx, ' (( / ( µ ( µ ( / / ( Γ( / µ Γ Γ Γ ( µ / ( Γ ( µ / µ ( / F /, /, /, x Γ / x µ ( / F µµ, ( /, µ /, x (6. where F is he hergeomeric fcio gie i (.5. Proof. Usig he forml (.9 wih r φ r r µ we ge, µ ( (, xx, ' ( ' J rx x (6. x x ' r dr. / / / µ ( r d o see he forml (6. we se (. wih /, ρ /, x d k -. Proosiio 6.. We he he followig forml coecig he weighed geerlized resole kerel o he weighed he kerel µ, (, xx, ' µ Γ ( µ e H, x, x ' d; e <. Proof. We se he forml µ ( Γ ( µ e <. (6.3 µ ; (6. e d Corollr 6.3. We he µ ( / ( µ ( / ( µ Γ Γ µ ( / Γ ( ( µ F ( /, µ ( /, /, x µ ( / Γ( µ ( / Γ( / Γ ( µ x Γ (( / Γ ( µ / x e F /, /, d F µµ, ( /, µ /, ' x x (6.5 Proof. We se he formls (6.3, (. d ( Commeries d Alicios The sbjec of sd of his er is sied he meeig oi of he ril differeil eqios d he secil fcios of he mhemicl hsics. Firsl exlici solios of he followig ril differeil eqios re gie i erms of he hergeomeric fcios. d * x (, x (, ; ( x, I I x x C I (, ( ; * V (, x (, ; (, V x x I I lim V, x V x, V C I ( x x * (, x I I, C ( I,, ;, x x,,, x x ;,, ( H ( P ( W (, x, x ' ( δ; x I ( Secodl some old d ew formls iolig he hergeomeric fcios re gie b comrig hese solios. The formls (5. d (6. gies he Llce rsform of Kmmer hergeomeric fcio wih rgme /x d exed he well kow forml giig he Llce rsform of he exoeil wih he rgme /x. ( x / e e d ( / i ( H ( x x / Im > (7. The formls (5. gies he Forier rsform of he Gss hergeomeric fcio wih rgme /x d exed he kow forml ( i H ( x x e ( s x x / ds is ' ' (7. x d o he bes of or kowledge here is o sch relio i he mhemicl lierre. Noe h he formls (7. d (7. re coseqeces reseciel of he forml

7 Ieriol Jorl of Pril Differeil Eqios d Alicios 78 ( H ( z z i Γ iz ( ( / / / e d (see Erdel e l [],. 83. H z α α z / / i i ( α i e e d (7.3 (7. Imz > d Imα z > (see Mgs e l [5],. 8. We c lso derie he formls (7. d (7. from (5.3 (.9, (.5 d (5.5, (., (.5. We fiish his secio b he followig corollr. Corollr 7.. We he he followig forml coecig he Hkel d he hergeomric fcios F. ( ( ( Γ H ( x ( x/ i x F,,, Γ x x/ F,,, i (7.5 Proof. B comrig he formls (.5 d (5. wih we obi (7.5. emrk 7.. B sig he forml z F ;,, Γ z J z (7.6 we c dedce from he forml (7.5 he followig clssicl forml ( ( si i H x i J x J x e efereces (7.7 [] Cor,. d Hilber, D., Mehods of Mhemicl hsics ol. Wile 989. [] Erdeli, A. Mgs, F; Oberheiger,F. d Tricomi, F. G. Higher Trscedel Fcios Tome II New York, Toroo, Lodo, INC 95. [3] Erdeli, A. Mgs, Oberheiger,F. d Tricomi, F. G., Higher Trscedel Fcios, Tble of iegrl Trsforms Tome II New York, Toroo, Lodo, INC 95. [] G. B. Folld, Irodcio o ril differeil eqios, Priceo iersi ress, Priceo N. J [5] W. Mgs, F. Oberheiger, d. P. Soi, Formls d Theorems for he secil fcios of Mhemicl hsics, Sriger-Verlog New-York 966. [6] Srichrz ober S. A gide o disribio heor d Forier rsform, Sdies i dced mhemics CC ress, Boc rco A Arbor lodo oko 993. [7] MICHAEL, E. Tlor, Pril Differeil Eqios, Bsic heor I, Alied Mhemicl Scieces 5, Sriger-Verlg 996.