I t n e a r ti o o J r n eri R s r e c m t n ( I u - ss e u -, A li I E D R F A T C IO U O F THE E I E D H TZ- T ION
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1 eaioa Joua o Egieeig each A Maagee (JERM) SSN : , Voue-3, ue- 4, Api 26 GENERALZE FRATONAL ALULUS OF TE GENERALZE URWTZ- L ER ZETA FUNTON J iea aiya, Jea R a A bac Thi pape ea wih he eivaio o he geeaie acioa ieeiaio a geeaie acioa iegaio o he geeaie uwi- Lech Z ea ucio eie a uie by Sivaava e a [ 6] R epeeaio o uch eaio ae obaie i e o Riea- L iovie iega M aheaic Subjec aiicaio: - 336, 26A33 Γ λ+ Γ ; a \ ; e Te- Geeaie uwi- L ech Zea ucio, Riea- L iouvie acioa iega, Saigo acioa i ega NTROUTON Thi pape i evoe o he iveigaio o he geeaie uwi-l ech Zea ucio eie by S ivaava e a [6] ( (,,),, (,, a, whe whe * ) i he ; \ whe, a a, i! ( ), ; * : ( ) Pochhae ybo o eie by ; ; i beig ueoo coveioay ha, () yie he geeaie uwi- L ech Zea ucio eie by Li a Sivaava [ 6] (,,),, (,, (,), (,, a ( 2) Fuhe we e, () yie he geeaie uwi-l ech Zea ucio uie by Gag e a [3] a Jaov e a [5] a oow: (,,),, (,, a,, (,,! ( 3) whe a, () yie he uwi-lech Zea ucio uie by G oya a Laha [4, p, Equaio eaiebeow (5)] a M aucip eceive Mach 29, 26 J iea aiya, epae o Maheaic & Saiic, Jai Naai V ya Uiveiy, johpu J ea Ra, epae o Maheaic & Saiic, Jai Naai V ya Uiveiy, johpu 55
2 Geeaie F acioa acuu o he Geeaie uwi-lech Zea Fu cio (,,),, (,, a! * (,, ( 4) L ea [ 6, eq 35, p 498] he geeaie uwi- Lech Zea ucio i e o - ucio (,,),, (,, ( ( ) ) ( ),3 3,3 (, ;),(, ;),( a,; (,),(, ;),( a,; ( 5) uhe eai abou uwi-l ech Zea ucio he eae i e ee o he Sivaaava [3], Sivaaava e a [ 4] a Saea e a [] R oow (See [9]) F RATONAL ALULUS OPERATORS he Riea-Liouvie e a igh- ie acioa cacuu opeao ae eie a, Γ a - 6, ( 7) ( ) =, ( 8) - =, 9 ( ) whee R e i he iega pa o R e,, a he geeaie acioa cacuu opeao give by Saigo [ 8] ae eie a,β, η () 2F, ; ; () e R ( ) 56
3 eaioa Joua o Egieeig each A Maagee (JERM) SSN : , Voue-3, ue- 4, Api 26,β -, η - () ;,β, η () 2 F, ; ; (),β -, η () ; ( ),β, η -, - β, η () () -, - β -, η - () ; ( 2),β, η () -, - β, η -,-β -, η () ; 2 F a,b,c; i he Gau hypegeoeic ucio eie o cope paaee a, b, c, b y he hypegeoeic eie ( 3) c,, a b c! 2 F a, b ;c; ( 4) w ih covege abouey o a o, i ovege o R c a b ; ( ee [,2]) a a ccou o hypegeoeic ucio [See ( 7, 2) ] eaie N oe 3: equaiie aociae wih Čebyšev ucioa o Saigo acioa iegaio opeao eie ae give by Saea e a [] The geeaie acioa cacuu opeao (5), (6), (7) a (8) c oicie i wih he Riea-L iouvie opea o (6) ( 9) o :,-,η ( ) (,-,η ( ) ( ),,-,η ( ) (,-,η ( ) ( ), ( 5) ( 6) ), ), ( 7) ( 8) 57
4 Geeaie F acioa acuu o he Geeaie uwi-lech Zea Fu cio 3 RESULTS REQURE N TE SEQUEL L ea we have [9] λ () Γ Γλ Γλ η β λ βγλ η whee,,,,, a, λβ ( 9) L ea 2 we have [9],β, η λ () Γ Γβ λ Γη λ λγ β η λ λβ ( 2) whee,,,,, i,( ) L ea 3 Saea e a [ 3] i ouce oowig o,,β, β, γ ρ Γ( )Γ( ) ( ) ρ γ () Γ(ρ)Γ( )Γ( ) ( 2) w hee ( ), ( a b), ( a a b g), ( ) Lea 3 Saea e a [ 3] iouce oowig o Γ Γρ Γ Γ,,β, β, γ ρ Γ ρ γ () ( ) ( 22) w hee ( b ), ( a b g), ( a a g), ( ) Theoe : L e, ; \,,,,a, a,, uhe e,β, η w ih hee ho he oua (,, ),,,, ( ),5 (, ;),(, ;),(, ;),(, ;), (a,; 5, 5 b (,),(, ;),(, ;),(, ;)( a,; (23) Poo :- U ig equaio (), (5), () a (9), we have (,, ),,,, a ( ) b! 58
5 eaioa Joua o Egieeig each A Maagee (JERM) SSN : , Voue-3, ue- 4, Api 26 a b! = a b! = ( ) ( a b )! = ) ( a b ( )! ( ) Theoe i hu pove,5 (, ;),(, ;),(, ;),(, ;), (a,; 5, 5 b (,),(, ;),(, ;),(, ;),( a,; i Theoe, i give he eu i e o Riea- L iovie acioa iega opeao a oow: (,, ),,,, ( ),4 b (, ;),(, ;),(, ;),(a,; (,),(, ;),(, ;),( a,; i Theoe, 2 i give he oowig ooay i e o geeaie uwi-lech Zea ucio,,,, ( ) (,),(, ;),(, ;),(, ;),( a,;,5 (,),(, ;),(, ;),(, ;), (a,; 5, 5 b Theoe 2: Le, ; \,,,,a, i, uhe e,β, η w ih, ( ) hee ho he oua T he poo (,, ),,,, ( ) ca be eveope o iia ie o ha o Theoe (,),(, ;),(, ;),(, ;),( a,; ( 24),5 (, ;),(, ;),(, ;),(, ;), (a,; 5, 5 b i Theoe 2, i give he eu i e o Riea- L iovie acioa iega opeao a oow : 59
6 Geeaie F acioa acuu o he Geeaie uwi-lech Zea Fu cio 2 (,, ),,,, ( ),4 b (, ;),(, ;),(, ;), (a,; (,),(, ;),(, ;),( a,; i Theoe 2, i give he oowig ooay i e o geeaie uwi-lech Zea ucio 2 2 T he,,,, ( ) oowig wo Theoe ca be pove i a iia ae (,),(, ;),(, ;),(, ;),( a,;,5 (,),(, ;),(, ;),(, ;), (a,; 5, 5 b T heoe 3: Le, ; \,, uhe e w ih,,,a,, a, hee ho he oua + (,, ),,,, ( ),5 (, ;),(, ;),(, ;),(, ;), (a,; 5, 5 b (,),(, ;),(, ;),(, ;),( a,; (25) i Theoe 3, i give he eu i e o Riea-L iovie acioa iega opeao a oow : 3,,,,,, ( ),4 b (, ;),(, ;),(, ;), (a,; (,),(, ;),(, ;),( a,; i Theoe 3, i give he oowig ooay i e o geeaie uwi-lech Zea ucio 3 2,,,, ( ),5 (,),(, ;),(, ;),(, ;), (a,; 5, 5 b (,),(, ;),(, ;),(, ;),( a,; Theoe 4: Le, ; \,, uhe e,β, η w ih,,, a,, i,( ) hee ho he oua,,,,,, ( ),5 (, ;),(, ;),(, ;),(, ;), (a,; 5, 5 b (,),(, ;),(, ;),(, ;),( a,; (26) i Theoe 3, i give he eu i e o Riea- L iovie acioa iega opeao a oow : 6
7 eaioa Joua o Egieeig each A Maagee (JERM) SSN : , Voue-3, ue- 4, Api 26 4,,,,,, ( ),4 b (, ;),(, ;),(, ;), (a,; (,),(, ;),(, ;),( a,; i Theoe 3, i give he oowig ooay i e o geeaie uwi-lech Zea ucio 42,,,, ( ),5 (,),(, ;),(, ;),(, ;), (a,; 5, 5 b (,),(, ;),(, ;),(, ;),( a,; A KNOWLEGEMENT Thi wo i uppoe by a Po- ocoa Feowhip o he Naioa Boa o ighe Maheaic (NBM), epae o Aoic E egy, ia o i auho (Jiea aiy R EFERENES [] A Eéyi, W Magu, F Obeheige, FG Ticoi, ighe Ta- ceea Fucio, Vo McGaw- i, New Yo - Tooo - Loo (953); R epie: Kiege, Meboue, Foia (98) [2] A Eéyi, W Magu, F Obeheige, FG Ticoi, ighe Taceea Fucio, Vo McGaw- i, New Yo - Tooo Loo (954); R epie: Kiege, Meboue, Foia (98) [3] M Gag, K Jai, a SL Kaa, A uhe uy o geea uwi-l ech ea ucio, Ageba G oup Geo [4] SP Goya a RK Laha, O he geeaie Z ea ucio a he geeaie Labe ucio, G aia Saeh (997), pp (28), pp 3 39 [5] Jaov, TK Pogáy, a RK Saea, Eee geea uwi- Lech Zea ucio a Mahieu ( a, λ)- eie, App Mah Le, Vo 24 (8), ( 2), [ 6] S - Li a M Sivaava, Soe aiie o he uwi- Lech Zea ucio a aociae acioa eivaive a ohe iega epeeaio, App Mah opu 54 (24), pp [7] A M Mahai, R K Saea a J aubo, The - ucio, Theoy a Appicaio, Spige, N ew Yo, 2 [ 8] M Saigo, A ea o iega opeao ivovig he Gau hypegeoeic ucio, Mah p Kyuhu Uiv (978), ] [ 9] S G Sao, AA Kiba, O Maichev, Facioa ega a eivaive Theoy a A ppicaio Go o a Beach, New Yo ( 993) [ ] R K Saea, T K Pogáy, R Saea a J aov, O geeaie uwi Lech Zea iibuio occuig i aiica ieece Aca U iv Sapieiae, Maheaica, 3, (2) [ ] RK Saea, J Ra, J aiya a TK Pogay, equaiie aociae wih Čebyšev ucioa o Saigo acioa iegaio opeao, ega T ao a Spec Fuc Vo 22 No 9 Sep ( 2), [ 2 ] RK Saea, J Ra, J aiya, Geeaie acioa cacuu o geeaie ui-ie Miag- Lee ucio, Joua o ia Aca Mah Vo 34 No 2 (22), [ 3] R K Saea, J hoi, J aiya a Kua, Facioa ieeiaio o he pouc o appe ucio F3 a uivaiabe - ucio, ou K oea M ah S oc 3 ( 26), N o, p p 5 29 [ 4] M Sivaava, A ew aiy o he λ- Geeaie uwi-l ech Zea ucio wih a ppicaio, App Mah Sci 8, No 4, (24), [ 5] M Sivaava, S G abouy a A Baya, Epaio oua o a eee uwi- Lech ea ucio obaie via acioa cacuu, Avace i i eece E quaio 24, 24:69 [ 6] M Sivaava, R K Saea, T K Pogáy a R Saea, ega a copuaioa epeeaio o he eee uwi- Lech ea ucio e Ta a Spec Fucvo 22 No 7 ( 2),
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