F(F \m 1,m 2, ), which is suitable for large, small, r N and 0<q<1, where. is the incomplete Gamma function ratio and
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1 A Asyptotic Expasio fo th No-tal -Distibtio y Jia azah ahoo Dpatt of Mathatics ollg of Ecatio 6 Abstact A asyptotic xpasio is iv fo th o-ctal -istibtio (\ ) hich is sitabl fo lag sall N a << h is th icoplt Gaa fctio atio a. This fo has so avatags ov pvios asyptotic xpasios i this gio of th paat spac i hich ps o all th paats a. Th avatag of this xpasio is that a algoith bas o it ca b o asily t fo paticla accacy its a fo paticla paat ags. 6 (\ ) << N
2 . toctio Th o-ctal -istibtio \ is fi by (y 959; Wal ) as th fo. f X a X a ipt ao vaiabls a X is a o ctal chisa istibtio ith gs of fo a o ctality paat a X is a ctal chi-sa istibtio ith gs of fo th th vaiabl X is sai to hav a o-ctal -Distibtio ith X gs of fo (positiv itgs) a o-ctal paat a it. Th istibtio fctio is giv by \ () h is icoplt ta fctio. \ t t t hich is th clativ istibtio fctio (c..f.) of o-ctal -Distibtio Th asyptotic xpasio as sti by oth sachs ho o i o fil hich as th folloig. Th asyptotic xpasio fo th atio of to Gaa fctios iv by (ils 966; L 969; z 987). A spcial cas of th asyptotic xpasio fo a atio of pocts of gaa fctios iv by (iihig ). galiz a fola hich as stat by (Digl 973) fist pov by (Pais 99) a ctly cosi by (Oliv 995). Th spcial fctios a thi appoxiatios ha b sti by (L 969). Th icoplt laplac itgals: Uifo asyptotic xpasio ith applicatio to th icoplt bta fctio sti by (T 987). Asyptotic xpasios of th officits i asyptotic sis soltios of lia ifftial atios Mthos a applicatios of aalysis iv by (Olv 994). Th Uifo asyptotic xpasios of itgals sti by (T 995) by sig xapls of stiltjis o o asyptotic of spcial fctios. A Uifo
3 3 asyptotic xpasio fo th Jacobi polyoials ith xplicit ai iv by (Wog & Zhag 996). Th vali asyptotic xpasios fo th axi lilihoo stiato of th paat of a statioay Gassia stogly pt pocss sti by (Liba Rossa & Zc 3). A ifo asyptotic xpasios fo icoplt Ria zta fctios iv by (Dst 4). Th ifo asyptotic xpasios fo hypogotic fctios ith lag paats sti by (Daalhis 5).. Divatio of a Asyptotic Expasio fo th No tal - Distibtio. W iv a asyptotic xpasio of \ thogh to stags: ist Stag: W shall iv th asyptotic xpasio of h > a N stat fo th ta fctio hich has th fola. t t t. () Th by sig th sbstittio t= - a t=- - obtai. (3) A sig th fact that Sih hav Sih (4) h. No xpa Sih i pos of as h Sih (5)
4 4 Lt h. Th last ality follos by (Dioat & Mois99). Wh a th xpasio cofficits of Sih a hich ca b xpss of th galiz olli polyoials (L969).y sbstittio atio (5) i (4) gt. A sig Watso s La obtai th asyptotic xpasio. (6) Sco Stag: this stag iv th asyptotic xpasio of t t t (7) h > < < a N a th tasfo th xpssio fo it as th sa i atio () a chagig itgatio ts to obtai. Sih (8) h as bfo by sig (5) hav. (9)
5 5 Lt = th a so fo (9) hav () h (..) is icoplt gaa fctio atio. W ca poc by sig th cc latios fo (..) to xpss i ts of. This givs R. () h hav s th fola (6) to cacl ot th factos ltiplyig th oth t R is a obl satio ov a sial ts obtai by xpssig i ts of. To obtai th asyptotic xpasio i to oig this s. ist it (8) i th fo Sih. () tgat th fist itgal by pats tic as follos : Sih Sih
6 6 Sih Sih. (3) th itgal i (3) o sbtact th sco t i xpasio of Sih a a a cospoig itgal so that th itgal i (3) bcos Sih.(4) Th fist of ths itgals is th itgat by pats tic pocig to fth itgat ts valat at a a itgal of a foth ivativ. this itgal a fth t fo th xpasio of Sih 3 is sbtact fo th ifftiat pat a a cospoig itgal a o spaatly. This poc is coti ifiitly. Th spaat itgals statig fo th os o th ight of () a (4) a togth to giv as i () so that (5) h Sih = h i th satio is to b itpt as lagst itg as i itg ivisio. Th atitis satisfy th sipl cc fola. (6) W ca xpss ictly i ts of a fo xapl. ov fo clos to valatio
7 7 i this ay ca la to lag oig os o sbtactio a so is btt valat fo its po sis xpasio i. No h sbstitt th fola (5) i atio () gt \. y sig th itity gt \. (7) hich is a asyptotic xpasio fo th o-ctal -istibtio.
8 Rfcs iihig W () A asyptotic xpasios fo a atio of pocts of gaa fctios. Physialischs stitt Uivsitat ilby Philosophg. 69/ ilbg Gay. 3.pf. Dioato A R & Mois A (99) Sigificat igit coptatio of th icoplt bta fctio atios. AM Tas. Math. Softa 8: Digl R (973) Asyptotic xpasios: thi ivatio a itptatio (Acaic pss. Loo). Dst T M (4) Uifo asyptotic xpasios fo icoplt ia zta fctios. Dpatt of Mathatics a Statistics stat UivsityU.S. A. pblicatios ils J L (966) Aot o th asyptotic xpasio of th atio of to gaa fctios. Poc. Eibgh Math. Soc.5: MR 34:379. z L (987) Eo bos fo asyptotic xpasios of th atio of to gaa fctios. SAM J. Math. Aal 8: MR 88:33. y S (959) Th aalysis of vaiac. Joh ily a sos c. N Yo. Liba O Rossa J & Zc D M (3) Vali asyptotic xpasios fo th axi lilihoo stiato of th paat of a statioay. Gassia Stogly Dpt Pocss. L Y L (969) Th spcial fctios a thi appoxiatios. Vol. Acaic Pss N Yo MR 39:339. Ol Daalhis A (5) Uifo asyptotic xpasios fo hypgotic fctios ith lag paats. Olv W J (994) Asyptotic xpasios of th cofficits i asyptotic sis soltios of lia ifftial atios Mthos a Applicatios of Aalysis : -3. Olv W J (995) O asyptotic xpasio of a atio of gaa fctios. Poc. Royal ish Aca. A95: 5-9. Pais R (99) Soothig of th stos phoo sig llibas itgals. J. opt Appl. Math. 4: T N M (987) coplt laplac itgals: ifo asyptotic xpasio ith applicatio to th icoplt bta fctio. SAM J. Math. Aal. 8: MR 89f: 436. T N M (995) Uifo asyptotic xpasios of itgals:a slctio of pobls. Wal () a-oo o statistical istibtio fo xpitalist lists. Uivsity of Stochol. Wog R & Zhag J M (996) Aifo asyptotic xpasio fo th Jacobi polyoials ith xplicit ai. Appl. Aal. 6:
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