pii-punaravrutti samikaran

ganitat pii-punaravrutti samikaran heey anukramanche eka resheey samikaran aahe jethe gunank anukram bahupadi mhanoon darshavile jaau shakatat. pii-punaravrutti samikarane hii bahupadi gunank asaleli rekheey punaravrutti samikarane (kinwa rekheey punaravrutti sambandh kinwa rekheey farak samikarane) aahet. hii samikarane ganitachya vividh kshetranmadhye, visheshtah sanyojanat mahattvachi bhoomika bajavtat. yaa samikarananche nirakaran ganana anukramanna holonomic, pii-recursive kinwa dii-finite mhanatat.
1980 chya dashakachya uttaradhapasun, yaa samikaranansathi upay shodhanyasathi pahile algoridam vikasit kele gele. sergei aaa. abramov, marco petkovsek aanee marqa vanne hoise yanni bahupadi, tarksangat, hypergiometric aanee dii 'alembertian upay shodhanyasathi algoridamche varnan kele.
vyakhya.
sootra 1 heey vaishishtyapurn shunyache kshetra asoo dya (udaharanarth sootra 2), sootra 4 sathee sootra 3 bahupadi, sootra 5 a kram aanee sootra 6 adnyat kram. samikaranrupi _ 7 laa bahupadi gunank asalele rekheey punaravrutti samikaran mhanatat (yaa lekhatil serve punaravrutti samikarane yaa swarupachi aahet). jarr sootra _ 8 aanee sootra _ 9 donhi shunyetar asateel, tarr sootra _ 10 laa samikaranacha kram mhanatat. jarr sootra _ 11 shoonya asel tarr samikaranala sajatiy mhanatat, anyatha tyala asamaroop mhanatat.
heey sootra _ 12 mhanoon dekhil lihile jaau shakate jethe sootra _ 13 heey bahupadi gunank asalele resheey punaravrutti operator aahe aanee sootra _ 14 heey shift operator aahe, i.eae. sootra _ 15.
bund swarupache upay.
sootra _ 16 kinwa samatulya sootra _ 17 heey bahupadi gunank asalele punaravrutti samikaran asoo dya. yaa samikaranachya upayanchi ganana karnare anek algoridam aahet. heey algoridam bahupadi, tarksangat, hypergiometric aanee dii 'alembertian upayanchi ganana karoo shakatat. eksandh samikaranache uttar resheey punaravrutti operatorchya karnaldware dile jaateh sootra _ 18. anukramanchya jaageche upasthan mhanoon yaa karnala aadhar aahe. sootra _ 19 haah sootra _ 20 chaa aadhar asoo dya, tarr aniyantrit sthirank sootra _ 22 sathee aupcharik berries sootra _ 21 laa sajatiy samasya sootra _ 23 chey samanya samadhan mhanatat. jarr sootra _ 24 heey sootra _ 17, i chey vishisht samadhan asel.eae. sootra _ 26, nantar sootra _ 27 heey dekhil eksandh samasyeche nirakaran aahe aanee tyala eksandh samasyeche samanya nirakaran mhanatat.
bahupadi upay.
1980 chya dashakachya uttaradhat sergei aaa. abramovne eka algoridamche varnan kele aahe jeey punaravrutti samikaranache samanya bahupadi samadhan shodhate, i.eae. sootra _ 28, bahupadi ujavya hatachya bajuchya formulas _ 29. tyane (aanee kahee varshannantar marco petkovsekne) bahupadi upayansathi eka degree dilee. asha prakare resheey samikarananchya pranalicha vichar karoon samasya sodavata yete. 1995 madhye abramov, bronstein aanee petkovsek yanni dakhavun dile qee punaravrutti samikaranachya shakti malikechya samadhanacha vishisht shaktichya aadharavar vichar karoon bahupadi prakaran adhik karyakshamatene sodavale jaau shakate (i.eae. samanya aadhar sootra nahee _ 30).
adhik samanya upay shodhanyasathi itar algoridam (udaa.jee. tarksangat kinwa atibhautikiy upay) dekhil bahupadi upayanchi ganana ganana algoridamvar avalambun asatat.
tarkshuddha upay.
1989 madhye sergei aaa. abramovne eka samanya tarksangat upay darshavila, i.eae. bahupadi ujavya bajuchya sutrasah sootra _ 31, sarvatrik vibhajakachi kalpana vaparoon shodhale jaau shakate. sarvatrik chhed heey bahupadi sootra aahe, jase qee pratyek tarksangat samadhanacha chhed sootra _ 33 vibhajit karato. abramovne heey dakhavun dile qee yaa sarvatrik vibhajakachi ganana keval pahile aanee shevatche gunank bahupadi sootra _ 8 aanee sootra _ 9 vaparoon kashee kelee jaau shakate. yaa sarvatrik chhedala sootra _ 37 chya adnyat chhedasathi pratisthapit karoon serve tarksangat upay rupantarit samikaranachya serve bahupadi upayanchi ganana karoon shodhale jaau shakatat.
hypergiometric solution.
jarr salag done sandnyanche gunottar sootra _ 39, i madheel tarksangat karya asel tarr anukram sootra _ 38 laa hypergiometric mhanatat.eae. sootra _ 40. heey asey aahe jarr aanee fakt jarr kram haah bahupadi gunankasah pahilya kramankachya punaravrutti samikaranacha upay asel. hypergiometric anukramancha sanch haah anukramanchya jaagecha upkshetra nahee kaaran too jodanikhali bund kelela nasato.
1992 madhye marco petkovsekne punaravrutti samikaranache samanya hypergiometric samadhan midvinyasathi eka algoridam deela jethe ujavya bajuche sootra _ 41 heey hypergiometric anukramanchi berries aahe. algoridam tarksangat karyachya gosper-petkovsek samanya-swarupacha vapar karato. yaa vishisht pratinidhitvasaha rupantarit samikaranachya bahupadi upayancha vichar karane punha purese aahe.
marqa vanne hoijamule eka vegala aanee adhik karyaksham drushtikon aahe. pahilya aanee shevatchya gunank bahupadi sootra _ 8 aanee sootra _ 9-jyala ekatmata mhanatat-yanchi mule vicharat ghet, pratyek hypergiometric anukram sootra _ 44 madhye formula _ 45 chey pratinidhitva aahe, kahee formulyasathi _ 46 madhye formula _ 47 formula _ 48 aanee formula _ 49 aahe. yethe formula _ 50 heey gamma function aanee formula _ 51 heey feeled formula _ 1 chey bijganitiy samapan darshavate. mug formula _ 53 heey samikaranache ekatmata asane aavashyak aahe (i.eae. sootra _ 8 kinwa sootra _ 9) chee mule. yavyatirikt, ghatak sootra _ 56 sathee maryada mojata yetat. nishchit mulyanchya sootra _ 57 sathee, eka unsatz tayar karane shakya aahe jeey sootra _ 58 sathee umedvar dete. eka vishisht sootra _ 58 sathee, abramovchya algoridamdware tarksangat karya sootra _ 60 midvinyasathi punha eka unsatz tayar karata yete. serve shakyata vicharat ghetalyas punaravrutti samikaranache samanya uttar milate.
dii 'alembertian upay.
kahee hypergiometric anukram sutrasathi sootra _ 62 asalyas anukram sootra _ 37 laa dii 'alembertian mhanatat sootra _ 63 aanee sootra _ 64 mhanje sootra _ 65 jethe sootra _ 66 farak operator darshavite, i.eae. sootra _ 67. heey asey aahe jarr aanee fakt jarr pahilya kramankache rekheey punaravrutti operator sootra _ 68 tarksangat gunankasah asel tarr sootra _ 69.
1994 madhye abramov aanee petkovsek yanni eka algoridamche varnan kele jeey punaravrutti samikaranachya samanya dii 'alembertian samadhanachi ganana karate. haah algoridam hypergiometric solutionschi ganana karato aanee punaravrutti samikaranacha kram punaravrutti mhanoon qamee karato.
udaharane.
swakshari kelele kramparivartan matricks.
aakar sootra _ 70 chya swakshari kelelya kramparivartan matrixchi sankhya anukram sootra _ 71 dware varnan kelee jaau shakate. swakshari kelele kramparivartan matricks haah eka chauras matricks aahe jyachi pratyek ranget aanee pratyek stambhat achookpane eka shunyetar nond aahe. shunyetar nondi sootra _ 72 asoo shakatat. kram bahupadi gunank formula _ 73 saha rekheey punaravrutti samikaranadvare aanee prarambhik mulyanche sootra _ 74 dware nirdharit keela jaato. hypergiometric upay shodhanyasathi algoridam lagoo karoon ekhadi vyakti kahee sthir sootra _ 76 sathee samanya hypergiometric solution formula _ 75 shodhu shakate. prarambhik mulyancha vichar karoon, anukram sootra _ 77 swakshari kelelya kramparivartan matrixchya sankhyeche varnan karate.
hastakshep.
sootra 79 ghatkansah sanchachya involution sutranchi sankhya _ 38 punaravrutti samikaran formula _ 80 dware dilee jaate, udaharanarth petkovsekchya algoridammadhe heey pahane shakya aahe qee yaa punaravrutti samikaranasathi konatehi bahupadi, tarksangat kinwa hypergiometric samadhan nahee.
anuprayog.
sootra _ 81 laa hypergiometric mhanatat jarr sootra _ 82 jethe sootra _ 83 heey sootra _ 79 aanee sootra _ 85 madheel tarksangat karye darshavate. hypergiometric berries hii formula _ 86 chee maryadit berries aahe jithe sootra _ 81 hypergiometric aahe. zeelbergercha creative telescope algoridam asha hypergiometric berijache bahupadi gunankansah punaravrutti samikaranat rupantar karoo shakato. heey samikaran nantar sodavale jaau shakate, udaharanarth hypergiometric solutionsche resheey sanyojan, jyala formula _ 11 chey closed form solution mhanatat.

