IWANNH D. STAMPOLA MAJHMATIKOU MELETH TWN RIZWN TWN ASSOCIATED ORJOGWNIWN q-poluwnumwn DIDAKTORIKH DIATRIBH TMHMA MAJHMATIKWN SQOLH JETIKWN EPISTHMWN PANEPISTHMIO PATRWN PATRA 2004
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EuqaristÐec Aisjˆnomai thn anˆgkh na ekfrˆsw thn eugnwmosônh mou kai tic jermèc euqaristðec mou ston epiblèponta thc paroôshc didaktorik c diatrib c kajhght k. P. D. SiafarÐka gia thn upìdeixh tou jèmatoc aut c, thn ousiastik kai suneq kajod ghs tou kaj' ìlh th diˆrkeia ekpìnhshc kai suggraf c thc, kaj c kai gia tic polôtimec sumboulèc kai upodeðxeic tou. EpÐshc, ja jela na euqarist sw ta ˆlla dôo mèlh thc trimeloôc sumbouleutik c epitrop c, ton kajhght k. E. K. Ufant kai thn epðkouro kajhg tria k. Q. G. Kokologiannˆkh, gia tic eôstoqec parathr seic touc se diˆfora episthmonikˆ jèmata kaj c kai gia thn parakoloôjhsh thc ergasðac mou kai thn ousiastik sumbol touc sthn ekpìnhs thc. Akìmh, ja jela na euqarist sw th lèktora k. E. N. PetropoÔlou gia thn ˆristh sunergasða kai gia tic qr simec kai ousiastikèc sumboulèc thc katˆ th diˆrkeia suggraf c thc didaktorik c mou diatrib c. Tèloc, ja jela na euqarist sw to 'Idruma Kratik n Upotrofi n (I.K.U.) gia thn oikonomik upost rixh pou mou pareðqe katˆ th diˆrkeia ekpìnhshc thc paroôshc diatrib c.
Perieqìmena Eisagwg. Istorik anadrom kai sôgqrona probl mata twn orjogwnðwn poluwnômwn................................2 Efarmogèc twn orjogwnðwn poluwnômwn.............. 7.3 Mèjodoi gia th melèth twn riz n twn associated orjogwnðwn q- poluwnômwn.............................. 0.4 Skopìc kai diˆrjrwsh thc paroôshc diatrib c........... 2 Sunarthsiak Analutik Mèjodoc gia th melèth twn riz n orjogwnðwn poluwnômwn 5 2. Eisagwg............................... 5 2.2 Melèth thc monotonðac twn riz n kai sunart sewn pou perièqoun tic rðzec twn associated orjogwnðwn q-poluwnômwn......... 7 2.2. MonotonÐa thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza twn associated orjogwnðwn q- poluwnômwn.......................... 7 2.2.2 MonotonÐa thc mikrìterhc rðzac kai sunart sewn pou perièqoun th mikrìterh rðza twn associated orjogwnðwn q- poluwnômwn.......................... 8 2.3 Melèth thc kurtìthta thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza associated orjogwnðwn q-po-luwnômwn 22 2.4 AjroÐsmata Newton twn riz n twn associated orjogwnðwn q-poluwnômwn 23 3 MonotonÐa thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza twn associated orjogwnðwn q-poluwnômwn 25 3. Eisagwg............................... 25 3.2 Apotelèsmata............................. 27 3.2. Associated q-pollaczek polu numa.............. 27 3.2.2 Associated continuous q-jacobi polu numa......... 35 i
ii PERIEQŸOMENA 3.2.3 Associated q-laguerre polu numa.............. 39 3.2.4 Associated Al-Salam-Carlitz II polu numa......... 4 3.2.5 Associated q-meixner polu numa.............. 42 3.2.6 q-lommel polu numa..................... 44 3.3 ApodeÐxeic............................... 46 3.3. Associated q-pollaczek polu numa.............. 46 3.3.2 Associated continuous q-jacobi polu numa......... 6 3.3.3 Associated q-laguerre polu numa.............. 66 3.3.4 Associated Al-Salam-Carlitz II polu numa......... 68 3.3.5 Associated q-meixner polu numa.............. 69 3.3.6 q-lommel polu numa..................... 73 4 MonotonÐa thc mikrìterhc rðzac kai sunart sewn pou perièqoun th mikrìterh rðza twn associated orjogwnðwn q-poluwnômwn 77 4. Eisagwg............................... 77 4.2 Apotelèsmata............................. 78 4.2. Associated Askey-Wilson polu numa............ 78 4.2.2 Associated continuous q-jacobi polu numa......... 86 4.2.3 Associated continuous q-laguerre polu numa........ 9 4.2.4 Associated big q-jacobi polu numa............. 93 4.2.5 Associated big q-laguerre polu numa............ 97 4.2.6 Associated big q-legendre polu numa............ 98 4.3 ApodeÐxeic............................... 00 4.3. Associated Askey-Wilson polu numa............ 00 4.3.2 Associated continuous big q-hermite polu numa...... 08 4.3.3 Associated continuous q-jacobi polu numa......... 09 4.3.4 Associated big q-jacobi polu numa............. 3 5 Kurtìthta thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza associated orjogwnðwn q-poluwnômwn 2 5. Eisagwg............................... 2 5.2 Apotelèsmata............................. 22 5.3 ApodeÐxeic............................... 23 6 AjroÐsmata Newton twn riz n twn associated orjogwnðwn q- poluwnômwn 27 6. Eisagwg............................... 27 6.2 KÔria apotelèsmata.......................... 28 6.2. Associated q-laguerre polu numa.............. 28
PERIEQŸOMENA iii 6.2.2 Associated q-charlier polu numa.............. 3 6.2.3 Associated continuous q-ultraspherical polu numa..... 32 6.2.4 q-lommel polu numa..................... 34 6.2.5 Associated q-meixner polu numa.............. 36 6.2.6 Associated Al-Salam Carlitz I polu numa.......... 4 Synopsis 43 BibliografÐa 47
Kefˆlaio Eisagwg. Istorik anadrom kai sôgqrona probl mata twn orjogwnðwn poluwnômwn H istorða twn orjogwnðwn poluwnômwn arqðzei ton 8o ai na ìtan o Andrien- Marie Legendre (752-833) eis gage ta gnwstˆ s mera wc polu numa Legendre P n (x) sthn ergasða tou Recherches sur l attraction des spheroides homogenes to 783, ìpou kai upolìgise tic suntetagmènec thc dônamhc se sust mata peristrof c stic treic diastˆseic. EpÐshc, o Legendre parousðase kai ta proshrthmèna polu numa Legendre P n (m) (x) = ( x 2 ) m/2 d m P n(x) ta opoða apoteloôn dx m lôseic thc exðswshc Laplace se sfairikèc suntetagmènec me th mèjodo qwrismoô twn metablht n. Mia deôterh oikogèneia orjogwnðwn poluwnômwn eis qjh apì ton Charles Hermite (822-90) sthn 5 ìpou kai prospˆjhse na brei ta anaptôgmata pragmatik n sunart sewn se seirèc poluwnômwn. H epìmenh, me seirˆ emfˆnishc, oikogèneia orjogwnðwn poluwnômwn gnwstˆ wc polu numa Laguerre L a n, ofeðlei to ìnoma thc ston Edmond Nicolás Laguerre (834-886) o opoðoc ta eis gage sthn 84 katˆ th melèth tou oloklhr matoc e x x dx. Ta polu numa Hermite kai Legendre eis qjhsan sth fusik me thn exðswsh tou Schrödinger wc aktinikèc lôseic gia to dunamikì tou Coulomb kai ton isotropikì armonikì talantwt. To 859 o Karl Gustav Jacob Jacobi (804-85), sthn 69 xekin ntac apì thn upergewmetrik sunˆrthsh tou Gauss parousðase ta gnwstˆ s mera wc po-
2 KEFALAIO. EISAGWGH lu numa Jacobi P (a,β) n (x) = Γ(n + a + ) Γ(a + )n! 2F ( n, n + a + b + a + ) x. 2 Eidikèc peript seic twn poluwnômwn Jacobi apoteloôn ta polu numa Ultraspherical Gegenbauer, ta polu numa Legendre kaj c kai ta polu numa Chebyshev pr tou kai deutèrou eðdouc. Ta polu numa Laguerre kai Hermite mporoôn epðshc na grafoôn me th qr sh genikeumènwn upergewmetrik n seir n p F q. Ta parapˆnw polu numa pou eðnai ta gnwstˆ s mera klasikˆ orjog nia polu numa apoteloôn èna shmantikì mèroc thc oikogèneiac twn eidik n sunart - sewn thc Majhmatik c Fusik c kai eðnai lôseic miac diaforik c exðswshc Sturm -Liouville deôterhc tˆxhc upergewmetrikoô tôpou σ(x)y (x) + τ(x)y (x) + λy(x) = 0, (.) ìpou σ(x) kai τ(x) eðnai polu numa to polô deutèrou kai pr tou bajmoô antðstoiqa kai λ eðnai mia stajerˆ, h opoða emfanðzetai katˆ th majhmatikopoðhsh problhmˆtwn thc atomik c kai purhnik c fusik c, thc hlektrodunamik c kai thc akoustik c, 30, 98. Ta klasikˆ orjog nia polu numa qarakthrðzontai apì orismènec koinèc idiìthtec, oi kuriìterec apì tic opoðec eðnai oi akìloujec:. EÐnai orjog nia sto diˆsthma a, b wc proc mia sunˆrthsh bˆrouc ρ(x), dhlad b a ρ(x)p n(x)p m (x)dx = 0, n, m = 0,, 2,..., n m. H sunˆrthsh ρ : a, b R + eðnai lôsh thc diaforik c exðswshc Pearson ρ(x)σ(x) = τ(x)ρ(x). 2. MporoÔn na ekfrasjoôn mèsw tou akìloujou tôpou tou Rodrigues d n P n (x) = B n ρ(x) dx n ρ(x)σn (x), n = 0,, 2,..., ìpou ρ(x) eðnai jetik sunˆrthsh se kˆpoio diˆsthma kai σ(x) eðnai polu numo anexˆrthto tou bajmoô n. 3. IkanopoioÔn mia diaforik exðswsh thc morf c σ(x) d2 dx 2 P n(x) + τ(x) d dx P n(x) + λ n P n (x) = 0,
.. ISTORIKH ANADROMH, PROBLHMATA 3 ìpou σ(x) kai τ(x) eðnai polu numa to polô deutèrou kai akrib c pr tou bajmoô antðstoiqa kai λ n stajerˆ. 4. IkanopoioÔn mia anadromik sqèsh tri n ìrwn thc morf c xp n (x) = α n P n+ (x) + β n P n (x) + γ n P n (x) P (x) = 0, P 0 (x) =. 5. H akoloujða twn twn parag gwn twn orjogwnðwn poluwnômwn P n (x) eðnai epðshc mia akoloujða orjogwnðwn poluwnômwn èstw {P n(x)} n=0. H idiìthta aut apodeðqjhke apì ton Sonin to 887. To 935 h idiìthta aut apodeðqjhke kai apì ton Hahn 44 o opoðoc sumperièlabe epiplèon kai ta polu numa Bessel en to 937 45 o Ðdioc apèdeixe ìti ta klasikˆ orjog nia polu numa eðnai ta mìna twn opoðwn oi parˆgwgoi opoiasd pote tˆxhc {P (k) n (x)} n=0, k apoteloôn epðshc akoloujða orjogwnðwn poluwnômwn. H jewrða twn q-analìgwn q-epektˆsewn klasik n sunart sewn kai sqèsewn q a basðzetai sthn apl parat rhsh ìti lim = a. Gia to lìgo autì (kai exaitðac thc {bˆshc} q) o arijmìc a = lim onomˆzetai basikìc arijmìc a. q q q a q q Basizìmenoc sthn parapˆnw parat rhsh to 846-847 o Edward Heine (82-889) stic ergasðec tou 47, 48 parousðase mia genðkeush thc upergewmetrik c ( α, β sunˆrthshc tou Gauss 2 F γ ), x thn ( ) α, β 2φ ; q γ z (α; q) n (β; q) n = z n, (γ; q) n (q; q) n ìpou (α; q) n = ( q)( αq) ( α n q) (gia perissìterec plhroforðec pˆnw n=0 stic basikèc upergewmetrikèc seirèc bl. 39). To 884 o Markov 87 parousðase mia pr th oikogèneia orjogwnðwn q-poluwnômwn ta opoða eðnai orjog nia wc proc mia katanom da(x), ìpou h a(x) eðnai mia bhmatik sunˆrthsh me metabol j(x) = q x, sta shmeða q x, x =, 2,.... EpÐshc, o L. T. Rogers douleôontac pˆnw se metasqhmatismoôc twn q-upergewmetrik n seir n 05-08 parousðase ta gnwstˆ s mera wc continuous q-hermite polu- numa kaj c kai ta continuous q-ultraspherical polu numa ta opoða apoteloôn q-anˆloga twn poluwnômwn Hermite kai Ultraspherical antðstoiqa.
4 KEFALAIO. EISAGWGH Mia pr th susthmatik melèth pˆnw sta orjog nia q-polu numa, ègine apì ton W.Hahn to 949 46. Se analogða me ta klasikˆ orjog nia polu numa eis gage ton telest L q,w f(x) = f(qx + w) f(x) (q )x + w, q, w R+, kai anaz thse akoloujðec poluwnômwn {P n (x)} n=0 tètoiec ste: na eðnai epðshc akoloujða orjogwnðwn poluw-. H akoloujða {L q,w P n (x)} n=0 nômwn. 2. Ta polu numa L q,w P n (x) na ikanopoioôn mia exðswsh diafor n thc morf c σ(x)l 2 q,wp n (x) + τ(x)l q,w P n (x) + λ n P n (x) = 0, n 0, (.2) ìpou σ(x) kai τ(x) polu numa to polô deutèrou kai akrib c pr tou bajmoô antðstoiqa. 3. Ta polu numa P n (x) na mporoôn na ekfrasjoôn mèsw thc sqèshc ρ(x)p n (x) = L n q,wx 0 (x)x (x) X n (x)ρ(x), ìpou X 0 (x) polu numo anexˆrthto tou n, X i+ (x) = X i (qx + w) kai ρ(x) sunˆrthsh anexˆrthth tou n. 4. Oi ropèc µ n = xn da(x) twn P n (x) na ikanopoioôn thn anadromik sqèsh µ n = a + bqn c + dq n µ n, ad bc 0. Sth sunèqeia to endiafèron pˆnw sta orjog nia q-polu numa ˆrqise na exasjeneð mèqri to 985 pou oi Andrews kai Askey sunèqisan thn èreuna pˆnw sta orjog nia q-polu numa douleôontac me q-upergewmetrikèc seirèc 7. Sugkekrimèna èdeixan ìti ìla ta klasikˆ orjog nia polu numa prokôptoun, me katˆllhlouc metasqhmatismoôc twn paramètrwn kai lambˆnontac to ìrio q, apì ta q-racah ta Askey-Wilson polu numa (pou eis qjhsan stic 8, 23), ta opoða orðzontai me th qr sh twn 4 φ 3 q-upergewmetrik n seir n. apoteloôn to apokaloômeno sq ma Askey (bl. 76). Ta polu numa autˆ Mia diaforetik prosèggish sto jèma twn orjogwnðwn q-poluwnômwn ègine apì touc Nikiforov kai Uvarov to 983 95 oi opoðoi basðsthkan sto gegonìc ìti ta
.. ISTORIKH ANADROMH, PROBLHMATA 5 q-polu numa ikanopoioôn mia exðswsh q-diafor n deôterhc tˆxhc ìpwc h (.2) (bl. epðshc 96). H prosèggish aut twn Nikiforov kai Uvarov katal gei epðshc sthn upergewmetrik anaparˆstash twn orjogwnðwn poluwnômwn. Sthn 0 gðnetai prospˆjeia enopoðhshc twn dôo sqhmˆtwn (gia perissìterec plhroforðec epð twn orjogwnðwn q-poluwnômwn bl. 4, 39, 96). Se pollèc peript seic sth fusik gðnetai qr sh posot twn pou orðzontai se èna diakritì sônolo. Oi posìthtec autèc eðnai dunatìn na perigrafoôn apì e- xis seic diafor n se diafìrwn eid n plègmata oi opoðec apoteloôn prosèggish thc (.) 96. 'Otan h prosèggish gðnetai se èna omoiìmorfo plègma, prokôptei mia exðswsh diafor n, lôseic thc opoðac eðnai ta orjog nia polu numa diakrit c metablht c (polu numa Hahn, Meixner, Kravchuk kai Charlier), en ta orjog nia q-polu numa prokôptoun wc lôseic exis sewn diafor n se mh omoiìmorfa plègmata. Ta orjog nia q-polu numa, ìpwc kai ta klasikˆ orjog nia polu numa, ikanopoioôn mia anadromik sqèsh tri n ìrwn thc morf c xp n (x; ν q) = α n (ν q)p n+ (x, ν q) + β n (ν q)p n (x; ν q) + γ n (ν q)p n (x; ν q), n = 0,,..., 0 < q <, P (x; ν q) = 0, P 0 (x; ν q) =. (.3) Ta associated orjog nia (q-)polu numa apoteloôn mia endiafèrousa genðkeush twn orjogwnðwn (q-)poluwnômwn kai orðzontai ap' thn anadromik sqèsh (.3) an antikatast soume touc suntelestèc α n, β n kai γ n me α n+r, β n+r kai γ n+r antðstoiqa, ìpou r > 0 r >. Gia efarmogèc aut n twn poluwnômwn bl. parˆgrafo.2. Lìgw twn poll n efarmog n touc se diˆforouc tomeðc twn majhmatik n kai thc fusik c, ta orjog nia polu numa, ta orjog nia q-polu numa kaj c kai ta associated orjog nia q-polu numa prosèlkusan to endiafèron poll n ereunht n. Shmantikì mèroc thc èreunac pˆnw sta anwtèrw afier jhke stic rðzec aut n exaitðac tou rìlou pou èqoun se diˆfora probl mata thc fusik c. IdiaÐtero endiafèron parousiˆzei h melèth thc sumperiforˆc twn riz n twn anwtèrw kathgori n orjogwnðwn poluwnômwn wc sunart sewn miac paramètrou. Gia to skopì autì ektìc apì th melèth thc monotonðac kai thc kurtìthtac twn riz n eðnai aparaðthth kai h melèth sunart sewn pou tic perièqoun lambˆnontac ètsi èna mètro gia to rujmì metabol c twn riz n kaj c kai frˆgmata aut n.
6 KEFALAIO. EISAGWGH Sthn kateôjunsh aut, kai ìson aforˆ sth monotonða, eðnai shmantik h eôresh miac katˆllhlhc jetik c kai diaforðsimhc sunˆrthshc (thc bèltisthc a- n eðnai dunatìn) f(a), tètoiac ste ta ginìmena f(a)ρ n,k (a), k n na èqoun thn antðjeth monotonða apì tic rðzec ρ n,k (a) twn upì melèth orjogwnðwn poluwnômwn, lambˆnontac ètsi èna mètro gia to rujmì metabol c touc. prìblhma autì diereun jhke gia ta klasikˆ orjog nia polu numa stic ergasðec 8, 82, 5, 60, 63, 55, 22, 29, 24, 89. To To prìblhma thc kurtìthtac twn riz n twn klasik n orjogwnðwn poluwnômwn èqei diereunhjeð stic ergasðec 23, 25, 27, 28, 78, 83, 26. Tèloc idiaðtero endiafèron parousiˆzei o upologismìc twn rop n thc katanom c twn riz n twn orjogwnðwn poluwnômwn 6. Gia na gðnei autì eðnai shmantikìc o upologismìc twn rop n gôrw apì thn arq thc puknìthtac twn riz n twn poluwnômwn 3. 'Oson aforˆ sta klasikˆ orjog nia polu numa, eðnai gnwstì ìti ikanopoioôn mia diaforik exðswsh deôterhc tˆxhc en ta antðstoiqa associated kai ta co-recursive associated orjog nia polu numa ikanopoioôn mia diaforik exðswsh tètarthc tˆxhc 09, 25. H idiìthtˆ touc aut qrhsimopoi jhke stic ergasðec 4, 5, 6, 42, 90, 9, 04, 09, 25, 26 gia ton upologismì twn rop n thc katanom c twn riz n twn poluwnômwn aut n mèsw twn suntelest n thc diaforik c exðswshc pou ikanopoioôn. Epiplèon, qrhsimopoi ntac diaforetikèc mejìdouc, o upologismìc twn rop n thc katanom c twn riz n isodônama twn ajroismˆtwn Newton twn riz n twn klasik n orjogwnðwn poluwnômwn èqei gðnei stic ergasðec 3, 5, 6, 20, 40, 4, 68, 85, 9, 04, twn hmiklasik n orjogwnðwn poluwnômwn stic 90, 24, twn associated kai twn co-recursive associated orjogwnðwn poluwnômwn stic 68, 92, 09, twn scaled co-recursive associated orjogwnðwn poluwnômwn stic 52, 93, twn quasiorthogonal poluwnômwn thc klasik c kathgorðac (classical class) sthn 26 kai twn relativistic poluwnômwn sthn 94. Se antðjesh me tic rðzec twn klasik n orjogwnðwn poluwnômwn polô lðga eðnai gnwstˆ gia tic rðzec twn associated orjogwnðwn q-poluwnômwn. Kˆpoia apotelèsmata sqetikˆ me th monotonða twn riz n twn poluwnômwn Al-Salam-Carlitz dðnontai sthn 65 kai gia tic rðzec twn continuous q-jacobi poluwnômwn sthn 9. 'Oson aforˆ sthn katanom twn riz n touc, sthn 2 meletˆtai h diakrit puknìthta twn riz n (dhlad o arijmìc twn riz n anˆ monˆda diast matoc) kaj c kai to asumptwtikì thc ìrio, miac oikogèneiac orjogwnðwn q-poluwnômwn h opoða perilambˆnei wc eidikèc peript seic ta perissìtera apì ta gnwstˆ orjog -
.2. EFARMOGŸES TWN ORJOGWNŸIWN POLUWNŸUMWN 7 nia q-polu numa. Sthn paroôsa diatrib xekin ntac apì thn anadromik sqèsh pou ikanopoioôn ta associated orjog nia q-polu numa kai qrhsimopoi ntac mia sunarthsiak analutik mèjodo, dðnontai apotelèsmata sqetikˆ me th monotonða, thn kurtìthta kai ta ajroðsmata Newton twn riz n orismènwn oikogenei n associated orjogwnðwn q-poluwnômwn. Apì autˆ lambˆnontac to ìrio q kai me katˆllhlouc metasqhmatismoôc twn paramètrwn prokôptoun antðstoiqa apotelèsmata gia tic rðzec twn associated kai twn klasik n orjogwnðwn poluwnômwn. Ta apotelèsmata autˆ genikeôoun, enopoioôn kai belti noun prohgoômena apotelèsmata..2 Efarmogèc twn orjogwnðwn poluwnômwn Ta teleutaða qrìnia upˆrqei auxhmèno endiafèron sqetikˆ me ta orjog nia polu numa. Upˆrqoun dôo lìgoi gia thn tˆsh aut. O ènac eðnai oi efarmogèc touc se tomeðc thc fusik c kai twn majhmatik n ìpwc h kbantomhqanik, h fusik stere n swmˆtwn, h jewrða pijanot twn, h majhmatik statistik, ta suneq klˆsmata, oi proseggðseic Padé k.a. O deôteroc eðnai h megˆlh diˆdosh twn hlektronik n upologist n kai h qr sh twn orjogwnðwn poluwnômwn sthn arijmhtik anˆlush oôtwc ste na epilegoôn axiìpistec kai {oikonomikèc} arijmhtikèc mèjodoi. Mia shmantik efarmog twn riz n twn orjogwnðwn poluwnômwn eðnai h hlektrostatik ermhneða touc. O Stieltjes stic 4, 5, 6 apèdeixe ìti oi rðzec twn poluwnômwn Jacobi, Laguerre kai Hermite apoteloôn ta shmeða isorropðac n eleôjerwn monadiaðwn fortðwn se èna hlektrikì pedðo pou parˆgetai apì dôo stajerˆ fortða topojethmèna sta ˆkra tou diast matoc orjogwniìthtac. Pio sugkekrimèna, an jewr soume dôo fortða q, p sta shmeða kai + antðstoiqa kai n eleôjera monadiaða fortða, n 2, ta opoða apwjoôntai metaxô touc sômfwna me to nìmo tou logarijmikoô dunamikoô, sta shmeða x, x 2,..., x n, h enèrgeia tou sust matoc dðnetai apì th sqèsh E = ln(t ) ìpou T (x, x 2,..., x n ) = n ( x k ) p ( + x k ) q k= ν,µ=,...,n ν<µ x ν x µ. O Stieltjes èdeixe ìti h enèrgeia E tou sust matoc elaqistopoieðtai ìtan ta {x ν } n ν= eðnai oi rðzec twn poluwnômwn Jacobi P (a,b) n (x) ìpou a = 2p kai b = 2q.
8 KEFALAIO. EISAGWGH AntÐstoiqa, an jewr soume èna jetikì fortðo p sto 0 kai n monadiaða fortða n 2 sta shmeða x, x 2,..., x n sto diˆsthma 0, +, tètoia ste na ikanopoieðtai h sunj kh n (x + x 2 +... + x n ) K, ìpou K eðnai ènac prokajorismènoc jetikìc arijmìc, tìte h sunolik enèrgeia tou sust matoc dônetai apì th sqèsh E = ln(u ) ìpou U(x, x 2,..., x n ) = n x p k k= ν,µ=,...n ν<µ x ν x µ, kai elaqistopoieðtai ìtan ta {x ν } n ν= eðnai oi rðzec twn poluwnômwn Laguerre (cx) ìpou a = 2p kai c = K (n + a). L (a) n Epiplèon o Szegö 7, sel. 40 èdeixe ìti stic parapˆnw peript seic to shmeðo isorropðac eðnai monadikì pou shmaðnei ìti h isorropða eðnai eustaj c. Prìsfata oi Hendriksen kai van Rossum 50 kai oi Forrester kai Rogers 36 melèthsan epektˆseic thc hlektrostatik ermhneðac tou Stieltjes stic migadikèc rðzec orismènwn orjogwnðwn poluwnômwn kai stic rðzec sto monadiaðo kôklo. EpÐshc o Grünbaum 80 èdwse mia hlektrostatik ermhneða twn riz n ton poluwnômwn Koornwinder-Krall en sthn 62 o Ismail parousðase èna hlektrostatikì montèlo gia tic rðzec twn genik n (general) orjogwnðwn poluwnômwn. Apì fusik c apìyewc eðnai eôkolo na diapistwjeð ìti oi rðzec twn poluwnômwn Jacobi fjðnoun wc proc a kai auxˆnoun wc proc b afoô ìtan auxˆnei to a ta eleôjera monadiaða fortða apwjoôntai proc ta aristerˆ en ìtan auxˆnei to b apwjoôntai proc ta dexiˆ. Genikìtera ektìc apì thn kateôjunsh me thn opoða metakinoôntai ta eleôjera monadiaða fortða ìtan metabˆllontai ta stajerˆ fortða pou isodunameð me th melèth thc monotonðac twn riz n twn orjogwnðwn poluwnômwn, endiafèron parousiˆzei h eôresh enìc mètrou gia thn taqôthta metakðnhshc twn fortðwn, dhlad gia ton rujmì metabol c twn riz n. Autì mporeð na gðnei me thn eôresh miac jetik c diaforðsimhc sunˆrthshc tètoiac ste to ginìmenì thc me tic rðzec na èqei thn antðjeth monotonða ap' aut twn riz n. Sta biblða 96 kai 97 gðnetai susthmatik melèth twn orjogwnðwn poluwnômwn kai twn orjogwnðwn q-poluwnômwn kai dðnontai diˆforec efarmogèc touc. Pio sugkekrimèna sto 97 melet ntai probl mata sqetikˆ me ton prosdiorismì tou fˆsmatoc diakrit c enèrgeiac (discrete energy spectrum) kai twn antðstoiqwn kumatik n sunart sewn (wave functions) se probl mata pou mporoôn na lujoôn me th qr sh klasik n orjogwnðwn poluwnômwn. Sto pèmpto kefˆlaio autoô tou biblðou melet ntai probl mata ìpwc o armonikìc talantwt c, h kðnhsh fortðwn
.3. MEJODOI MELETHS TWN RIZWN 9 se èna pedðo kai oi lôseic twn exis sewn Schrödinger, Dirac kai Klein-Gordon gia to dunamikì tou Coulomb. Sta kefˆlaia 4-6 tou 96 dðnontai efarmogèc twn orjogwnðwn poluwnômwn sta upologistikˆ majhmatikˆ kai sth jewrða twn sqhmˆtwn diafor n, sthn sumpðesh kai apoj keush plhrofori n, sth jewrða pijanot twn kai jewrða kwdikopoðhshc (coding theory) kai se orismèna probl mata thc jewrðac our n (queuing theory). Sqetikˆ me tic efarmogèc twn poluwnômwn diakrit c metablht c (polu numa Hahn, Meixner, Kravchuk kai Charlier), anafèroume epðshc tic ergasðec 32 gia efarmogèc sth jewrða pijanot twn (kurðwc se probl mata thc jewrðac our n), 32, 7, 72, 73, 74 gia efarmogèc se diadikasðec gènnhshc kai janˆtou (birth and death processes) kai 86 gia efarmogèc sth jewrða kwdikopoðhshc. Ta associated orjog nia polu numa ektìc apì ta majhmatikˆ probl mata sta opoða emfanðzontai (bl. 9, 99) emfanðzontai kai se pollˆ probl mata thc fusik c. 'Etsi emfanðzontai se probl mata thc kbantomhqanik c 02, 2, 3, se diadikasðec gènnhshc kai janˆtou 2, 66, k.t.l.. EpÐshc oi rðzec touc brðskoun efarmog sthn kataskeu diadikasi n bèltisthc parembol c ( optimal interpolation processes) 88. Ta orjog nia q-polu numa èqoun epðshc pollèc efarmogèc se diˆforouc tomeðc twn majhmatik n kai thc fusik c. Endeiktikˆ anafèroume ta suneq klˆsmata, tic seirèc Euler, tic j ta sunart seic kai tic elleiptikèc sunart seic (bl. gia parˆdeigma 6, 34). Sth fusik paðzoun shmantikì rìlo sth gwniak orm ( angular momentum) kai sto q-anˆlogì thc, sthn q-schrödinger exðswsh kai stouc q-armonikoôc talantwtèc (bl. 2 kai tic anaforèc pou upˆrqoun ekeð). Epiplèon ta orjog nia q-polu numa sundèontai me touc suntelestèc Glebsch-Gordan (3j kai 6j sômbola) oi opoðoi èqoun shmantikèc efarmogèc sth fusik. Sthn monografða 4 parousiˆzetai analutikˆ h sôndesh twn q-poluwnômwn me th jewrða anaparˆstashc me qr sh omˆdwn kai tic q-ˆlgebrec kai tonðzetai o rìloc pou autèc èqoun sth fusik. Anafèroume endeiktikˆ to prìblhma tou Bloch hlektronðou se èna magnhtikì pedðo 38, 9, 20, 2, 22 ìpou h lôsh mhdenik c enèrgeiac (zero energy solution) ekfrˆzetai mèsw orjogwnðwn q-poluwnômwn kai idiaðtera mèsw twn poluwnômwn Askey-Wilson, en oi rðzec touc kajorðzoun to fˆsma kai ikanopoioôn tic exis seic Bethe-Ansatz. Tèloc ta orjog nia q-polu numa kai oi rðzec touc emfanðzontai sthn sunduastik anˆlush kai sth jewrða kwdikopoðhshc, 8, 43.
0 KEFALAIO. EISAGWGH.3 Mèjodoi gia th melèth twn riz n twn associated orjogwnðwn q-poluwnômwn 'Opwc èqei dh anaferjeð sthn parˆgrafo., ta klasikˆ orjog nia polu numa ikanopoioôn mia diaforik exðswsh Sturm-Liouville deôterhc tˆxhc en ta antðstoiqa associated orjog nia polu numa mia diaforik exðswsh tètarthc tˆxhc 2, 09, 25. H idiìthtˆ touc aut qrhsimopoi jhke gia th melèth thc monotonðac twn riz n touc stic 29, 8 kai gia ton upologismì twn rop n thc katanom c twn riz n touc stic 4, 5, 6, 42, 90, 9, 04, 09, 25, 26. EpÐshc h melèth thc monotonðac twn riz n twn orjogwnðwn poluwnômwn mporeð na gðnei me qr sh tou jewr matoc tou Markov 7, sel. 6 h efarmog tou opoðou ìmwc apaiteð thn Ôparxh enìc jetikoô kai suneqoôc mètrou orjogwniìthtac. Se antðjesh me ta klasikˆ orjog nia polu numa ta orjog nia q-polu numa eðnai oi poluwnumikèc lôseic miac exðswshc q-diafor n thc morf c σ(x)d q D /q y(x) + τ(x)d q y(x) + λ q,n y(x) = 0, ìpou σ(x) kai τ(x) eðnai polu numa to polô deutèrou kai pr tou bajmoô antðstoiqa kai D q eðnai ènac telest c q-diafor n pou orðzetai wc ex c f(qx) f(x), q, x 0, D q f(x) = (q )x f (0), x = 0. Epiplèon to mètro orjogwniìthtac twn associated orjogwnðwn q-poluwnômwn eðnai en gènei ˆgnwsto kai ˆra to je rhma tou Markov den mporeð na efarmosteð gia th melèth thc monotonðac twn riz n touc. 'Opwc epðshc anafèrjhke ta orjog nia q-polu numa den ikanopoioôn mia diaforik exðswsh allˆ mia exðswsh q-diafor n deôterhc tˆxhc en gia orismèna associated orjog nia q-polu numa eðnai gnwstì 35 ìti ikanopoioôn mia exðswsh q-diafor n tètarthc tˆxhc. Opìte mèjodoi gia thn melèth twn riz n twn poluwnômwn pou basðzontai stic diaforikèc exis seic pou autˆ ikanopoioôn, den eðnai dunatìn na efarmostoôn sthn perðptwsh twn associated orjogwnðwn q-poluwnômwn en den upˆrqoun antðstoiqec mèjodoi pou na basðzontai se exis seic diafor n. H mèjodoc pou ja qrhsimopoi soume gia th melèth thc monotonðac kai thc kurtìthtac twn riz n twn associated q-poluwnômwn, eðnai mia sunarthsiak analutik mèjodoc kai basðzetai sthn anadromik sqèsh tri n ìrwn pou ikanopoioôn autˆ
.4. SKOPŸOS KAI DIŸARJRWSH THS PAROŸUSHS DIATRIBŸHS ta orjog nia polu numa kai eis qjh apì touc Ifantis kai Siafarikas sthn 55 gia ta klasikˆ orjog nia polu numa kai stic 56, gia ta antðstoiqa associated orjog nia polu numa. EpÐshc gia ton upologismì twn ajroismˆtwn Newton twn riz n twn associated orjogwnðwn q-poluwnômwn h sunarthsiak analutik mèjodoc pou ja qrhsimopoihjeð parousiˆsthke sthn 52 gia ton upologismì twn ajroismˆtwn Newton twn riz n twn scaled co-recursive associated orjogwnðwn poluwnômwn..4 Skopìc kai diˆrjrwsh thc paroôshc diatrib c Skopìc thc paroôshc diatrib c eðnai h melèth thc monotonðac kai thc kurtìthtac twn riz n orismènwn associated orjogwnðwn q-poluwnômwn kaj c kai sunart - sewn pou perièqoun tic rðzec autèc. EpÐshc upologðzontai ta ajroðsmata Newton twn riz n touc mèqri tètarthc tˆxhc. 'Opwc anafèrjhke h sunarthsiak mèjodoc pou qrhsimopoieðtai eis qjh apì touc Ifantis kai Siafarikas sthn 55 gia ta klasikˆ orjog nia polu numa kai stic 56, gia ta antðstoiqa associated orjog nia polu numa kai perigrˆfetai sto kefˆlaio 2. Sto trðto kefˆlaio meletˆtai h monotonða kai dðnontai diaforikèc anisìthtec gia th megalôterh rðza twn associated continuous q-jacobi poluwnômwn, twn associated q-laguerre poluwnômwn, twn associated continuous q-ultraspherical poluwnômwn, twn associated Al-Salam-Carlitz II poluwnômwn kai twn associated q-meixner poluwnômwn eidik perðptwsh twn opoðwn eðnai ta associated q-charlier polu numa. Ta parapˆnw associated orjog nia q-polu numa prokôptoun apì ta antðstoiqa polu numa pou emfanðzontai sto q-anˆlogo tou sq matoc Askey bl. 76. EpÐshc dðnontai antðstoiqa apotelèsmata gia th megalôterh rðza twn associated q-pollaczek poluwnômwn kaj c kai gia th megalôterh rðza twn q-lommel poluwnômwn apì ta opoða prokôptoun apotelèsmata gia thn pr th rðza thc Jackson q-bessel sunˆrthshc J ν (2) (x; q). Lambˆnontac to ìrio q kai me katˆllhlouc metasqhmatismoôc twn paramètrwn brðskoume antðstoiqa apotelèsmata gia th megalôterh rðza orismènwn oikogenei n associated orjogwnðwn poluwnômwn. 'Ola ta apotelèsmata autoô tou kefalaðou pou aforoôn sth megalôterh rðza twn associated orjogwnðwn q-poluwnômwn eðnai prwtìtupa, upˆrqoun sthn erga-
2 KEFALAIO. EISAGWGH sða 77 kai belti noun prohgoômena apotelèsmata twn antðstoiqwn oikogenei n associated orjogwnðwn poluwnômwn. Sto tètarto kefˆlaio meletˆtai h mikrìterh rðza wc proc th monotonða thc kai dðnontai diaforikèc anisìthtec gia aut, orismènwn associated orjogwnðwn q- poluwnômwn ta opoða emfanðzontai sto sq ma Askey. Apì tic diaforikèc autèc anisìthtec prokôptoun frˆgmata gia th mikrìterh rðza kaj c kai gia to aristerì ˆkro tou diast matoc orjogwniìthtac twn upì melèth poluwnômwn. Pio sugkekrimèna melet ntai ta associated Askey-Wilson, associated continuous q-jacobi kai associated big q-jacobi polu numa. Apì autˆ prokôptoun apotelèsmata kai gia ta associated continuous dual q-hahn, associated Al-Salam-Chihara, associated continuous big q-hermite, associated continuous q-laguerre, associated big q-laguerre kai associated big q-legendre polu numa. EpÐshc lambˆnontac to ìrio q brðskoume apotelèsmata pou dðnoun nèec plhroforðec gia ta associated Wilson, associated continuous dual Hahn, associated Jacobi, associated Laguerre, kai associated Legendre polu numa. 'Ola ta apotelèsmata autoô tou kefalaðou pou aforoôn sth mikrìterh rðza twn associated orjogwnðwn q-poluwnômwn kaj c kai sto aristerì ˆkro tou diast matoc orjogwniìthtˆc touc eðnai prwtìtupa kai upˆrqoun sthn ergasða 0. Sto pèmpto kefˆlaio meletˆtai h kurtìthta thc megalôterhc rðzac twn associated q-laguerre poluwnômwn, twn q-lommel poluwnômwn kai twn associated Al - Salam - Carlitz II poluwnômwn, kaj c epðshc kai h kurtìthta ginomènwn orismènwn sunart sewn me th megalôterh rðza twn associated q-laguerre poluwnômwn kai twn associated Al-Salam-Carlitz II poluwnômwn. Ta apotelèsmata tou kefalaðou autoô eðnai prwtìtupa kai upˆrqoun sthn ergasða 0. Tèloc sto èkto kefˆlaio prosdiorðzontai ta ajroðsmata Newton, mèqri tètarthc tˆxhc, twn riz n twn associated q-laguerre, twn associated q-charlier, twn associated continuous q-ultraspherical, twn q-lommel, twn associated q-meixner, kai twn associated Al-Salam-Carlitz I, II poluwnômwn. Apì autˆ, lambˆnontac to ìrio q, brðskoume wc eidikèc peript seic, ta ajroðsmata Newton twn riz n twn associated Laguerre, twn associated Charlier, twn associated Ultraspherical kai twn associated Meixner poluwnômwn pou èqoun brejeð prìsfata stic ergasðec 3, 4, 52, 68, 92, 93. EpÐshc, apì ta ajroðsmata Newton twn riz n twn q-lommel poluwnômwn, prokôptoun ta antðstoiqa apotelèsmata gia tic rðzec twn q-bessel sunart sewn kai apì autˆ ta gnwstˆ ajroðsmata Rayleigh 8 gia tic rðzec twn sunart sewn Bessel.
.4. SKOPŸOS KAI DIŸARJRWSH THS PAROŸUSHS DIATRIBŸHS 3 Ta apotelèsmata tou kefalaðou autoô pou aforoôn sta ajroðsmata Newton twn riz n twn associated orjogwnðwn q-poluwnômwn eðnai prwtìtupa kai upˆrqoun sthn ergasða 79.
Kefˆlaio 2 Sunarthsiak Analutik Mèjodoc gia th melèth twn riz n orjogwnðwn poluwnômwn 2. Eisagwg H sunarthsiak analutik mèjodoc pou ja qrhsimopoi soume gia th melèth twn riz n twn orjogwnðwn q-poluwnômwn basðzetai sto gegonìc ìti mia akoloujða poluwnômwn, {P n (x, ν q)}, eðnai orjog nia wc proc èna jetikì mètro orjogwniìthtac an kai mìno an ikanopoieð mia anadromik sqèsh tri n ìrwn thc morf c xp n (x; ν q) = α n (ν q)p n+ (x, ν q) + β n (ν q)p n (x; ν q) + γ n (ν q)p n (x; ν q), n = 0,,..., 0 < q <, P (x; ν q) = 0, P 0 (x; ν q) = (2.) me α n (ν q)γ n (ν q) > 0. H parˆmetroc ν eðnai opoiad pote parˆmetroc apì thn opoða mporeð na exart ntai ta orjog nia polu numa (to je rhma autì apodðdetai sun jwc ston Favard 3 allˆ eðnai palaiìtero, bl. sqìlia sthn 7). Gia to lìgo autì, sth sunèqeia ja diereun soume mìno tic peript seic ìpou ikanopoieðtai h sunj kh α n (ν q)γ n (ν q) > 0, n 0. Ta antðstoiqa r-associated orjog nia q-polu numa prokôptoun an sthn anadromik sqèsh (2.) antikatast soume ta α n (ν q), β n (ν q) kai γ n (ν q) me α n (r; ν q) = α n+r (ν q), β n (r; ν q) = β n+r (ν q) kai γ n (r; µ) = γ n+r (ν q) antðstoiqa, ìpou r > 0 r >. To r mporeð na jewrhjeð 5
6 KEFALAIO 2. SUNARTHSIAKH ANALUTIKH MEJODOS wc mða apì tic paramètrouc apì tic opoðec exart ntai ta associated orjog nia q-polu numa. α n (ν q) Jètontac Q n (x; ν q) = U n P n (x; ν q), ìpou U n = γ n (ν q) U n, U = 0, U 0 =, brðskoume ta antðstoiqa orjokanonikˆ polu numa Q n (x; ν q), ta opoða èqoun tic Ðdiec rðzec me ta polu numa P n (x; ν q) kai ikanopoioôn thn anadromik sqèsh xq n (x; ν q) = a n (ν q)q n+ (x; ν q) + b n (ν q)q n (x; ν q) + a n (ν q)q n (x; ν q), n = 0,,..., Q (x; ν q) = 0, Q 0 (x; ν q) =, (2.2) ìpou a n (ν q) = α n (ν q)γ n+ (ν q) kai b n (ν q) = β n (ν q), n = 0,,..., eðnai akoloujðec pragmatik n arijm n. H mèjodoc pou ja qrhsimopoi soume eðnai h akìloujh: 'Estw e j, j = 0,..., n mia orjokanonik bˆsh se ènan peperasmènhc diˆstashc q ro Hilbert H n. 'Estw epðshc, V kai V o dexiˆ metajèthc (shift telest c) kai o suzug c tou oi opoðoi orðzontai epð thc bˆshc e j apì tic akìloujec sqèseic Ve j = e j+, j = 0,..., n 2, Ve n = 0, V e j = e j, j =..., n, V e 0 = 0. 'Estw epðshc, oi diag nioi telestèc A(ν q) kai B(ν q) oi opoðoi orðzontai apì tic sqèseic A(ν q)e j = a j (ν q)e j kai B(ν q)e j = b j (ν q)e j, j = 0..., n. Tìte eðnai gnwstì (bl. 55) ìti oi rðzec x n,k, k =,..., n, diatetagmènec katˆ fjðnousa seirˆ wc proc k, twn poluwnômwn Q n (x; ν q) ta opoða orðzontai apì thn anadromik sqèsh (2.2) eðnai oi idiotimèc tou telest T(ν q) = A(ν q)v + VA(ν q) + B(ν q) me antðstoiqec idiosunart seic X k (ν q), dhlad (A(ν q)v + VA(ν q) + B(ν q))x k (ν q) = x n,k (ν q)x k (ν q) x n,k (ν q) = (A(ν q)v + VA(ν q) + B(ν q))x k (ν q), X k (ν q), (2.3) ìpou X k (ν q) =, k n kai antistrìfwc.
2.2. MONOTONIA TWN RIZWN TWN ORJOGWNIWN q-poluwnumwn 7 2.2 Melèth thc monotonðac twn riz n kai sunart sewn pou perièqoun tic rðzec twn associated orjogwnðwn q-poluwnômwn 'Otan oi telestèc A(ν q), B(ν q) eðnai omoiìmorfa fragmènoi wc proc ν se kˆpoio diˆsthma kai diaforðsimoi wc proc th norm tou H n, tìte ta idiodianôsmata kaj c kai oi antðstoiqec idiotimèc tou T(ν q) eðnai isqur c diaforðsimec wc proc ν 53, 6, 75 kai sômfwna me to je rhma twn Hellman-Feynmann 33, 49, 53 h parˆgwgoc wc proc ν twn idiotim n tou T(ν q), dhlad twn riz n x n,k (ν q) twn poluwnômwn Q n (x; ν q), dðnetai apì th sqèsh dx n,k (ν q) dν = (A (ν q)v + VA (ν q) + B (ν q))x k (ν q), X k (ν q), X k (ν q) =, (2.2.) ìpou A (ν q)e j = da j(ν q) e j = a dν j(ν q)e j, j = 0..., n (gia leptomèreiec bl. 53). B (ν q)e j = db j(ν q) e j = b dν j(ν q)e j, 2.2. MonotonÐa thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza twn associated orjogwnðwn q-poluwnômwn 'Oson aforˆ sth megalôterh idiotim x n, (ν q) tou telest T(ν q), eðnai gnwstì 37, 54, 00 ìti ìtan a j (ν q) > 0, b j (ν q) 0, j = 0,,..., n to antðstoiqo idiodiˆnusma X n, (ν q) eðnai austhrˆ jetikì, me thn ènnoia ìti X n, (ν q), e j > 0, j = 0,,..., n. Apì to apotèlesma autì kai th sqèsh (2.2.) prokôptei ìti an a j(ν q) > 0 kai b j(ν q) > 0 (a j(ν q) < 0 kai b j(ν q) < 0), j = 0,..., n, h megalôterh idiotim tou T(ν q), dhlad h megalôterh rðza twn poluwnômwn Q n (x; ν q), eðnai austhrˆ aôxousa (fjðnousa) sunˆrthsh tou ν. H eôresh miac jetik c sunˆrthshc f(ν q) tètoia ste to ginìmeno f(ν q)x n, (ν q) na eðnai fjðnousa (aôxousa) sunˆrthsh tou ν, mporeð na gðnei me ton akìloujo trìpo: An a j (ν q) > 0 kai b j (ν q) > 0, j = 0,..., n kai epiplèon d dν a j(ν q) = γ j (ν q)a j (ν q) kai d dν b j(ν q) = δ j (ν q)b j (ν q),
8 KEFALAIO 2. SUNARTHSIAKH ANALUTIKH MEJODOS tìte apì th sqèsh (2.2.) kai to gegonìc ìti oi sunist sec tou idiodianôsmatoc X n, (ν q) eðnai austhrˆ jetikoð arijmoð, prokôptei ìti, an upˆrqei mia sunˆrthsh h(ν q) 0 tètoia ste γ j (ν q) < h(ν q) kai δ j (ν q) < h(ν q) (γ j (ν q) > h(ν q) kai δ j (ν q) > h(ν q) ), j = 0,..., n tìte isqôei h akìloujh diaforik anisìthta d dν x n,(ν q) < h(ν q)x n, (ν q), ( d dν x n,(ν q) > h(ν q)x n, (ν q)), apì thn opoða prokôptei ìti to ginìmeno e h(ν q)dν x n, (ν q) eðnai mia fjðnousa (aôxousa) sunˆrthsh tou ν. 2.2.2 MonotonÐa thc mikrìterhc rðzac kai sunart sewn pou perièqoun th mikrìterh rðza twn associated orjogwnðwn q-poluwnômwn Ta orjog nia q-polu numa pou emfanðzontai sto sq ma twn Askey-Wilson 76, se pollèc peript seic ikanopoioôn mia anadromik sqèsh thc morf c xp n (x; ν q) = A n (ν q)p n+ (x; ν q)+ + λ(ν q) (A n (ν q) + B n (ν q))p n (x; ν q) + B n (ν q)p n (x; ν q), (2.2.2) ìpou ta A n (ν q), B n+ (ν q) eðnai jetikèc sunart seic gia n 0, B 0 (ν q) = 0, kai wc sun jwc upojètoume ìti 0 < q <. MporoÔme na ekmetalleutoôme th sugkekrimènh aut morf thc (2.2.2) gia na melet soume th mikrìterh rðza twn orjogwnðwn q-poluwnômwn. Sthn perðptwsh aut oi suntelestèc thc anadromik c sqèshc (2.2) pou ikanopoioôn ta antðstoiqa orjokanonikˆ q-polu numa eðnai a n (ν q) = A n (ν q)b n+ (ν q) kai b n (ν q) = λ(ν q) (A n (ν q) + B n (ν q)). Shmei noume ìti, eˆn T(ν q) eðnai o telest c pou orðzetai apì th sqèsh T(ν q) = Ã(ν q)v + VÃ(ν q) + B(ν q), ìpou oi diag nioi telestèc Ã(ν q) kai B(ν q) orðzontai apì tic sqèseic Ã(ν q)e j = ã j (ν q)e j kai B(ν q)e j = b j (ν q)e j, j = 0,,..., n, me ã j (ν q) = a j (ν q) = A j (ν q)b j+ (ν q) kai b j (ν q) = λ(ν q) b j (ν q) = A j (ν q) + B j (ν q), tìte
2.2. MONOTONIA TWN RIZWN TWN ORJOGWNIWN q-poluwnumwn 9 T(ν q) = A(ν q)v + VA(ν q) + λ(ν q) B(ν q) EÐnai gnwstì kai eôkolo na apodeiqjeð ìti eˆn E(ν q) eðnai idiotim tou T(ν q) me idiostoiqeðo f(ν q) tìte E(ν q) eðnai idiotim tou A(ν q)v +VA(ν q) B(ν q) me idiostoiqeðo Uf(ν q), ìpou U eðnai o isometrikìc telest c Ue j = ( ) j e j, j = 0,,..., n. Opìte oi idiotimèc E j (ν q) tou T(ν q) kai Ẽ j (ν q) tou T(ν q) sundèontai me th sqèsh Ẽ j (ν q) = λ(ν q) E j (ν q). Apì ta anwtèrw prokôptei ìti h megalôterh idiotim tou T(ν q) eðnai x n, (ν q) = λ(ν q) x n,n (ν q). Efìson ta b j (ν q) kai ã j (ν q) eðnai jetikˆ, eˆn oi sunart seic Γ j (ν q) = j (ν q) = A j (ν q) B j (ν q) A j (ν q), j 0, ν B j (ν q), j, ν eðnai jetikèc (arnhtikèc), tìte h x n, (ν q) eðnai aôxousa (fjðnousa) wc proc ν. 'Otan to λ(ν q) den exartˆtai apì to ν, prokôptei ìti h mikrìterh rðza x n,n (ν q) twn orjogwnðwn q-poluwnômwn eðnai fjðnousa (aôxousa) wc proc ν. Shmei netai epðshc ìti oi telestèc T(ν q) = A(ν q)v + VA(ν q) + B(ν q) kai ˆT(ν q) = Â(ν q)v + VÂ(ν q) + B(ν q) ìpou oi diag nioi telestèc A(ν q), Â(ν q) kai B(ν q) orðzontai apì tic sqèseic A(ν q)e j = a j (ν q)e j, Â(ν q)e j = â j (ν q)e j = a j (ν q)e j kai B(ν q)e j = b j (ν q)e j, j = 0,,..., n,, èqoun tic Ðdiec idiotimèc. H idiìthta aut apodeiknôetai qr simh se peript seic pou a j ( ν q) = a j (ν q) b j ( ν q) = b j (ν q). 'Oson aforˆ sth melèth thc monotonðac thc x n, (ν q), h opoða eðnai isodônamh me th melèth thc monotonðac thc x n,n (ν q) ìtan to λ(ν q) den exartˆtai apì to ν, ìpwc kai sthn perðptwsh thc megalôterhc rðzac eðnai polô shmantikì na broôme mia jetik sunˆrthsh F (ν q) tètoia ste to x n,(ν q) na èqei thn antðjeth monotonða apì thn x n, (ν q). Autì mac parèqei èna mètro gia to rujmì metabol c F (ν q) thc
20 KEFALAIO 2. SUNARTHSIAKH ANALUTIKH MEJODOS x n,n (ν q), kaj c epðshc kai frˆgmata gia thn apìstash anˆmesa stic x n,n (ν q) kai x n,n (ν 2 q) gia ν ν 2. Mia tètoia sunˆrthsh F (ν q) mporeð na brejeð me ton akìloujo trìpo. ParathroÔme ìti ã j (ν q) ã j (ν q) ν = 2 (Γ j(ν q) + j+ (ν q)), j 0, b j (ν q) ν = { Aj (ν q)γ j (ν q) gia j = 0, A j (ν q)γ j (ν q) + B j (ν q) j (ν q) gia j. Gia tic oikogèneiec twn orjogwnðwn q-poluwnômwn pou ja melet soume eðnai dunatìn na brejoôn koinˆ frˆgmata ètsi ste Γ j (ν q), j+ (ν q) < (>) ln F (ν q), j 0, (2.2.3) ν ã j (ν q) ã j (ν q), ν bj (ν q) b j (ν q) ν < (>) ln F (ν q), ν apì ìpou prokôptei ìti ν xn, (ν q) F (ν q) = ν λ(ν q) xn,n (ν q) < (>) 0. (2.2.4) F (ν q) Parat rhsh 2.2.. Apì thn anadromik sqèsh (2.2.2) prokôptei ìti h x n, (ν q) eðnai epðshc idiotim tou telest K(ν q) = Ā(ν q)v + V B(ν q) + B(ν q) ìpou kai Ā(ν q)e j = A j (ν q)e j, B(ν q)e j = B j (ν q)e j B(ν q)e j = b j (ν q)e j = (A j (ν q) + B j (ν q)e j, j = 0,,..., n, o opoðoc eðnai diag nia kurðarqoc gia A j (ν q) > 0, B j (ν q) 0. Opìte, x n, (ν q) 0 kai, wc ek toôtou, λ(ν q) x n,n (ν q) 0. Sthn pragmatikìthta, efìson bn (ν q) = A n (ν q) + B n (ν q) > 0, eðnai gnwstì ìti h megalôterh idiotim x n, (ν q) tou T (ν q) eðnai austhrˆ jetik 54.
2.2. MONOTONIA TWN RIZWN TWN ORJOGWNIWN q-poluwnumwn 2 H anisìthta (2.2.4) parèqei plhroforðec ìson aforˆ sto sqetikì rujmì aôxhshc (meðwshc) twn jetik n sunart sewn λ(ν q) x n,n (ν q) kai F (ν q). ParathroÔme ìti, ìtan to λ(ν q) den exartˆtai apì to ν, h melèth thc monotonðac thc x n,n (ν q) eðnai isodônamh me th melèth thc antðstrofhc monotonðac thc jetik c sunˆrthshc λ(ν q) x n,n (ν q). Parat rhsh 2.2.2. Sthn perðptwsh twn r-associated orjogwnðwn q-poluwnômwn mporeð na gðnei mia parìmoia anˆlush, an antikatast soume ta A n (ν q), B n (ν q) me A n (r; ν q) = A n+r (ν q), B n (r; ν q) = B n+r (ν q). Efìson genikˆ gia r 0 eðnai B 0 (r; ν q) 0, prèpei na broôme frˆgmata gia ìlec tic sunart seic Γ j (r; ν q) = thc morf c A j (r; ν q) A j (r; ν q), j (r; ν q) = ν Γ j (r; ν q), j (r; ν q) < (>) B j (r; ν q) B j (r; ν q), j 0. ν ln F (r; ν q), j 0. (2.2.5) ν Parat rhsh 2.2.3. Ta orjog nia q-polu numa apoteloôn q-anˆloga twn oikogenei n twn klasik n orjogwnðwn poluwnômwn, me thn ènnoia ìti lambˆnontac to ìrio q kai me katˆllhlec allagèc paramètrwn apì ta orjog nia q- polu numa prokôptoun ta antðstoiqa klassikˆ orjog nia polu numa. Autì e- xasfalðzei ìti se merikèc peript seic to ìrio gia q tou T n (ν q) dðnei ton telest T n (ν) pou sqetðzetai me mia oikogèneia klasik n orjogwnðwn poluwnômwn. Efìson oi idiotimèc enìc diagwnopoi simou telest o opoðoc eðnai suneq c wc proc mia parˆmetro, eðnai epðshc suneqeðc sunart seic aut c thc paramètrou 75, to ìrio q thc x n,k (ν q) parèqei tic rðzec x n,k (ν) twn antðstoiqwn klasik n orjogwnðwn poluwnômwn. Autì mac epitrèpei, apì ta apotelèsmata gia tic rðzec twn orjogwnðwn q-poluwnômwn, na broôme antðstoiqa apotelèsmata gia tic rðzec twn klasik n poluwnômwn. 'Etsi, gia parˆdeigma, ìson aforˆ stic diaforikèc anisìthtec pou ikanopoioôntai apì opoiad pote rðza twn orjogwnðwn q-poluwnômwn, mporoôme na broôme antðstoiqec diaforikèc anisìthtec gia tic antðstoiqec rðzec twn klasik n orjogwnðwn poluwnômwn. Wc gnwstìn h diaforik anisìthta, λ(ν q) xn,n (ν q) < (>) 0, 0 < q < (2.2.6) ν F (ν q) shmaðnei ìti h λ(ν q) x n,n(ν q) F (ν q) eðnai mia fjðnousa (aôxousa) sunˆrthsh tou ν. Opìte, eˆn lim λ(ν q) = λ(ν) kai lim F (ν q) = F (ν), tìte h λ(ν) x n,n(ν) eðnai q q F (ν)
22 KEFALAIO 2. SUNARTHSIAKH ANALUTIKH MEJODOS mia mh aôxousa (mh fjðnousa) sunˆrthsh tou ν. 'Ara, ìtan oi λ(ν), F (ν) kai o telest c T n (ν) eðnai diaforðsimec sunart seic tou ν èqoume th diaforik anisìthta λ(ν) xn,n (ν) ( ) 0. ν F (ν) Parat rhsh 2.2.4. H melèth thc monotonðac thc mikrìterhc rðzac x n,n (ν q) miac oikogèneiac orjogwnðwn q-poluwnômwn {Q n (x; ν q)} n=0 mac parèqei plhroforðec sqetikˆ me to aristerì ˆkro α(ν q) tou diast matoc orjogwniìthtac aut n twn poluwnômwn. Efìson isqôei α(ν q) = lim x n,n (ν q), h monotonða thc x n,n (ν q) n parèqei th mh austhr monotonða tou α(ν q). Epiplèon, eˆn h anisìthta (2.2.6) isqôei gia ν > ν 0, tìte λ(ν q) x n,n (ν q) F (ν q) kai lambˆnontac to ìrio n, brðskoume ìti λ(ν q) α(ν q) F (ν q) < (>) λ(ν 0 q) x n,n (ν 0 q), ν > ν 0, F (ν 0 q) ( ) λ(ν 0 q) α(ν 0 q), ν > ν 0. F (ν 0 q) Eˆn to α(ν 0 q) eðnai gnwstì, h parapˆnw anisìthta mac dðnei èna frˆgma gia to α(ν q) ìtan ν > ν 0. IdiaÐtero endiafèron parousiˆzei to gegonìc autì sthn perðptwsh twn r-associated orjogwnðwn q-poluwnômwn gia ν = r. Ekmetalleuìmenoi to gegonìc ìti gia r = 0 pou antistoiqeð sta orjog nia q-polu numa to diˆsthma orjogwniìthtac eðnai gnwstì, mporoôme na broôme frˆgmata gia to aristerì ˆkro a(r q) tou diast matoc orjogwniìthtac twn r-associated orjogwnðwn q-poluwnômwn to opoðo eðnai en gènei ˆgnwsto gia r 0. 2.3 Melèth thc kurtìthta thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza associated orjogwnðwn q-poluwnômwn Gia th megalôterh idiotim λ max (c, p; q) tou telest T(ν q) eðnai gnwstì 53 ìti isqôei h akìloujh sqèsh: d 2 λ max (ν q) dν 2 ( d2 A(ν q) V + V d2 A(ν q) dν 2 dν 2 + d2 B(ν q) dν 2 )X n, (ν q), X n, (ν q), (2.3.)
2.4. AJROISMATA NEWTON RIZWN ORJOGWNIWN q-poluwnumwn 23 ìpou d2 A(ν q) e dν 2 j = d2 a j (ν q) e dν 2 j, d2 B(ν q) e dν 2 j = d2 b j (ν q) e dν 2 j, j = 0,,..., n. Apì thn parapˆnw anisìthta kai to gegonìc ìti to idiodiˆnusma X n, (ν q) eðnai austhrˆ jetikì, ìtan a j (ν q) > 0, b j (ν q) 0, j = 0,,..., n, prokôptei ìti an ta ìtan a j (ν q) kai b j (ν q) eðnai kurtèc sunart seic tou ν, tìte kai h megalôterh idiotim tou T(ν q), dhlad h megalôterh rðza x n, (ν q) twn upì exètash poluwnômwn eðnai epðshc kurt sunˆrthsh tou ν. 2.4 AjroÐsmata Newton twn riz n twn associated orjogwnðwn q-poluwnômwn EÐnai gnwstì apì th jewrða telest n ìti an M eðnai ènac summetrikìc telest c se ènan pragmatikì q ro Hilbert peperasmènhc diˆstashc, ìpwc o q roc H n, n tìte to ˆjroisma Me j, e j eðnai anexˆrthto apì thn orjokanonik bˆsh e j, j= j =, 2,...n. 'Opwc kai sthn 52 eðnai dunatìn na apodeiqjeð ìti, eˆn x n,j (ν q), j =, 2,..., n, eðnai to pl rec orjokanonikì sôsthma idiodianusmˆtwn tou telest T (ν q) ston H n, tìte n T k (ν q)e j e j = j= n T k (ν q)x n,j (ν q), X n,j (ν q) = j= n x k n,j(ν q). (2.4.) j= Apì thn (2.4.), gia k =, 2, 3, 4, èqoume antistoðqwc n n x n,j (ν q) = b j (ν q), (2.4.2) j= j=0 n n 2 n x 2 n,j(ν q) = 2 a 2 j(ν q) + b 2 j(ν q), (2.4.3) j= j=0 j=0 n n 2 n 2 n x 3 n,j(ν q) = 3 a 2 j(ν q)b j (ν q) + 3 a 2 j(ν q)b j+ (ν q) + b 3 j(ν q), (2.4.4) j= j=0 j=0 j=0
24 KEFALAIO 2. SUNARTHSIAKH ANALUTIKH MEJODOS kai n n n 2 n 3 x 4 n,j(ν q) = b 4 j(ν q) + 2 a 4 j(ν q) + 4 a 2 j(ν q)a 2 j+(ν q)+ j= j=0 j=0 j=0 n 2 n 2 + 4 a 2 j(ν q)b 2 j(ν q) + 4 a 2 j(ν q)b 2 j+(ν q)+ j=0 j=0 n 2 + 4 a 2 j(ν q)b j (ν q)b j+ (ν q). j=0 (2.4.5) Parat rhsh 2.4.. Me th mèjodo aut eðnai dunatìn na upologistoôn ta ajroðsmata x k n,j(ν q) gia opoiod pote k, oi upologismoð ìmwc pou apaitoôntai n eðnai j= idiaðtera polôplokoi.
Kefˆlaio 3 MonotonÐa thc megalôterhc rðzac kai sunart sewn pou perièqoun th megalôterh rðza twn associated orjogwnðwn q-poluwnômwn 3. Eisagwg H diereônhsh thc sumperiforˆc twn riz n twn klasik n orjogwnðwn poluwnômwn wc sunart sewn miac paramètrou, èstw ρ n,k (a), k n, eðnai idiaðtera shmantik exaitðac thc sunˆfeiˆc thc me diˆfora fusikˆ fainìmena. 'Etsi gia parˆdeigma o Stieltjes stic 4, 5, 6 èdeixe ìti oi rðzec twn poluwnômwn Jacobi P n (a,b) (x) eðnai ta shmeða isorropðac n eleôjerwn monadiaðwn fortðwn sto (, ) se èna pedðo pou parˆgetai apì dôo fortða a + sto kai b + sto, ìpou ta 2 2 fortða apwjoôntai sômfwna me to nìmo tou logarijmikoô dunamikoô. AntÐstoiqa oi rðzec twn poluwnômwn Laguerre L (a) n (cx) eðnai ta shmeða isorropðac n eleôjerwn monadiaðwn fortðwn sto (0, + ) se èna pedðo pou parˆgetai apì èna fortðo a + 2 sto 0. Apì ta parapˆnw eðnai fanerì ìti h monotonða twn riz n twn orjogwnðwn poluwnômwn wc proc tic paramètrouc apì tic opoðec exart ntai antistoiqeð sthn kðnhsh twn eleôjerwn monadiaðwn fortðwn ìtan metabˆllontai ta stajerˆ fortða pou parˆgoun to pedðo. 25
26 KEFALAIO 3. MONOTONIA THS MEGALUTERHS RIZAS Hlektrostatikèc ermhneðec gia tic rðzec diafìrwn oikogenei n orjogwnðwn poluwnômwn èqoun dojeð kai apì touc Hendriksen kai van Rossum 50 oi opoðoi melèthsan epektˆseic thc hlektrostatik c ermhneðac tou Stieltjes stic migadikèc rðzec orismènwn orjogwnðwn poluwnômwn kai touc Forrester kai Rogers 36 oi o- poðoi melèthsan antðstoiqa probl mata sto monadiaðo kôklo. EpÐshc o Grünbaum 80 èdwse mia hlektrostatik ermhneða twn riz n ton poluwnômwn Koornwinder- Krall, en sthn 62 o Ismail parousðase èna hlektrostatikì montèlo gia tic rðzec twn genik n (general) orjogwnðwn poluwnômwn. IdiaÐtero endiafèron parousiˆzei epðshc h eôresh miac katˆllhlhc jetik c kai diaforðsimhc sunˆrthshc (thc bèltisthc an eðnai dunatìn) f(a), tètoiac ste ta ginìmena f(a)ρ n,k (a), k n na auxˆnoun na fjðnoun wc proc a lambˆnontac ètsi èna mètro gia to rujmì metabol c twn riz n. Sthn 8, o Laforgia diereônhse to parapˆnw prìblhma gia tic rðzec x n,k (λ), n 2, k n 2 twn poluwnômwn Ultraspherical ( Gegenbauer). Eidikìtera, sthn 8 apèdeixe ìti h sunˆrthsh λx n,k (λ) auxˆnei wc proc λ gia λ (0, ) kai sthn 82 èkane thn eikasða ìti autì isqôei gia ìla ta λ > 0. Argìtera oi Ahmed, Muldoon kai Spigler sthn 5, br kan ìti h sunˆrthsh λ + (2n 2 + )/(4n + 2) /2 x n,k (λ) auxˆnei wc proc λ, gia /2 < λ 3/2. Sth sunèqeia stic 60 kai 63, oi Ismail-Letessier-Askey diatôpwsan thn eikasða (ILAC) ìti h sunˆrthsh(λ + ) /2 x n,k (λ) auxˆnei wc proc λ, gia λ > /2. H eikasða aut apedeðqjh apì touc Ifantis kai Siafarikas sthn 55 gia th megalôterh jetik rðza kai sthn 22 apì ton Dimitrov gia ìlec tic jetikèc rðzec x n,k (λ) gia /2 < λ 9/2 /2 < λ < 3/2 + ν kai n > + (ν 2 + 3ν + 3/2) /2 ìpou ν N. Tèloc sthn 29, oi Elbert kai Siafarikas apèdeixan ìti h sunˆrthsh λ + (2n 2 + )/(4n + 2) /2 x n,k (λ) auxˆnei wc proc λ gia λ > /2. To apotèlesma autì epekteðnei to apotèlesma twn Ahmed, Muldoon kai Spigler kai apodeiknôei thn eikasða ILAC. Sthn 24 apedeðqjh ìti to apotèlesma autì eðnai bèltisto. Sthn 89, oi Natalini kai Palumbo èdwsan kˆpoia antðstoiqa apotelèsmata gia tic rðzec twn genikeumènwn poluwnômwn Laguerre. Sto kefˆlaio autì dðnontai apotelèsmata sqetikˆ me th monotonða kaj c kai diaforikèc anisìthtec pou ikanopoieð h megalôterh rðza orismènwn oikogenei n associated orjogwnðwn q-poluwnômwn. Sugkekrimèna sthn 3.2. meletˆtai h megalôterh rðza twn associated q-pollaczek poluwnômwn, ìpou ta q-pollaczek polu numa eis qjhsan stic 7, 58, 59 wc q-anˆloga twn poluwnômwn Pollaczek. Eidikèc peript seic twn associated q-pollaczek poluwnômwn eðnai ta associated
3.2. APOTELŸESMATA 27 continuous q-ultraspherical polu numa kai ta associated q-hermite polu numa gia ta opoða prokôptoun ˆmesa antðstoiqa apotelèsmata Stic 3.2.2-3.2.5, meletˆtai h monotonða kai dðnontai diaforikèc anisìthtec gia tic rðzec twn associated continuous q-jacobi poluwnômwn, twn associated q-laguerre poluwnômwn, twn associated Al-Salam-Carlitz II poluwnômwn kai twn associated q-meixner poluwnômwn eidik perðptwsh twn opoðwn eðnai ta associated q-charlier polu numa. Ta parapˆnw associated orjog nia q-polu numa prokôptoun apì ta antðstoiqa polu numa pou emfanðzontai sto q-anˆlogo tou sq matoc Askey 76. Tèloc sthn 3.2.6, meletˆtai h monotonða kai dðnontai anisìthtec gia th megalôterh rðza twn q-lommel poluwnômwn pou eis qjhsan sthn 57 se susqètish me thn Jackson q-bessel sunˆrthsh J ν (2) (x; q) 70. Apì autˆ prokôptoun antðstoiqa apotelèsmata gia thn pr th rðza thc J ν (2) (x; q). Ta parapˆnw apotelèsmata pou enopoioôn, genikeôoun kai belti noun prohgoômena apotelèsmata eðnai prwtìtupa kai perilambˆnontai sthn ergasða 77. 3.2 Apotelèsmata 3.2. Associated q-pollaczek polu numa Ta orjog nia associated q-pollaczek polu numa 64 ikanopoioôn thn anadromik sqèsh (2.2) me a n (r; λ, a, b q) = ( q n+r+ )( λ 2 q n+r ) > 0, 2 ( aλq n+r )( aλq n+r+ ) n 0, r >, (3.2.) kai q n+r b b n (r; λ, a, b q) = ( aλq n+r ). (3.2.2) kai eðnai orjog nia gia q r/2 λ q r/2 kai aλ < q r. Je rhma 3.2.. (MonotonÐa wc proc λ) (i) H megalôterh rðza x n, (r; λ, a, b q) me r >, twn associated q-pollaczek poluwnômwn fjðnei wc proc λ, gia: