Infinite Continued Fraction (CF) representations. of the exponential integral function, Bessel functions and Lommel polynomials

Transcription

1 Ifii Coiu Fraio CF rraio of h oial igral fuio l fuio a Lol olyoial Coiu Fraio CF rraio a orhogoal olyoial I hi io w rall h rlaio bw ifi rurry rlaio of olyoial orroig orhogoaliy a aroria ifii oiu fraio rraio a h orhogoaliy of olyoial origia i h hory of Coiu Fraio [SzG] I 95 J Favar [FaJ] rov h Thor : If hr i a qu of ral olyoial of h for oig a rurio rlaio a a h h olyoial ar orhogoal For a ifii CF w u h oaio [SzG] 5 * a a a b b b b H a uual h ovrg S i fi a h fii fraio obai fro * by oig a h r b W hav for 4 h rurry forula bb a b b a S b S S bs as

2 Th Chrioffl-Darbou rurry forula [SzG] a Thor : Th followig rlaio hol for ay hr ouiv orhogoal olyoial: ** C for 4 Hr a C ar oa a C If h high orffii of i o by w hav C Th rurry forula ** ugg h oiraio of h CF: *** C C C Thor [SzG] hor 5: L h o b a a b a fi o-raig fuio wih ifiily ay oi of ira i h fii or ifii irval Th ovrg S of *** ar ri by h forula / b a / S S Th followig ooiio io arial fraio hol for h ovrg of ***: S / whr a ar fi by b a

3 Ifii CF rraio I of h oial igral fuio Lagurr I hi io w rall h y ia of M Lagurr [LaE] o riv h CF rraio of h oial igral fuio Ei : Ei 5 / / /6 6 Hi bai ia i o buil aroria olyoial f a F :! of gr r wih / fulfillig h followig rlaio Thi la o i F O f! i F Ei li f whrby h olyoial f fulfill h rlaio Th fuio f f f y f a y f ar how o b h oluio of h iffrial quaio y y y

4 Th fuio u : y fulfill h rurry rlaio u u u Th obiaio wih h a rlaio for f la o u Ei a u fro whih h CF rraio follow 4

5 5 Ifii CF rraio II of h oial igral fuio Lgr I [NiN] 84 hr i giv a alraiv roof of a CF rraio for h oial igral fuio W rall i y rori uig h followig oaio: : : P a : I hol : a! P r! P La: I hol i P P P ii li P li iii iv v

6 6 Proof: For h roof of i-v w rfr o [NiN] 9 84 W ju io h rlaio arial igraio laig o whih i u i obiaio wih h followig iiy Corollary: [NiN] 84: i ii Corollary: Puig i follow Ei

7 So Prori of h l Fuio Th l fuio of h fir i a orr zro [WaG] i fi by i J :! i i li P J a J :! Th l quaio of orr zro a b irr a igular ourar of h rgular Shröigr quaio whih how o h o ha i logarihi igulariy bhavior i h for li g log CL Sigl [WaG] 5- gav o arly roof abou algbrai l fuio valu g h rov ha J z i o a algbrai ubr if i a raioal ubr a z i a algbrai ubr ohr ha zro Th aalyi of h oial igral fuio i li o h o of Lagurr olyoial Hri olyoial ar h igvalu oluio of h rgular Shröigr quaio Thy buil a orhogoal bai of L a fuio of Lagurr olyoial buil h orroig olyoial y of L alyig h variabl rafor y Th rurry forula for h l fuio of orr zro J [WaG] 9-6 ar giv by J z J z J z z I i u o r J z liarly i r of J z a J z Th offii i hi liar rlaio ar olyoial i whih ar h Lol olyoial fi by h z rurr rlaio J z J z J z z 7

8 Thr i a rurry rlaio for h Lol olyoial ilf i [WaG] a wll a [GWa] 9-65 for a 4 J J J / J Puig g : g a h : h : g h oifi Lol olyoial ar fi by r g g g g : g : / g -!! -! -! wih oly ral zro [WaG] 9-7 Th rlaio of h oifi Lol olyoial o h Lol olyoial ar giv by g g 8

9 Fro [WaG] w rall Eulr ivigaio i h zro of J : ig h abrviaio J ar aig o b : j w wri h zro of j i N J i h for j Th zro of Z a i hol for j / 4 N I orr o ria h all zro of J Eulr iffria logarihially h rou forula o olu log J rovi ha a h la ri i abolu ovrg Puig : a hag h orr of uaio rul io * J J J a o h forula * abov Eulr obai a y of quaio whih allow o alula h a fro ha o u h all valu of i Eulr alula / / / 48 9 / 47/ o u g La ig j h zro of Z y i J J h zro of J i hol J a j / 4 N ii J log J J j j iii log J for iv i h oly igr valu for h j v J J a J J r J J J 9

10 Th orhogoaliy rlaio of h Lol olyoial [ChT] h VI 6 [DiD] [SH] w uari i La Lol olyoial: ig j h zro of Z J y For h Lol olyoial i hol h followig orhogoaliy rlaio h h h h j j j j j j wih h g whrby j ar h zro of J Z a h rurry rlaio h h h h h W uari Hurwiz lii of l fuio [Hu] [WaG] i La Hurwiz lii for l fuio: I hol: J g li li h!!! r J li h z z li J z wih * g : -!! -! -!

11 fr [ChT] TS Chihara irouio o Orhogoal Polyoial Mahai a i liaio Goro a rah Nw Yor 978 [DiD] D Diio O Lol a l olyoial Pro r Mah So [FaJ] J Favar ur olyo Thbihff CaSi Pari [Hu] Hurwiz br i Nullll r l h Fuio Mahaih al [LaE] E Lagurr Sur l'iégral ull SoMah Fra [NiN] N Nil Habuh r Thori r Gaafuio Lizig G Tubr Vrlag 96 [SiC] CLSigl br iig wug iohaihr roiaio bh Pru a Wi Phyi-Mah [SzG] G Szgö Orhogoal Polyoial r Mah So Colloquiu Publiaio volu 98 [WaG] G N Wao Trai o h Thory of l Fuio Cabrig ivriy Pr Cabrig iio fir ublih 944 rri