Simple proofs of Bressoud s and Schur s polynomial versions of the. Rogers-Ramanujan identities. Johann Cigler
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1 Simple proofs of Bressoud s ad Schur s polyomial versios of the Rogers-Ramaua idetities Joha Cigler Faultät für Mathemati Uiversität Wie A-090 Wie, Nordbergstraße 5 Joha Cigler@uivieacat Abstract We give simple elemetary proofs of Bressoud s ad Schur s polyomial versios of the Rogers-Ramaua idetities Bressoud s idetity I [] George E Adrews ad Kimmo Erisso gave a simple proof of David Bressoud s ([] polyomial versio of the Rogers-Ramaua idetities I wat to show that their proof ca be further simplified by startig with the idetity ( ( = = + ( Ole Waraar has iformed me that this idetity has bee obtaied i [6], Lemma as limit case of Rogers -Dougall sum I [6] he already used ( to prove (a geeralizatio of Bressoud s idetity ( Christia Krattethaler has told me that ( ca be cosidered as limit case of Jacso s -Dixo summatio It is also a special case of Paule s trasformatio T of [4] A simple computer proof ca be give if we write the left had side of ( i the euivalet form ( + f(, = ( = + The the implemetatio Zeil of the Zeilberger algorithm gives f (, = f(,, from which ( is obvious if we observe that f(, = ( If you do t trust the computer set (,, ( a = ( + + ad + ( ( b (,, = a (,, ad verify that for > ( + ( (
2 a (,, a (,, = b (,, b (,, Here I give a elemetary proof of ( which uses oly the recurrece relatios for the biomial coefficiets: To this ed let ( ( + (, ( f = ( = = + = + ( From the recurrece relatio + + = + ( for the biomial coefficiets we also get ( + ( = f(, + (4 This follows from ( ( + ( ( = + + ( ( ( f(, ( = The last sum vaishes, because + defies a sig reversig ivolutio The other recurrece relatio gives + = + (5 ( + ( ( = ( + f(, = f(, Therefore we get ( + ( (, ( + f = ( = = + + ( + + = ( + + (6
3 This implies ( ( + + (, ( + f + = ( = + + ( ( + ( f(, ( + = + = f( ( ( + +, + f(, Therefore the seuece ( f (, satisfies the recurrece relatio ( for the biomial coefficiets ad the correspodig boudary values This proves Theorem The followig idetities hold: ( ( ( + ( = + + = ( + + = ( = + + (7 From (7 we obtai (5 ( ( + ( = + (8 The Vadermode formula m+ m = ( m ( (9 implies ( ( + = + (0 Therefore (8 reduces to Bressoud s idetity (5 ( = (
4 I the same way we get ( ( + ( ( = + + ( ( + = f(, = + This implies as above ( = + (5 + + ( As is well ow (cf eg [] by lettig i ( we get the first Rogers-Ramaua idetity 0 (5 = ( ( ( ( ( ( I the same way from ( we get the secod Rogers-Ramaua idetity 0 + (5 = ( ( ( ( ( (4 Schur s idetity The idetity which correspods to ( for Schur s polyomial versio is Theorem ( + + g (, = ( = = (5 + This idetity has bee obtaied i [] by other meas By usig (5 we get ( ( (, ( g+ = ( + = = + + 4
5 For the first sum we get agai by usig (5 ( + + ( + + ( ( = + + = = ( ( = g (, + (,, = + where ( + + (, = ( = 0, = + because + iduces a sig reversig ivolutio Therefore we have ( + + ( = g(, (6 = + The secod term i the above formula gives ( + + ( ( ( = + + = = ( + + ( ( = ( + g(, = = + + Let ow ( + + h (, = ( = + 5
6 The ( + + h (, = ( = + ( ( + + ( = ( + = = + + = (, + g(, = g(, Therefore we get the recursio g ( +, = g (, + g (, + g (, (7 0 It is easy to verify that g (,0 = = 0, (, 0 g = = for ad + g ( +, = = 0 for By this recurrece ad the iitial values g (, is uiuely determied for all Sice ( ( = + = + + we see that g (, = for all By summig over all ad usig the Vadermode formula we get ( + + ( 0 = = = 0 = + (5 + + (5 ( ( + = ( ( = + 5 = = + This gives Schur s ([5] polyomial versio of the first Rogers-Ramaua idetity (5 ( = + 5 (8 = 0 = 6
7 I the same way from (6 we get ( ( = 0 = = 0 = = = + (5 (5 ( ( + ( ( = = This is Schur s polyomial versio of the secod Rogers-Ramaua idetity, which is usually writte i the form (5 + ( = 5+ (9 = 0 = Refereces [] George E Adrews & Kimmo Erisso, Iteger Partitios, Cambridge Uiversity Press 004 [] David M Bressoud, Some idetities for termiatig -series, Math Proc Cambridge Phil Soc 89 (98, - [] Joha Cigler, -Fiboacci polyomials ad the Rogers-Ramaua idetities, Aals of Combiatorics 8 (004, [4] Peter Paule, O idetities of the Rogers-Ramaua type, J Math Aal Appl 07 (985, [5] Issai Schur, Ei Beitrag zur additive Zahletheorie ud zur Theorie der Kettebrüche, Ges Abh, 7-6 [6] S Ole Waraar, The geeralized Borwei coecture I The Burge trasform, i BC Berdt, K Oo (Eds, -Series with Applicatios to Combiatorics, Number Theory, ad Physics, Cotemp Math, Vol 9, AMS, Providece, RI, 00,
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