New Finite Rogers Ramanujan Identities

Frederic Jouhet2

Description

Niveau: Supérieur, Doctorat, Bac+8
New Finite Rogers-Ramanujan Identities Victor J. W. Guo1, Frederic Jouhet2 and Jiang Zeng3 1Department of Mathematics, East China Normal University, Shanghai 200062, People's Republic of China , 2Universite de Lyon, Universite Lyon 1, UMR 5208 du CNRS, Institut Camille Jordan, F-69622, Villeurbanne Cedex, France , 3Universite de Lyon, Universite Lyon 1, UMR 5208 du CNRS, Institut Camille Jordan, F-69622, Villeurbanne Cedex, France , Abstract. We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by spe- cialization or through Bailey's method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice. Keywords: Rogers-Ramanujan identities, Watson's transformation, Bailey chain, Bailey lattice AMS Subject Classifications (2000): 05A30; 33D15 1 Introduction The famous Rogers-Ramanujan identities (see [4]) may be stated as follows: 1 + ∞∑ k=1 qk 2 (1? q)(1? q2) · · · (1? qk) = ∞∏ n=0 1 (1? q5n+1)(1? q5n+4) , (1.1) 1 + ∞∑

• bailey's lemma


• ramanujan identities


• can iterate


• hand side


• called bailey chain


• watson's classical


• bailey chain


• theorems


• theorem can

Sujets

• Jordan
• Rogers
• Bailey pair
• Theorem

Informations

• Publié par: mijec
• Nombre de lectures: 41

Extrait

New Finite Rogers-Ramanujan Identities Victor J. W. Guo 1 ,Fre´de´ricJouhet 2 and Jiang Zeng 3 1 Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China [e-mail], http://math.ecnu.edu.cn/~jwguo 2 Universite´deLyon,Universite´Lyon1,UMR5208duCNRS,InstitutCamilleJordan, F-69622, Villeurbanne Cedex, France [e-mail], http://math.univ-lyon1.fr/~jouhet 3 Universite´deLyon,UniversiteLyon1,UMR5208duCNRS,InstitutCamilleJordan, ´ F-69622, Villeurbanne Cedex, France [e-mail], http://math.univ-lyon1.fr/~zeng

Abstract. We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson’s transformation formula by spe- cialization or through Bailey’s method, the second similar formula can be proved either by using the first formula and the q -Gosper algorithm, or through the so-called Bailey lattice. Keywords: Rogers-Ramanujan identities, Watson’s transformation, Bailey chain, Bailey lattice AMS Subject Classifications (2000): 05A30; 33D15

1 Introduction The famous Rogers-Ramanujan identities (see [4]) may be stated as follows: 1 + k ∞ = X 1 (1 − q )(1 − q q k 2 2 ) ∙ ∙ ∙ (1 − q k ) = ∞ Y 1 1 − q 5 n +4 ) , (1.1) n =0 (1 − q 5 n +1 )( ∞ k 2 + k ∞ 1 + k = X 1 (1 − q )(1 − q q 2 ) ∙ ∙ ∙ (1 − q k ) = n Y =0 (1 − q 5 n +2 ) 1 (1 − q 5 n +3 ) . (1.2) Throughout this paper we suppose that q is a complex number such that 0 < | q | < 1. For any integer n define the q -shifted factorial ( a ) n by ( a ) 0 = 1 and , . . . , ( a ) n = ( a ; q ) n := ( [ ( ( 1 1 − − a a ) q ( − 1 1 ) − (1 aq − ) ∙ a ∙ q ∙ − ( 2 1 ) ∙ − ∙∙ a ( q 1 n − − 1 ) a , q n )] − 1 ,n n = =1 − , 1 2 , − 2 , . . . . Note that ( q 1 ) = 0 if n < 0. For m ≥ 1 we will also use the compact notations: n ( a 1 , . . . , a m ) n := ( a 1 ) n ∙ ∙ ∙ ( a m ) n , ( a 1 , . . . , a m ) ∞ := lim ( a 1 , . . . , a m ) n . n →∞

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