Movatterモバイル変換


[0]ホーム

URL:


×

zbMATH Open — the first resource for mathematics

from until
Reset all

Examples

GeometrySearch for the termGeometry inany field. Queries arecase-independent.
Funct*Wildcard queries are specified by* (e .g.functions,functorial, etc.). Otherwise the search isexact.''Topological group'':Phrases (multi - words) should be set in''straight quotation marks''.
au: Bourbaki & ti: AlgebraSearch forauthorBourbaki andtitleAlgebra. Theand-operator & is default and can be omitted.
Chebyshev | TschebyscheffTheor-operator| allows to search forChebyshev orTschebyscheff.
Quasi* map* py: 1989The resulting documents havepublicationyear1989.
so:Eur* J* Mat* Soc* cc:14Search for publications in a particularsource with aMathematics SubjectClassificationcode in14.
cc:*35 ! any:ellipticSearch for documents about PDEs (prefix with * to search only primary MSC); the not-operator ! eliminates all results containing the wordelliptic.
dt: b & au: HilbertThedocumenttype is set tobooks; alternatively:j forjournal articles,a forbookarticles.
py: 2000 - 2015 cc:(94A | 11T)Numberranges when searching forpublicationyear are accepted . Terms can be grouped within( parentheses).
la: chineseFind documents in a givenlanguage .ISO 639 - 1 (opens in new tab) language codes can also be used.
st: c r sFind documents that arecited, havereferences and are from asingle author.

Fields

ab Text from the summary or review (for phrases use “. ..”)
an zbMATH ID, i.e.: preliminary ID, Zbl number, JFM number, ERAM number
any Includes ab, au, cc, en, rv, so, ti, ut
arxiv arXiv preprint number
au Name(s) of the contributor(s)
br Name of a person with biographic references (to find documents about the life or work)
cc Code from the Mathematics Subject Classification (prefix with* to search only primary MSC)
ci zbMATH ID of a document cited in summary or review
db Database: documents in Zentralblatt für Mathematik/zbMATH Open (db:Zbl), Jahrbuch über die Fortschritte der Mathematik (db:JFM), Crelle's Journal (db:eram), arXiv (db:arxiv)
dt Type of the document: journal article (dt:j), collection article (dt:a), book (dt:b)
doi Digital Object Identifier (DOI)
ed Name of the editor of a book or special issue
en External document ID: DOI, arXiv ID, ISBN, and others
in zbMATH ID of the corresponding issue
la Language (use name, e.g.,la:French, orISO 639-1, e.g.,la:FR)
li External link (URL)
na Number of authors of the document in question. Interval search with “-”
pt Reviewing state: Reviewed (pt:r), Title Only (pt:t), Pending (pt:p), Scanned Review (pt:s)
pu Name of the publisher
py Year of publication. Interval search with “-”
rft Text from the references of a document (for phrases use “...”)
rn Reviewer ID
rv Name or ID of the reviewer
se Serial ID
si swMATH ID of software referred to in a document
so Bibliographical source, e.g., serial title, volume/issue number, page range, year of publication, ISBN, etc.
st State: is cited (st:c), has references (st:r), has single author (st:s)
sw Name of software referred to in a document
ti Title of the document
ut Keywords

Operators

a & bLogical and (default)
a | bLogical or
!abLogical not
abc*Right wildcard
ab cPhrase
(ab c)Term grouping

See also ourGeneral Help.

On series expansions of Capparelli’s infinite product.(English)Zbl 1160.11355

Summary: Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers PhD thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof.
Most combinatorial Rogers-Ramanujan type identities (e.g., the Göllnitz-Gordon identities, Gordon’s combinatorial generalization of the Rogers-Ramanujan identities, etc.) have an analytic counterpart. The main purpose of this paper is to provide two new series representations for the infinite product associated with Capparelli’s conjecture. Some additional related identities, including new infinite families are also presented.

MSC:

11P81 Elementary theory of partitions
05A17 Combinatorial aspects of partitions of integers
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities

Cite

References:

[1]Alladi, K.; Andrews, G. E.; Gordon, B., Refinements and generalizations of Capparelli’s conjecture on partitions, J. Algebra, 174, 636-658 (1995) ·Zbl 0830.05005
[2]Andrews, G. E., The Theory of Partitions, Encyclopedia Math. Appl., vol. 2 (1998), Addison-Wesley: Cambridge Univ. Press, Reissued ·Zbl 0906.05004
[3]Andrews, G. E., Multiple series Rogers-Ramanujan type identities, Pacific J. Math., 114, 267-283 (1984) ·Zbl 0547.10012
[4]Andrews, G. E., Generalized Frobenius partitions, Mem. Amer. Math. Soc., 49, 301 (1984) ·Zbl 0544.10010
[5]Andrews, G. E., \(q\)-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Reg. Conf. Ser. Math., vol. 66 (1986), Amer. Math. Soc: Amer. Math. Soc Providence, RI ·Zbl 0594.33001
[6]Andrews, G. E., Schur’s theorem, Capparelli’s conjecture and \(q\)-trinomial coefficients, (Proc. Rademacher Centenary Conf., 1992. Proc. Rademacher Centenary Conf., 1992, Contemp. Math., vol. 166 (1994), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 141-154 ·Zbl 0811.05001
[7]G.E. Andrews, private communication, August 18, 2003; G.E. Andrews, private communication, August 18, 2003
[8]Bailey, W. N., Some identities in combinatory analysis, Proc. London Math. Soc. (2), 49, 421-425 (1947) ·Zbl 0041.03403
[9]Bailey, W. N., On identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2), 50, 1-10 (1948) ·Zbl 0031.39203
[10]S. Capparelli, Vertex operator relations for affine Lie algebras and combinatorial identities, PhD thesis, Rutgers, 1988; S. Capparelli, Vertex operator relations for affine Lie algebras and combinatorial identities, PhD thesis, Rutgers, 1988
[11]Capparelli, S., A construction of the level 3 modules for the affine algebra \(A_2^{(2)}\) and a new combinatorial identity of the Rogers-Ramanujan type, Trans. Amer. Math. Soc., 348, 2, 481-501 (1996) ·Zbl 0862.17017
[12]Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge Univ. Press ·Zbl 0695.33001
[13]H. Göllnitz, Einfache partitionen, Diplomarbeit W.S., Göttingen, 1960; H. Göllnitz, Einfache partitionen, Diplomarbeit W.S., Göttingen, 1960
[14]Gordon, B., Some continued fractions of the Rogers-Ramanujan type, Duke J., 31, 741-748 (1965) ·Zbl 0178.33404
[15]Lepowsky, J.; Milne, S. C., Lie algebraic approaches to classical partition identities, Adv. Math., 29, 1, 15-59 (1978) ·Zbl 0384.10008
[16]Lepowsky, J.; Milne, S. C., Lie algebras and classical partition identities, Proc. Natl. Acad. Sci. USA, 75, 2, 578-579 (1978) ·Zbl 0379.17003
[17]Lepowsky, J.; Wilson, R. L., Construction of the affine Lie algebra \(A_1^{(1)}\), Commun. Math. Phys., 62, 43-53 (1978) ·Zbl 0388.17006
[18]Lepowsky, J.; Wilson, R. L., The Rogers-Ramanujan identities: Lie theoretic interpretation and proof, Proc. Natl. Acad. Sci. USA, 78, 699-701 (1981) ·Zbl 0449.17010
[19]Lepowsky, J.; Wilson, R. L., A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities, Adv. Math., 45, 21-72 (1982) ·Zbl 0488.17006
[20]Lepowsky, J.; Wilson, R. L., A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Invent. Math., 77, 199-290 (1984) ·Zbl 0577.17009
[21]Lepowsky, J.; Wilson, R. L., The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities, Invent. Math., 79, 417-442 (1985) ·Zbl 0577.17010
[22]MacMahon, P. A., Combinatory Analysis, vol. 2 (1918), Cambridge Univ. Press ·JFM 46.0118.07
[23]Meurman, A.; Primc, M., A basis of the basic \(sl(3,C)^~\)-module, Commun. Contemp. Math., 3, 4, 593-614 (2001) ·Zbl 1004.17003
[24]Petkovšek, M.; Wilf, H.; Zeilberger, D., \(A=B (1996)\), Peters ·Zbl 0848.05002
[25]Propp, J., Some variants of Ferrers diagrams, J. Combin. Theory A, 52, 98-128 (1989) ·Zbl 0682.05007
[26]Rogers, L. J., Second memoir on the expansion of certain infinite products, Proc. London Math. Soc., 25, 318-343 (1894)
[27]Schur, I., (Ein Beitrag zur additeven Zahlentheorie und zur Theorie der Kettenbrüche (1917), Sitz.ber. Berlin. Akad), 302-321 ·JFM 46.0201.01
[28]Slater, L. J., Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2), 54, 147-167 (1952) ·Zbl 0046.27204
[29]Tamba, M.; Xie, C., Level three standard modules for \(A_2^{(2)}\) and combinatorial identities, J. Pure Appl. Algebra, 105, 1, 53-92 (1995) ·Zbl 0854.17029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
© 2025FIZ Karlsruhe GmbHPrivacy PolicyLegal NoticesTerms & Conditions
  • Mastodon logo
 (opens in new tab)

[8]ページ先頭

©2009-2025 Movatter.jp