Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols

被引:27
作者
Bringmann, Kathrin [2 ]
Lovejoy, Jeremy [1 ]
Osburn, Robert [3 ]
机构
[1] Univ Paris 07, LIAFA, F-75205 Paris 13, France
[2] Univ Cologne, Math Inst, D-50931 Cologne, Germany
[3] Univ Coll Dublin, Sch Math Sci, Dublin 4, Ireland
基金
爱尔兰科学基金会;
关键词
FROBENIUS REPRESENTATION; PARTITIONS; CONJUGATION; SERIES; DYSONS;
D O I
10.1093/imrn/rnp131
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions.
引用
收藏
页码:238 / 260
页数:23
相关论文
共 33 条
[1]   Partitions:: At the interface of q-series and modular forms [J].
Andrews, GE .
RAMANUJAN JOURNAL, 2003, 7 (1-3) :385-400
[2]   DYSONS CRANK OF A PARTITION [J].
ANDREWS, GE ;
GARVAN, FG .
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 1988, 18 (02) :167-171
[3]   Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks [J].
Andrews, George E. .
INVENTIONES MATHEMATICAE, 2007, 169 (01) :37-73
[4]   The number of smallest parts in the partitions of n [J].
Andrews, George E. .
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 2008, 624 :133-142
[5]  
[Anonymous], 1988, LOST NOTEBOOK OTHER
[6]   Relations between the ranks and cranks of partitions [J].
Atkin, AOL ;
Garvan, FG .
RAMANUJAN JOURNAL, 2003, 7 (1-3) :343-366
[7]   Some observations on Dyson's new symmetries of partitions [J].
Berkovich, A ;
Garvan, FG .
JOURNAL OF COMBINATORIAL THEORY SERIES A, 2002, 100 (01) :61-93
[8]  
BRINGMANN K, ANN MATH IN PRESS
[9]  
BRINGMANN K, 2007, INT MATH RES NOTICES, P1
[10]  
BRINGMANN K, MATH RES LE IN PRESS