Abstract
Motivated by work of Ramanujan, Freeman Dyson defined the rank of an integer partition to be its largest part minus its number of parts. If $N(m,n)$ denotes the number of partitions of $n$ with rank $m$, then it turns out that \[ R(w;q):=1+\!\sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty} \!\!\! N(m,n)w^mq^n \! =\! 1+\!\sum_{n=1}^{\infty}\!\frac{q^{n^2}} {\prod_{j=1}^{n}(1\!-\!(w\!+\!w^{-1})q^j\!+ q^{2j})}. \] We show that if $\zeta\neq 1$ is a root of unity, then $R(\zeta;q)$ is essentially the holomorphic part of a weight $1/2$ weak Maass form on a subgroup of $\operatorname{SL}_2(\mathbb Z)$. For integers $0\leq r\lt t$, we use this result to determine the modularity of the generating function for $N(r,t;n)$, the number of partitions of $n$ whose rank is congruent to $r\pmod t$. We extend the modularity above to construct an infinite family of vector valued weight $1/2$ forms for the full modular group $\operatorname{SL}_2(\mathbb Z)$, a result which is of independent interest.
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JOURNAL = {J. London Math. Soc. $(2)$ },
VOLUME = {2},
YEAR = {1936},
PAGES = {55--80},
NOTE = {reprinted in {\it Ramanujan\/{\rm :} Essays and Surveys},
{\it Hist. Math.} {\bf 22},
{325--347},
{Amer. Math. Soc.},
{Providence, RI},
2001},
MRCLASS = {11B65 (01A70 33D15)},
MRNUMBER = {1862757},
ZBLNUMBER={013.11502},
} -
[Watson2] G. N. Watson, "The mock theta functions, II," Proc. London Math. Soc., vol. 42, pp. 274-304, 1937.
@article {Watson2,
author = {Watson, G. N.},
title={The mock theta functions, II},
journal={Proc. London Math. Soc.},
volume={42},
year=1937, pages={274-304}
} -
[WW] W. E. T. and G. N. Watson, Modern Analysis, Fourth ed., Cambridge: Cambridge Univ. Press, 1927.
@book{WW,
author = {E. T. Whittaker and G. N. Watson},
title = {Modern Analysis},
edition = {Fourth},
publisher = {Cambridge Univ. Press},
address={Cambridge},
year = {1927},
JFMNUMBER={53.0180.04},
} -
[Zwegers] S. P. Zwegers, "Mock $\theta$-functions and real analytic modular forms," in $q$-Series with Applications to Combinatorics, Number Theory, and Physics, Providence, RI: Amer. Math. Soc., 2001, pp. 269-277.
@incollection {Zwegers, MRKEY = {1874536},
AUTHOR = {Zwegers, S. P.},
TITLE = {Mock {$\theta$}-functions and real analytic modular forms},
BOOKTITLE = {{$q$}-Series with Applications to Combinatorics, Number Theory, and Physics},
venue = {Urbana, {IL},
2000},
SERIES = {Contemp. Math.},
NUMBER = {291},
PAGES = {269--277},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2001},
MRCLASS = {11F27 (33D15)},
MRNUMBER = {2003f:11061},
MRREVIEWER = {Bruce C. Berndt},
ZBLNUMBER = {1044.11029},
} -
[ZwegersPhD] S. P. Zwegers, "Mock theta functions," PhD Thesis , Universiteit Utrecht, 2002.
@phdthesis{ZwegersPhD,
author = {Zwegers, S. P.},
title={Mock theta functions},
school={Universiteit Utrecht},
year=2002 }