1 Introduction and statement of results
We recall that a partition of an integer n is a non-increasing sequence
$\lambda = \lambda _1+ \dots + \lambda _\ell $
of positive integers whose parts, denoted by
$\lambda _j$
(
$1 \leq j \leq \ell $
), sum up to n. Restrictions on partitions concerned with the parity of the parts in the partition have played a major role in the history of partition theory; see, e.g., Andrews’ article [Reference Andrews3] for an excellent overview of many ways parity has appeared in partition theory. In Andrews’ recent works in this direction [Reference Andrews4, Reference Andrews5], he introduced the notion of “partitions with parts separated by parity” in connection with mock theta functions. A partition
$\lambda $
has parts separated by parity
Footnote
1
if each even part
$\lambda _j$
is larger than each odd part
$\lambda _k$
, or if each odd part
$\lambda _j$
is larger than each even part
$\lambda _k$
. Andrews used the notation
$p_{\mathrm {xy}}^{\mathrm {zw}}(n)$
to count various kinds of partitions of n with parts separated by parity. In his notation, we let
$\{\mathrm {x},\mathrm {z}\}=\{\mathrm {e},\mathrm {o}\}$
, where the case
$\mathrm {z} = \mathrm {e}, \mathrm {x} = \mathrm {o}$
(resp.,
$\mathrm {z} = \mathrm {o}, \mathrm {x} = \mathrm {e}$
) signifies that even parts are larger than odd parts (resp., that odd parts are larger than even parts). Andrews then allowed
$\mathrm {y}$
and
$\mathrm {w}$
to denote either
$\mathrm {u}$
or
$\mathrm {d}$
, which represent unrestricted and distinct, respectively, and signify whether parts of parity
$\mathrm {x}$
and
$\mathrm {z}$
are unrestricted or must be distinct, respectively. Andrews studied the eight possible functions built this way; e.g.,
$p_{\mathrm{od}}^{\mathrm{eu}}(n)$
counts the number of partitions of n with even parts larger than odd parts which have all odd parts distinct.
Andrews’ papers [Reference Andrews4, Reference Andrews5] and the follow-up work of the first author and Jennings–Shaffer [Reference Bringmann and Jennings-Shaffer12] focused on the generating functions
$$ \begin{align*} F_{\mathrm{xy}}^{\mathrm{zw}}(q) := \sum_{n \geq 0} p_{\mathrm{xy}}^{\mathrm{zw}}(n) q^n. \end{align*} $$
These generating functions have connections to a wide variety of modular-type objects. To state some of these results, we define the q-Pochhammer symbol
$$ \begin{align*} \left( a \right)_n = \left( a;q \right)_n := \prod_{k=0}^{n-1} \left( 1 - aq^k \right), \ \ \ \ \ \left( a_1, a_2, \dots, a_m; q \right)_n := \prod_{k=1}^m \left( a_k; q \right)_n, \end{align*} $$
for
$n \in \mathbb {N}_0 \cup \{\infty \},\,m\in \mathbb {N}$
. As a first example, Andrews proved [Reference Andrews4, Equation (2.1)] that
$$ \begin{align*} F_{\mathrm{eu}}^{\mathrm{ou}}(q) = \dfrac{1}{(1-q)\left( q^2;q^2 \right)_\infty}, \end{align*} $$
which is essentially a modular form of weight
$-\frac {1}{2}$
. The papers [Reference Andrews5, Reference Bringmann and Jennings-Shaffer12] both proved (see also notation and comments from [Reference Bringmann, Craig and Nazaroglu10, Reference Bringmann, Craig and Nazaroglu11]) that
$$ \begin{align*} F_{\mathrm{od}}^{\mathrm{eu}}(q) = \dfrac{1}{\left( q^2;q^2 \right)_\infty}\left( 1 - \dfrac{\sigma(-q)}{2} + \dfrac{\left( -q;-q \right)_\infty}{2} \right), \end{align*} $$
where
$$ \begin{align*} \sigma(q) := \sum_{n \geq 0} \dfrac{q^{\frac{n(n+1)}{2}}}{\left( -q;q \right)_n} \end{align*} $$
is the famous Ramanujan
$\sigma $
-function from his lost notebook [Reference Ramanujan22]. It was shown by Andrews–Dyson–Hickerson [Reference Andrews, Dyson and Hickerson6] and Cohen [Reference Cohen13] that
$\sigma (q)$
is related to Maass forms. The variety of generating functions in [Reference Andrews5] also connects to modular forms and mock modular forms. Andrews’ work has given rise to many other examples, looking at further generating function identities [Reference Bringmann and Jennings-Shaffer12], asymptotic formulas [Reference Bringmann, Craig and Nazaroglu10, Reference Bringmann, Craig and Nazaroglu11], and connections with mock theta functions [Reference Fu and Tang17]. Although the definitions of each
$p_{\mathrm {xy}}^{\mathrm {zw}}(n)$
are quite similar, the properties that arise from their generating functions are very different (including modular forms, mock modular forms, false modular forms, and mock Maass theta functions like
$\sigma (q)$
); this suggests a rich theory yet to be fully understood.
In this article, we extend this framework to overpartitions. An overpartition [Reference Corteel and Lovejoy14] is a partition where the first instance of each part may possibly be overlined. For instance, there are 8 overpartitions of 3, given by
$$ \begin{align*} 3,\ \overline 3,\ 2+1,\ \overline 2 + 1,\ 2 + \overline 1,\ \overline 2 + \overline 1,\ 1 + 1 + 1,\ \overline 1 + 1 + 1. \end{align*} $$
Overpartitions have played numerous roles in q-series and combinatorics (see, e.g., [Reference Bessenrodt and Pak8, Reference Lovejoy20] and the references therein), mathematical physics (see, e.g., [Reference Fortis, Jacob and Mathieu16]), symmetric functions (see, e.g., [Reference Brenti9]), representation theory (see, e.g., [Reference Kang and Kwon19]), and algebraic number theory (see, e.g., [Reference Lovejoy21]). The goal of this article is to pursue this theme in the context of partitions with parts separated by parity.
We define two variations of overpartitions with parts separated by parity. In both cases, we say that an overpartition
$\lambda $
has parts separated by parity
Footnote
2
if each even
$\lambda _j$
is larger than each odd
$\lambda _k$
, or if each odd
$\lambda _j$
is larger than each even
$\lambda _k$
. Because allowing the distinct parts to be overlined introduces powers of two that are both complicated and unenlightening, we introduce an additional constraint in order to create an alternative family of overpartitions. If we impose the restriction that any parts required to be distinct cannot be overlined, then we call these modified. We let
$\overline {p}_{\mathrm {xy}}^{\mathrm {zw}}\left ( n\right )$
and
$\underline {p}_{\mathrm {xy}}^{\mathrm {zw}}\left ( n\right )$
denote the functions which count the number of overpartitions with parts separated by parity and the modified variation, respectively. For instance, we have that
$\overline {p}_{\mathrm{od}}^{\mathrm{eu}}(3) = 6$
and
$\underline {p}_{\mathrm{od}}^{\mathrm{eu}}(3) = 3$
, respectively, since the corresponding overpartitions are
$$ \begin{align*} 3 ,\ 2+1,\ \overline 3,\ \overline 2 + 1,\ 2 + \overline 1,\ \overline 2 + \overline 1 \end{align*} $$
and
$$ \begin{align*} 3,\ 2+1,\ \overline 2 + 1. \end{align*} $$
The difference between the two cases is that the overpartitions counted by
$\overline {p}_{\mathrm{od}}^{\mathrm{eu}}(3)$
are permitted to have overlines on both the even and odd parts, whereas the overpartitions counted by
$\underline {p}_{\mathrm{od}}^{\mathrm{eu}}(3)$
are only permitted to have overlines on the even parts.
In this article, we consider the 16 generating functions
$$ \begin{align*} \overline{F}_{\mathrm{xy}}^{\mathrm{zw}}(q) := \sum_{n \geq 0} \overline{p}_{\mathrm{xy}}^{\mathrm{zw}}(n) q^n, \ \ \ \ \ \underline{F}_{\mathrm{xy}}^{\mathrm{zw}}(q) := \sum_{n \geq 0} \underline{p}_{\mathrm{xy}}^{\mathrm{zw}} (n) q^n. \end{align*} $$
We also need the Jacobi theta function
$$ \begin{align*} \Theta\left( \tau\right) := \sum_{n \in \mathrm{z}Z} q^{n^2} \ \ \ \ \ (q:=e^{2\pi i\tau}). \end{align*} $$
This is a modular form of weight
$\frac {1}{2}$
. Our first results are formulas for the generating functions
$\overline {F}_{\mathrm {xy}}^{\mathrm {zw}}(q)$
.
Theorem 1.1 The following identities hold:
$$ \begin{align} \overline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q) &= \frac{\left( -q^2; q^2\right) _\infty}{2\left( q^2; q^2\right) _\infty} \left( \Theta^2(\tau) + 1 \right), \end{align} $$
$$ \begin{align} \overline{F}_{\mathrm{ou}}^{\mathrm{eu}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} + \frac{2q}{1-q} \frac{\left( -q^2; q^2\right) _\infty}{\left( q^2; q^2\right) _\infty} \sum_{n \geq 0} \frac{\left( -q, q^2; q^2\right) _{n}q^{2n}}{\left(q^3,-q^2; q^2\right)_{n}} , \end{align} $$
$$ \begin{align} \overline{F}_{\mathrm{ed}}^{\mathrm{od}}(q) &= \left( -2q;q^2 \right)_\infty - \dfrac{q}{1-q} \left( -2q^2;q^2 \right)_\infty + \dfrac{q}{1-q} \left( -2q;q^2 \right)_\infty, \end{align} $$
$$ \begin{align} \overline{F}_{\mathrm{od}}^{\mathrm{ed}}(q) &= \left( -2q^2;q^2 \right)_\infty + \dfrac{q}{1-q} \left( 3 \left( -2q^2;q^2 \right)_\infty - \left( -2q;q^2 \right)_\infty \right), \end{align} $$
$$ \begin{align} \overline{F}_{\mathrm{eu}}^{\mathrm{od}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{2^n q^{n^2}}{\left( -q^2;q^2 \right)_n}, \end{align} $$
$$ \begin{align} \overline{F}_{\mathrm{od}}^{\mathrm{eu}}(q) &= - \dfrac{\left( -2q, q^2; q^2 \right)_\infty}{\left( -q^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{(-1)^n q^{2n}}{\left( 2q; q^2 \right)_{n+1}} + 2 \sum_{n \geq 0} \dfrac{(-1)^n 2^n q^{n^2+2n}}{\left( 1 + q^{2n+2} \right) \left( 2q; q^2 \right)_{n+1}}, \end{align} $$
$$ \begin{align}\overline{F}_{\mathrm{ed}}^{\mathrm{ou}}(q) &= \frac{\left(-q^2;q^2\right)_\infty}{\left(q^2;q^2\right)_\infty} + 2q\left(-2q^2;q^2\right)_\infty \sum_{n\ge0} \frac{(-1)^n q^{2n}}{\left(2q;q^2\right)_{n+1}} \nonumber\\&\quad + \frac{2q\left(-q;q^2\right)_\infty}{\left(q^3;q^2\right)_\infty} \sum_{n\ge0} \frac{(-1)^n 2^n\left(-q;q^2\right)_n q^{n^2+3n}}{\left(2q,-q^2;q^2\right)_{n+1}}, \end{align} $$
$$ \begin{align} \overline{F}_{\mathrm{ou}}^{\mathrm{ed}}(q) &= \left( -2q^2;q^2 \right)_\infty + \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right)_n \left( -2q^{2n+2};q^2 \right)_\infty}{\left( q;q^2 \right)_{n+1}} q^{2n+1}. \end{align} $$
Remark We note a connection to modular forms. In particular, (1.1) yields a sum of weakly holomorphic modular forms of weights
$-\frac 12$
and
$\frac 12$
.
We next turn to
$\underline {F}^{\mathrm {zw}}_{\mathrm {xy}}(q)$
. We note that in cases where odd and even parts are either both unrestricted or both distinct, the generating functions boil down to either
$\overline {F}$
or F, respectively; that is,
$$ \begin{align*} \underline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q) = \overline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q), \ \ \ \underline{F}^{\mathrm{eu}}_{\mathrm{ou}}(q) = \overline{F}^{\mathrm{eu}}_{\mathrm{ou}}(q), \ \ \ \underline{F}_{\mathrm{ed}}^{\mathrm{od}}(q) = F_{\mathrm{ed}}^{\mathrm{od}}(q), \ \ \ \underline{F}_{\mathrm{od}}^{\mathrm{ed}}(q) = F_{\mathrm{od}}^{\mathrm{ed}}(q). \end{align*} $$
These are treated either in Theorem 1.1 or in previous works [Reference Andrews5, Reference Bringmann and Jennings-Shaffer12]. Thus, we focus on the remaining four examples of interest, in which one parity is restricted and one is unrestricted. In order to state this theorem, we recall the third order mock theta function
$$ \begin{align*} \phi(q) := \sum_{n \geq 0} \dfrac{q^{n^2}}{\left( -q^2;q^2 \right)_n}. \end{align*} $$
We prove the following result on the remaining modified generating functions.
Theorem 1.2 The following identities hold:
$$ \begin{align} \,\underline{F}_{\mathrm{eu}}^{\mathrm{od}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \phi(q), \end{align} $$
$$ \begin{align} \underline{F}_{\mathrm{od}}^{\mathrm{eu}}(q) &= \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} - 2q \left( -q; q^2 \right)_\infty \sum_{n\ge0}\frac{q^n}{\left(-q^2;q^2\right)_{n+1}}\nonumber\\ &\quad + 4q \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{(-1)^n q^{n^2 + 2n}}{\left( 1 + q^{2n+2} \right) \left( q;q^2 \right)_{n+1}}, \end{align} $$
$$ \begin{align} \underline{F}_{\mathrm{ed}}^{\mathrm{ou}}(q) &= - 2q \left( -q^2; q^2 \right)_\infty \sum_{n \geq 0} \dfrac{q^{n}}{\left( -q^2;q^2 \right)_{n+1}} + \dfrac{\left( -q;q^2 \right)_\infty}{\left( q;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{ (-1)^n \left( -q;q^2 \right)_{n+1} q^{n^2+n}}{\left( -q^2, q; q^2 \right)_{n+1}}, \end{align} $$
$$ \begin{align} {\underline{F}}_{\mathrm{ou}}^{\mathrm{ed}}(q) &= \left( -q^2;q^2 \right)_\infty \left( 1 + 2q \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right)_n q^{2n} }{\left( q, -q^2;q^2 \right)_n}\right). \end{align} $$
The article is organized as follows. In Section 2, we provide several known q-series identities which we use in the proof of our main results. In Sections 3 and 4, we prove the results regarding standard and modified overpartitions with parts separated by parity, respectively. In Section 5, we discuss future directions for work.
2 Preliminaries
In this section, we recall various identities we use in this article. Andrews’ book [Reference Andrews2] or Fine’s book [Reference Fine15] are excellent introductions to q-series transformations of these types.
2.1 The Heine transformation
We begin by stating one of Heine’s transformations.
Lemma 2.1 [Reference Andrews2, Corollary 2.3]
For
$|q|,|t|<1$
and
$0 < |b| < 1$
, we have
$$ \begin{align*} \sum_{n\geq0} \frac{\left( a, b\right) _n t^n}{\left( q, c\right) _n} = \frac{\left( b, at\right) _\infty}{\left( c, t\right) _\infty} \sum_{n\geq0} \frac{\left( \frac{c}{b}, t\right) _nb^n}{\left( q, at\right) _n}. \end{align*} $$
We also use another Heine transformationFootnote 3 (see [Reference Gasper and Rahman18, (III.2)]).
Lemma 2.2 For
$|q|, |t| <1$
and
$ 0 < |c| < |b| < 1$
, we have
$$ \begin{align*} \sum_{n\ge0} \frac{(a,b)_n t^n}{(q,c)_n} = \dfrac{\left(\frac{c}{b},bt\right)_\infty}{(c,t)_\infty} \sum_{n\ge0} \frac{\left( \frac{abt}{c},b\right)_n\left( \dfrac{c}{b} \right)^n}{(q,bt)_n}. \end{align*} $$
Lemma 2.3 [Reference Andrews, Subbarao and Vidyasagar7, Equation (4.1)]
We haveFootnote 4
$$ \begin{align*} \sum_{n \geq 0} \dfrac{(x)_nq^n}{(y)_n} = \dfrac{q(x)_\infty}{y\left( 1 - \frac{xq}{y} \right) (y)_\infty} + \dfrac{1 - \frac{q}{y}}{1 - \frac{xq}{y}}. \end{align*} $$
2.2 Identities from the lost notebook
A final identity which we require emerges from the early work on Ramanujan’s Lost Notebook. In particular, we use the following very general transformation formula due to Andrews that is used frequently to prove identities for partial theta functions.
Lemma 2.4 [Reference Andrews1, Theorem 1]
We have
$$ \begin{align*} \sum_{n\ge0} \frac{(B, -Abq)_n q^n}{(-aq, -bq)_n}&= \frac{-a^{-1} (B,-Abq) _\infty}{(-ab, -bq)_\infty} \sum_{n\ge0} \frac{\left( A^{-1}\right) _n\left( \frac{Abq}{a} \right) ^{n}}{\left( - \frac{B}{a} \right) _{n+1}}\\&\quad + (1+b) \sum_{n \ge0} \frac{\left( -a^{-1}\right) _{n+1} \left( - \frac{ABq}{a} \right) _n \left( -b\right) ^n}{\left( -\frac{B}{a}, \frac{Abq}{a} \right) _{n+1}}.\nonumber \end{align*} $$
Letting
$A\mapsto \frac {A}{b}$
,
$B\mapsto -Bq^2$
,
$q\mapsto q^2$
, and then
$b\to 0$
, we obtain the following.
Lemma 2.5 [Reference Andrews1, Equation (3.9)]
We have
$$ \begin{align*} \sum_{n \geq 0} \dfrac{\left( -Bq^2, -Aq^2; q^2 \right)_nq^{2n}}{\left( -aq^2;q^2 \right)_n} & = \dfrac{-a^{-1} \left( -Bq^2, -Aq^2; q^2 \right)_\infty}{\left( -aq^2;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{\left( \frac{Aq^2}{a} \right)^n}{\left( \frac{Bq^2}{a}; q^2 \right)_{n+1}} \\& \quad + \sum_{n \geq 0} \dfrac{\left( -a^{-1}; q^2 \right)_{n+1} \left( \frac{AB}{a} \right)^n q^{n^2+3n}}{\left( \frac{Bq^2}{a}, \frac{Aq^2}{a}; q^2 \right)_{n+1}}. \end{align*} $$
3 Proof of Theorem 1.1
In this section, we prove all claimed identities for the functions
$\overline {F}_{\mathrm {xy}}^{\mathrm {zw}}$
.
3.1 Proof of (1.1)
We may build up a generating function for
$\overline {F}_{\mathrm{eu}}^{\mathrm{ou}}$
by summing over the cases where
$2n$
is the size of the maximum even part of the overpartition for each
$n \in \mathbb {N}$
. There are two possibilities: either the overpartition has a marked part of size
$2n$
or it only has unmarked parts of size
$2n$
. Both cases yield the same expression for the generating function, so the construction is a multiple of two. This gives
$$ \begin{align*} \overline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q) &= 2 \frac{\left( -q;q^2\right) _\infty}{\left( q;q^2\right) _\infty} \sum_{n \geq 0 } \frac{\left( -q^2; q^2\right) _{n-1} \left( q;q^2\right) _nq^{2n}}{\left( q^2, -q; q^2\right) _n} \\ &= \frac{\left( -q;q^2\right) _\infty} {\left( q;q^2\right) _\infty} \sum_{n \geq 0 } \frac{\left( -1, q;q^2\right) _nq^{2n}}{\left( q^2, -q; q^2\right) _n}. \end{align*} $$
Using Lemma 2.1 with
$q \mapsto q^2$
,
$a = -1$
,
$b = q$
,
$c = -q$
, and
$t = q^2$
, we obtain
$$ \begin{align} \overline{F}_{\mathrm{eu}}^{\mathrm{ou}}(q) = 2\frac{\left( -q^2; q^2\right) _\infty}{\left( q^2; q^2\right) _\infty} \sum_{n \geq 0} \frac{q^n}{1 + q^{2n}}. \end{align} $$
By work of Kronecker (see [Reference Warnaar23])
$$ \begin{align*} 1+4\sum_{n\ge 1} \frac{(-1)^n q^n}{1+q^{2n}} = \frac{(q,q)_\infty}{(-q,-q)_\infty}. \end{align*} $$
Replacing
$q$
by
$-q$
gives
$$ \begin{align*} \sum_{n\ge1} \frac{q^n}{1+q^{2n}} = \frac{1}{4}\left(\Theta^2(\tau)-1\right). \end{align*} $$
Plugging this into (3.1) gives the claim.
3.2 Proof of (1.2)
The construction of the generating function
$\overline {F}_{\mathrm{ou}}^{\mathrm{eu}}(q)$
is analogous to that of
$\overline {F}_{\mathrm{eu}}^{\mathrm{ou}}(q)$
. Namely, we sum over the instances where the largest part is odd, and we distinguish two cases based on whether there exists an overlined part of that size. As with
$\overline {F}_{\mathrm{eu}}^{\mathrm{ou}}(q)$
, these cases turn out the same generating function. Moreover, we account separately for the situation where there are no odd parts with the leading term. We obtain the identity
$$ \begin{align*} \overline{F}_{\mathrm{ou}}^{\mathrm{eu}}(q) = \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} + \frac{2q}{1-q} \frac{\left( -q^2; q^2\right) _\infty}{\left( q^2; q^2\right) _\infty} \sum_{n \geq 0} \frac{\left( -q, q^2; q^2\right) _{n}q^{2n}}{\left( q^3,-q^2; q^2\right)_{n}}. \end{align*} $$
This completes the proof.
3.3 Proof of (1.3)
We again follow Section 3.1, splitting into two cases where the largest even part is either marked or not and including a leading term to account for the case where there are no even parts, and we obtain
$$ \begin{align*} \overline{F}_{\mathrm{ed}}^{\mathrm{od}}(q) &= \left(-2q; q^2\right)_\infty + 2 q^2 \left( -2q^3; q^2\right) _\infty \sum_{n\ge 0} \frac{\left( -2q^2; q^2\right) _nq^{2n}}{\left( -2q^3; q^2\right) _n}. \end{align*} $$
Using Lemma 2.3 with
$q \mapsto q^2$
and then
$x = -2q^2$
and
$y = -2q^3$
, gives the claim.
3.4 Proof of (1.4)
We again follow Section 3.1, splitting into two cases depending on whether the largest odd part is either marked or not and including a leading term to account for the case where there are no odd parts, and we obtain
$$ \begin{align*} \overline{F}_{\mathrm{od}}^{\mathrm{ed}}(q) = \left(-2q^2; q^2\right) _\infty + 2 q \left( -2q^2; q^2\right) _\infty \sum_{n \ge 0} \frac{\left( -2q; q^2\right) _n q^{2n}}{\left( -2q^2; q^2\right) _n}. \end{align*} $$
Using Lemma 2.3 with
$q \mapsto q^2$
, and then
$x = -2q$
and
$y = - 2q^2$
, we obtain the claim.
3.5 Proof of (1.5)
We again follow Section 3.1, splitting into two cases depending on whether the largest even part is either marked or not, and obtain
$$ \begin{align*} \overline{F}_{\mathrm{eu}}^{\mathrm{od}}(q) = 2 \sum_{n \geq 0} \dfrac{\left( -q^2;q^2 \right)_{n-1} \left( -2q^{2n+1};q^2 \right)_\infty q^{2n}}{\left( q^2;q^2 \right)_n} = \left( -2q;q^2 \right)_\infty \sum_{n \geq 0} \dfrac{\left( -1;q^2 \right)_nq^{2n}}{\left( -2q,q^2;q^2 \right)_n}. \end{align*} $$
Letting
$q \mapsto q^2$
,
$t = -\frac {2q}{a}$
,
$a\to \infty $
,
$b=q^2$
, and
$c=q$
in Lemma 2.1, we conclude the claimed formula for
$\overline {F}_{\mathrm{eu}}^{\mathrm{od}}(q)$
.
3.6 Proof of (1.6)
We again follow Section 3.1, splitting into two cases depending on whether the largest odd part is either marked or not and including a leading term to account for the case where there are no odd parts. We obtain
$$ \begin{align*} \overline{F}_{\mathrm{od}}^{\mathrm{eu}}(q) &=\frac{\left(-q^2; q^2\right)_\infty}{\left(q^2; q^2\right)_\infty} + 2 \sum_{n \ge 0} \frac{\left( -2q; q^2\right) _n \left( -q^{2n+2}; q^2\right) _\infty q^{2n+1}}{\left( q^{2n+2}; q^2\right) _\infty} \\ &= \frac{\left(-q^2; q^2\right)_\infty}{\left(q^2; q^2\right)_\infty} + 2q \frac{\left( -q^2; q^2\right) _\infty}{\left( q^2; q^2\right) _\infty} \sum_{n \ge 0} \frac{\left( -2q,q^2; q^2\right) _nq^{2n}}{\left( -q^2; q^2\right) _n}. \end{align*} $$
Applying Lemma 2.5 with
$a=1$
,
$A = -1$
, and
$B = \frac {2}{q}$
, gives (1.6).
3.7 Proof of (1.7)
We again follow Section 3.1, splitting into two cases depending on whether the largest even part is either marked or not and including a leading term to account for the case where there are no even parts. We obtain
$$ \begin{align*} \overline{F}_{\mathrm{ed}}^{\mathrm{ou}}(q) &= \frac{\left(-q^2; q^2\right)_\infty}{\left(q^2; q^2\right)_\infty} + 2 \sum_{n \ge 0} \frac{\left( -2q^2; q^2\right) _n \left( -q^{2n+3}; q^2\right) _\infty q^{2n+2}}{\left( q^{2n+3}; q^2\right) _\infty},\\ &= \frac{\left(-q^2; q^2\right)_\infty}{\left(q^2; q^2\right)_\infty} + 2q^2 \frac{\left( -q^3; q^2\right) _\infty}{\left( q^3; q^2\right) _\infty} \sum_{n \ge 0} \frac{\left( -2q^2, q^3; q^2\right) _nq^{2n}}{\left( -q^3; q^2\right) _n}. \end{align*} $$
Using Lemma 2.5 with
$B=2$
,
$A = -q$
, and
$a=q$
, we obtain
$$ \begin{align*} \sum_{n \geq 0} \dfrac{\left( -2q^2, q^3; q^2 \right)_nq^{2n}}{\left( -q^3; q^2 \right)_n} &= - \dfrac{\left( -2q^2, q^3; q^2 \right)_\infty}{q\left( -q^3; q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{(-1)^n q^{2n}}{\left( 2q; q^2 \right)_{n+1}} \\&\quad + \left( 1+\dfrac{1}{q} \right) \sum_{n \geq 0} \dfrac{(-1)^n 2^n \left(-q;q^2 \right)_n q^{n^2+3n}}{\left( 2q, -q^2; q^2 \right)_{n+1}}. \end{align*} $$
Plugging this into the formula for
$\overline {F}_{\mathrm{ed}}^{\mathrm{ou}}(q)$
then gives the claim.
3.8 Proof of (1.8)
We again follow Section 3.1, splitting into two cases depending on whether the largest odd part is either marked or not and including a leading term to account for the case where there are no odd parts. This quickly yields the claimed result, namely
$$ \begin{align*} \overline{F}_{\mathrm{ou}}^{\mathrm{ed}}(q) &= \left( -2q^2;q^2 \right)_\infty + \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right)_n \left( -2q^{2n+2};q^2 \right)_\infty q^{2n+1}}{\left( q;q^2 \right)_{n+1}} \\ &= \left( -2q^2;q^2 \right)_\infty \left( 1 + \dfrac{2q}{1-q} \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right) _nq^{2n}}{\left( q^3, -2q^2;q^2 \right) _n} \right). \end{align*} $$
4 Proof of Theorem 1.2
In this section, we prove Theorem 1.2.
4.1 Proof of (1.9)
We modify our argument from Section 3.1 and obtain
$$ \begin{align*} \underline{F}_{\mathrm{eu}}^{\mathrm{od}}(q) = 2 \left( -q;q^2 \right)_\infty \sum_{n \geq 0} \dfrac{\left( -q^2;q^2 \right)_{n-1}q^{2n}}{\left( q^2, -q;q^2 \right)_n} = \left( -q;q^2 \right)_\infty \sum_{n \geq 0} \dfrac{\left( -1;q^2 \right)_nq^{2n}}{\left( q^2, -q;q^2 \right)_n}. \end{align*} $$
We again use Lemma 2.1 with
$q \mapsto q^2$
,
$t = - \frac {q}{a}$
,
$a\to \infty $
,
$b = q^2$
, and
$c = -q^2$
to conclude the claim.
4.2 Proof of (1.10)
As above, we have
$$ \begin{align*} \underline{F}_{\mathrm{od}}^{\mathrm{eu}}(q) = \dfrac{\left( -q^2;q^2 \right)_\infty}{\left( q^2;q^2 \right)_\infty} \left( 1 + 2 q \sum_{n \geq 0} \dfrac{\left( -q;q^2 \right)_n \left( q^2;q^2 \right)_nq^{2n}}{\left( -q^2;q^2 \right)_n} \right). \end{align*} $$
Using Lemma 2.5 with
$B = \frac {1}{q}$
,
$A = -1$
, and
$a=1$
, we have
$$ \begin{align*} \sum_{n \geq 0} \dfrac{\left( -q, q^2; q^2 \right)_n}{\left( -q^2; q^2 \right)_n} q^{2n} &= - \dfrac{\left( -q, q^2; q^2 \right)_\infty}{\left( -q^2; q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{(-1)^n q^{2n}}{\left( q;q^2 \right)_{n+1}} + 2 \sum_{n \geq 0} \dfrac{(-1)^n q^{n^2 + 2n}}{\left( 1 + q^{2n+2} \right) \left( q;q^2 \right)_{n+1}}. \end{align*} $$
We next claim that the first summation can be written
$$ \begin{align*} \sum_{n \geq 0} \dfrac{(-1)^n q^{2n}}{\left( q;q^2 \right)_{n+1}} = \sum_{n\ge0}\frac{q^n}{\left(-q^2;q^2\right)_{n+1}}. \end{align*} $$
Changing variables
$q \mapsto q^{\frac 12}$
, the claim is equivalent to
$$ \begin{align*} \frac{1}{1-q^{\frac{1}{2}}} \sum_{n\ge0} \frac{(-1)^n q^n}{\left(q^{\frac{3}{2}};q\right)_n} = \frac{1}{1+q} \sum_{n\ge0} \frac{q^{\frac{n}{2}}}{\left(-q^2;q\right)_n}. \end{align*} $$
Using Lemma 2.2 with
$a = 0, b = q, c = -q^2$
, and
$t=q^{\frac 12}$
, we obtain the claim. In turn, we complete the proof.
4.3 Proof of (1.11)
We follow the argument in Section 4.1. Namely, we split into two terms based on whether the largest even part is marked or not and include a leading term to account for the case where there are no even parts. This gives
$$ \begin{align*} \underline{F}_{\mathrm{ed}}^{\mathrm{ou}}(q) &= \dfrac{\left( -q;q^2 \right)_\infty}{\left( q;q^2 \right)_\infty} \left( 1 + 2 \sum_{n \geq 1} \dfrac{\left( q;q^2 \right)_n \left( -q^2;q^2 \right)_{n-1}q^{2n}}{\left( -q;q^2 \right)_n} \right) = \dfrac{\left( -q;q^2 \right)_\infty}{\left( q;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{\left( q, -1;q^2 \right)_n q^{2n}}{\left( -q;q^2 \right)_n}. \end{align*} $$
Using Lemma 2.5 with
$B = -\frac {1}{q}$
,
$A=\frac {1}{q^2}$
, and
$a = \frac {1}{q}$
, we have
$$ \begin{align*} \sum_{n \geq 0} \dfrac{\left( q, -1; q^2 \right)_nq^{2n}}{\left( -q;q^2 \right)_n} = - \dfrac{q\left( q, -1; q^2 \right)_\infty}{\left( -q;q^2 \right)_\infty} \sum_{n \geq 0} \dfrac{q^{n}}{\left( -q^2;q^2 \right)_{n+1}} + \sum_{n \geq 0} \dfrac{(-1)^n \left( -q;q^2 \right)_{n+1} q^{n^2+n}}{\left( -q^2, q; q^2 \right)_{n+1}}, \end{align*} $$
from which the desired result follows.
5 Future work
In this article, we produce a number of new formulas for overpartitions with parts separated by parity. These new formulas, along with connections to modular and mock modular forms, suggest a number of applications and open questions:
-
(1) What are the asymptotics growth rates of the various functions counting overpartitions with parts separated by parity?
-
(2) Which families of overpartitions with parts separated by parity are connected to modular forms, mock modular forms, or some other modular object?
-
(3) Are there two-variable extensions for these generating functions which have connections to two-variable modular objects like Jacobi forms?
-
(4) Do any of these overpartition families possess Ramanujan-type congruences?
Acknowledgments
This article was partially written while the second author was visiting the Max Planck Institute for Mathematics, whose hospitality she acknowledges. The authors also thank Koustav Banerjee for directing our attention toward Lemma 2.4, which improved some of their identities, and for many discussions about these identities. The authors also thank the referee for carefully reading their article. The views expressed in this article are those of the authors and do not reflect the official policy or position of the U.S. Naval Academy, Department of the Navy, the Department of Defense, or the U.S. Government.
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