1 Introduction
A partition
$\pi $
of a positive integer n is a finite weakly decreasing sequence of positive integers
$\pi _1\geq \pi _2\geq \cdots \geq \pi _r$
such that
$\sum _{i=1}^r\pi _i=n$
. The
$\pi _i$
are called the parts of the partition
$\pi $
. Fraenkel and Peled [Reference Fraenkel, Peled and Nowakowski9] originally defined the minimal excludant for any set S of positive integers as the least positive integer not in S. In 2019, Andrews and Newman [Reference Andrews and Newman3] defined the minimal excludant of an integer partition
$\pi $
as the least positive integer missing from the partition, denoted by
$\textrm {mex}(\pi )$
. For example, there are five partitions of
$4$
:
$4$
with
$\textrm {mex}(\pi )=1$
;
$3+1$
with
$\textrm {mex}(\pi )=2$
;
$2+2$
with
$\textrm {mex}(\pi )=1$
;
$2+1+1$
with
$\textrm {mex}(\pi )=3$
;
$1+1+1+1$
with
$\textrm {mex}(\pi )=2$
. Andrews and Newman [Reference Andrews and Newman3, Theorem 1.1] established an elegant identity involving the quantity
$\sigma \textrm {mex}(n)$
, which denotes the sum of minimal excludants over all the partitions of n. More precisely, they proved that
$$ \begin{align} \sum_{n=0}^\infty\sigma\textrm{mex}(n)q^n=(-q;q)_\infty^2=\sum_{n=0}^\infty Q_2(n)q^n, \end{align} $$
where
$Q_2(n)$
denotes the number of partition pairs of n into distinct parts. Throughout the rest of this paper, we always assume that q is a complex number and adopt the standard notation:
$$ \begin{align*} (a;q)_\infty &:=\prod_{j=0}^\infty(1-aq^{\,j}),\\ (a_1,a_2,\ldots,a_m;q)_\infty &:=(a_1;q)_\infty(a_2;q)_\infty\cdots(a_m;q)_\infty. \end{align*} $$
Interestingly, (1.1) was derived earlier by Grabner and Knopfmacher [Reference Grabner and Knopfmacher11, (4.2)] under a different terminology. Recently, Ballantine and Merca [Reference Ballantine and Merca4] also proved (1.1) by employing purely combinatorial arguments.
Quite recently, Baruah et al. [Reference Baruah, Bhoria, Eyyunni and Maji5] investigated a refinement of the arithmetic function
$\sigma \textrm {mex}(n)$
by considering the parity of the minimal excludant. More specifically, in [Reference Baruah, Bhoria, Eyyunni and Maji5, (1.2) and (1.3)] they defined the two functions
$$ \begin{align} \sigma_o\textrm{mex}(n)=\sum_{\substack{\pi\vdash n\\ \textrm{mex}(n)~\textrm{odd}}}\textrm{mex}(\pi)\quad\textrm{and}\quad \sigma_e\textrm{mex}(n)=\sum_{\substack{\pi\vdash n\\ \textrm{mex}(n)~\textrm{even}}}\textrm{mex}(\pi). \end{align} $$
For instance, with
$n=4$
,
$\sigma _o\textrm {mex}(4)=1+1+3=5$
and
$\sigma _e\textrm {mex}(4)=2+2=4$
. By some q-series manipulations, Baruah et al. [Reference Baruah, Bhoria, Eyyunni and Maji5, Theorem 2.1] proved the following two partition identities which can be viewed as a refinement of (1.1): for any
$n\geq 0$
,
$$ \begin{align} \sigma_o\textrm{mex}(n)=Q_2^e(n)\quad\textrm{and}\quad\sigma_e\textrm{mex}(n) &=Q_2^o(n), \end{align} $$
where
$Q_2^e(n)$
and
$Q_2^o(n)$
denote the number of partition pairs of n into distinct parts with an even number of parts and an odd number of parts, respectively. As a consequence of (1.3), [Reference Baruah, Bhoria, Eyyunni and Maji5, Theorem 2.2] gives the following three congruences modulo
$4$
and
$8$
for
$\sigma _o\textrm {mex}(n)$
and
$\sigma _e\textrm {mex}(n)$
:
$$ \begin{align} \begin{aligned}\sigma_o\textrm{mex}(2n+1) &\equiv0\pmod{4},\\ \sigma_o\textrm{mex}(4n+1) &\equiv0\pmod{8},\end{aligned} \end{align} $$
$$ \begin{align}\ \ \kern1pt\quad\sigma_e\textrm{mex}(4n) &\equiv0\pmod{4}. \end{align} $$
Based on numerical evidence, Baruah et al. proposed the following conjecture.
Conjecture 1.1 [Reference Baruah, Bhoria, Eyyunni and Maji5, Conjecture 6.1]
For any
$n\geq 0$
,
$$ \begin{align} \sigma_o\textrm{mex}(8n+1) &\equiv0\pmod{16}, \end{align} $$
$$ \begin{align} \quad\, \sigma_e\textrm{mex}(8n) &\equiv0\pmod{8}. \end{align} $$
The main purpose of this paper is to confirm (1.6) and (1.7).
The rest of this paper is constructed as follows. In Section 2, we first collect some necessary identities, and next introduce some notation, terminology and theorems in the theory of modular forms. The proof of Theorem 1.2 is presented in Section 3. We conclude this paper with two remarks.
2 Preliminaries
To prove (1.6) and (1.7), we first need the following identities.
Lemma 2.1 (Jacobi’s triple product identity, [Reference Andrews and Berndt1, Lemma 1.2.2])
We have
$$ \begin{align} \sum_{n=-\infty}^\infty a^{n(n+1)/2}b^{n(n-1)/2}=(-a,-b,ab;ab)_\infty,\quad \text{where } |ab|<1. \end{align} $$
For notational convenience, we denote
$$ \begin{align*} J_{a,b}:=(q^a,q^{b-a},q^b;q^b)_\infty,\quad\overline{J}_{a,b}:=(-q^a,-q^{b-a},q^b;q^b)_\infty, \quad J_a:=J_{a,3a}=(q^a;q^a)_\infty. \end{align*} $$
Lemma 2.2. We have
$$ \begin{align} \dfrac{1}{J_1} &=\dfrac{1}{J_2^2}(\overline{J}_{6,16}+q\overline{J}_{2,16}), \end{align} $$
$$ \begin{align} J_1^2 &=\dfrac{J_2J_8^5}{J_4^2J_{16}^2}-2q\dfrac{J_2J_{16}^2}{J_8}, \end{align} $$
$$ \begin{align} \dfrac{1}{J_1^2} &=\dfrac{J_8^5}{J_2^5J_{16}^2}+2q\dfrac{J_4^2J_{16}^2}{J_2^5J_8}, \end{align} $$
$$ \begin{align} J_1^4 &=\dfrac{J_4^{10}}{J_2^2J_8^4}-4q\dfrac{J_2^2J_8^4}{J_4^2}, \end{align} $$
$$ \begin{align} \dfrac{1}{J_1^4} &=\dfrac{J_4^{14}}{J_2^{14}J_8^4}+4q\dfrac{J_4^2J_8^4}{J_2^{10}}, \end{align} $$
$$ \begin{align} \dfrac{J_1^2}{J_2^2} &=\dfrac{J_2^{22}}{J_1^{14}J_4^8}-16q\dfrac{J_4^8}{J_1^6J_2^2}. \end{align} $$
Proof. The identity (2.2) appears in [Reference Andrews, Berndt, Chan, Kim and Malik2, Lemma 4.1]. The identities (2.3)–(2.6) follow from [Reference Berndt6, page 40, Entries 25(i), (ii), (v) and (vi)] (see also [Reference Yao and Xia16, Lemmas 2.2 and 2.3]). It follows immediately from [Reference Berndt6, page 40, Entry 25(vii)] that
$$ \begin{align} \dfrac{J_2^{20}}{J_1^8J_4^8}-16q\dfrac{J_4^8}{J_2^4}=\dfrac{J_1^8}{J_2^4}. \end{align} $$
Multiplying by the factor
$J_2^2/J_1^6$
on both sides of (2.8) yields (2.7).
Lemma 2.3 [Reference Tang15, (2.10)]
We have
$$ \begin{align} J_1^3 &\equiv\overline{J}_{28,64}-3q\overline{J}_{20,64}+5q^3\overline{J}_{12,64}-7q^6\overline{J}_{4,64}\pmod{16}. \end{align} $$
Next, we collect some notation and terminology on the theory of modular forms. The full modular group is given by
$$ \begin{align*} \Gamma=\textrm{SL}_2(\mathbb{Z}) =\left\{\begin{pmatrix}a &b\\ c &d\end{pmatrix}\colon a,b,c,d\in\mathbb{Z},\text{and } ad-bc=1\right\}, \end{align*} $$
and for a positive integer N, the congruence subgroup
$\Gamma _1(N)$
is defined by
$$ \begin{align*} \Gamma_1(N)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in\Gamma\colon a\equiv d\equiv 1\pmod{N}, c\equiv 0\pmod{N}\right\}. \end{align*} $$
We denote by
$\gamma $
the matrix
$(\begin {smallmatrix}a &b\\ c &d \end {smallmatrix})$
, if not specified otherwise. Let
$\gamma $
act on
$\tau \in \mathbb {C}$
by the linear fractional transformation
$$ \begin{align*} \gamma\tau = \frac{a\tau+b}{c\tau+d} \quad \text{and} \quad \gamma\infty =\lim_{\tau\rightarrow \infty} \gamma\tau. \end{align*} $$
Let k be a positive integer and
$\mathbb {H}=\{\tau \in \mathbb {C}\colon \mathrm {Im}(\tau )>0\}$
. A holomorphic function
$f\colon \mathbb {H}\rightarrow \mathbb {C}$
is called a modular form with weight k for
$\Gamma _1(N)$
if it satisfies the following two conditions:
-
(1)
$f(\gamma \tau )=(c\tau +d)^kf(\tau )$
for all
$\gamma \in \Gamma _1(N)$
; -
(2) for any
$\gamma \in \Gamma $
,
$(c\tau +d)^{-k}f(\gamma \tau )$
has a Fourier expansion of the form where
$$ \begin{align*} (c\tau+d)^{-k}f(\gamma\tau)=\sum_{n=n_\gamma}^\infty a(n)q_{w_\gamma}^n, \end{align*} $$
$a(n_\gamma )\neq 0$
,
$n_\gamma \geq 0$
,
$q_{w_\gamma }=e^{2\pi i\tau /w_{\gamma }}$
and
$w_\gamma $
is the minimal positive integer h such that
$$ \begin{align*} \begin{pmatrix}1 & h\\ 0 & 1\end{pmatrix}\in\gamma^{-1}\Gamma_1(N)\gamma. \end{align*} $$
For a modular form
$f(\tau )$
of weight k with respect to
$\Gamma _1(N)$
, the order of
$f(\tau )$
at the cusp
$a/c\in \mathbb {Q}\cup \{\infty \}$
is defined by
$$ \begin{align*} \mathrm{ord}_{a/c}(f)=n_\gamma \end{align*} $$
for some
$\gamma \in \Gamma $
such that
$\gamma \infty =a/c$
;
$\mathrm {ord}_{a/c}(f)$
is well defined (see [Reference Diamond and Shurman8, page 72]). If the orders of f at all cusps are strictly greater than
$0$
, then f is called a cusp form for
$\Gamma _1(N)$
.
Let
$q=e^{2\pi i\tau }$
and
$\tau \in \mathbb {H}$
. The Dedekind eta-function
$\eta (\tau )$
is defined by
$$ \begin{align*} \eta(\tau)=q^{{1}/{24}}\prod_{n=1}^\infty(1-q^n). \end{align*} $$
The function
$\eta ^{24}(\tau )$
is a cusp form with weight
$12$
for
$\Gamma $
and also for
$\Gamma _1(N)$
for any positive integer N. For a positive integer
$\delta $
and a residue class
$g\pmod {\delta }$
, the generalised Dedekind eta-function
$\eta _{\delta ,g}(\tau )$
is defined by
$$ \begin{align*} \eta_{\delta,g}(\tau)=q^{{\delta}P_2{({g}/{\delta})}/2} \prod_{\substack{n>0\\ n \equiv g \pmod{\delta}}}(1-q^n) \prod_{\substack{n>0\\ n \equiv -g\pmod{\delta}}}(1-q^n), \end{align*} $$
where
$$ \begin{align*} P_2(t)=\{t\}^2-\{t\}+\tfrac{1}{6} \end{align*} $$
is the second Bernoulli function and
$\{t\}$
is the fractional part of t (see, for example, [Reference Robins, Andrews, Bressoud and Parson12, Reference Schoeneberg13]). Notice that
$$ \begin{align*} \eta_{\delta, 0}(\tau)=\eta^2(\delta\tau)\quad\text{and}\quad\eta_{\delta, {\delta}/{2}}(\tau)=\frac{\eta^2({\delta}\tau/2)}{\eta^2(\delta\tau)}. \end{align*} $$
A generalised eta-quotient is a function of the form
$$ \begin{align} \prod_{\delta|N\atop 0\leq g<\delta}\eta_{\delta,g}^{r_{\delta,g}}(\tau), \end{align} $$
where
$N\geq 1$
and
$$ \begin{align*} r_{\delta,g}\in \begin{cases} \dfrac{1}{2}\mathbb{Z} &\text{if}\; g=0\; \text{or}\; g=\dfrac{\delta}{2};\\[3pt] \mathbb{Z} &\text{otherwise}. \end{cases} \end{align*} $$
Although the work of Robins [Reference Robins, Andrews, Bressoud and Parson12, Theorem 3] which gives a criterion for a generalised eta-quotient to be modular is for the zero weight case, the following theorem is true for nonzero weight as well (see [Reference Cotron, Michaelsen, Stamm and Zhu7, Theorem 2.5]).
Theorem 2.4. If
$k=\tfrac 12\sum _{\delta |N} r_{\delta ,0}\in \mathbb {Z}$
and
$f(\tau )=\prod _{\delta | N,\, 0\leq g<\delta }\eta _{\delta , g}^{r_{\delta , g}}(\tau )$
is a generalised eta-quotient such that
$$ \begin{align*} \sum_{\substack{\delta|N\\0\leq g<\delta}}\delta P_2{\left(\frac{g}{\delta}\right)}r_{\delta,g}\equiv0\pmod{2} \end{align*} $$
and
$$ \begin{align*} \sum_{\substack{\delta|N\\0\leq g<\delta}}\frac{N}{\delta}P_2{\left(0\right)}r_{\delta,g}\equiv0\pmod{2}, \end{align*} $$
then
$$ \begin{align*} f(\gamma\tau)=(c\tau+d)^{2k}f(\tau) \end{align*} $$
for all
$\gamma \in \Gamma _1(N)$
.
We can obtain a formula for the order of a generalised eta-quotient at the cusp of
$\Gamma _1(N)$
by [Reference Frye, Garvan, Blümlein, Schneider and Paule10, Theorem 2.3].
Theorem 2.5. The order of the function
$$ \begin{align*}f(\tau)= \prod_{\substack{\delta|N\\ 0<g\leq\lfloor\frac{\delta}{2}\rfloor}} \eta_{\delta, g}^{a_{\delta, g}}(\tau)\end{align*} $$
at the cusp
$a/c$
is given by
$$ \begin{align} \sum_{\substack{\delta|N\\ 0<g\leq\lfloor\frac{\delta}{2}\rfloor}} w_\gamma a_{\delta,g}{\bigg(\frac{e^2}{2\delta} \bigg(\frac{ag}{e}-\bigg\lfloor\frac{ag}{e}\bigg\rfloor-\frac{1}{2}\bigg)^2-\frac{h^2}{6\delta} \bigg(\frac{a\delta}{h}-\bigg\lfloor\frac{a\delta}{h} \bigg\rfloor-\frac{1}{2}\bigg)^2\bigg)}, \end{align} $$
where
$\gamma $
satisfies
$\gamma \infty =a/c$
,
$e=\gcd (\delta ,c)$
and
$h=\gcd (3\delta ,c)$
.
The following theorem of Sturm [Reference Sturm, Chudnovsky, Chudnovsky, Cohn and Nathanson14, Theorem 1] plays an important role in proving congruences using the theory of modular forms.
Theorem 2.6. Let
$\Gamma '$
be a congruence subgroup of
$\Gamma $
, and let k be an integer and
$g(\tau )=\sum _{n=0}^\infty c(n)q^n$
a modular form of weight k for
$\Gamma '$
. For any given positive integer u, if
$c(n)\equiv 0\pmod {u}$
holds for all
$n\leq ({1}/{12})k[\Gamma :\Gamma ']$
, then
$c(n)\equiv 0\pmod {u}$
holds for any
$n\geq 0$
.
There is an explicit formula for the index [Reference Diamond and Shurman8, page 13]:
$$ \begin{align*} [\Gamma: \Gamma_1(N)]=N^2\cdot\prod_{\substack{p|N\\p~\text{is~prime}}}{\bigg(1-\frac{1}{p^2}\bigg)}. \end{align*} $$
3 Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. In [Reference Baruah, Bhoria, Eyyunni and Maji5], Baruah et al. stated without proof the following identity:
$$ \begin{align} \sum_{n=0}^\infty\sigma_o\textrm{mex}(8n+1)q^n &=(\overline{J}_{3,8}\overline{J}_{7,16}+q^2\overline{J}_{1,8}\overline{J}_{1,16}) \nonumber\\ &\quad\quad\times {\bigg({-}\dfrac{J_2^2J_4^5}{J_1^7J_8^2}-\dfrac{J_4^5}{J_1^3J_8^2}+2\dfrac{J_2^7J_4^8}{J_1^{13}J_8^4} +8q\dfrac{J_2^{11}J_8^4}{J_1^{13}J_4^4}\bigg)}\nonumber\\ &\quad+2q(\overline{J}_{1,8}\overline{J}_{7,16}+q\overline{J}_{3,8}\overline{J}_{1,16}) {\bigg({-}\dfrac{J_2^4J_8^2}{J_1^7J_4}+\dfrac{J_2^2J_8^2}{J_1^3J_4}+4\dfrac{J_2^9J_4^2}{J_1^{13}}\bigg)}. \end{align} $$
For the sake of completeness, we present a proof of (3.1) here.
According to [Reference Baruah, Bhoria, Eyyunni and Maji5, (4.25)],
$$ \begin{align} \sum_{n=0}^\infty\sigma_o\textrm{mex}(4n+1)q^n &=\overline{J}_{3,8}\cdot J_4^2\cdot\dfrac{1}{J_1}\cdot\dfrac{1}{J_1^2} {\bigg(2\dfrac{J_4^5}{J_2J_8^2}\cdot\dfrac{1}{J_1^2}-\dfrac{1}{J_2^2}\cdot J_1^4-1\bigg)}. \end{align} $$
From (2.1),
$$ \begin{align} \overline{J}_{3,8} &=(-q^3,-q^5,q^8;q^8)_\infty=\sum_{n=-\infty}^\infty q^{4n^2+n}\nonumber\\ &=\sum_{n=-\infty}^\infty q^{4(2n)^2+2n}+\sum_{n=-\infty}^\infty q^{4(2n-1)^2+(2n-1)}\nonumber\\ &=\sum_{n=0}^\infty q^{16n^2+2n}+\sum_{n=-\infty}^\infty q^{16n^2-14n+3}=\overline{J}_{14,32}+q^3\overline{J}_{2,32}. \end{align} $$
Substituting (2.2), (2.4), (2.5) and (3.3) into (3.2), after simplification,
$$ \begin{align} \sum_{n=0}^\infty\sigma_o\textrm{mex}(8n+1)q^n &=(\overline{J}_{3,8}\overline{J}_{7,16}+q^2\overline{J}_{1,8}\overline{J}_{1,16})\nonumber\\ &\quad\quad\times{\bigg({-}\dfrac{J_2^{12}J_4}{J_1^{11}J_8^2}+8q\dfrac{J_2^2J_4^3J_8^2}{J_1^7}-\dfrac{J_2^2J_4^5}{J_1^7J_8^2} +2\dfrac{J_2^7J_4^8}{J_1^{13}J_8^4}+8q\dfrac{J_2^{11}J_8^4}{J_1^{13}J_4^4}\bigg)}\nonumber\\ &\quad+2q(\overline{J}_{1,8}\overline{J}_{7,16}+q\overline{J}_{3,8}\overline{J}_{1,16})\nonumber\\ &\quad\quad\times{\bigg(2\dfrac{J_4^9}{J_1^7J_8^2}-\dfrac{J_2^{14}J_8^2}{J_1^{11}J_4^5}-\dfrac{J_2^4J_8^2}{J_1^7J_4} +4\dfrac{J_2^9J_4^2}{J_1^{13}}\bigg)}. \end{align} $$
$$ \begin{align} -\dfrac{J_2^{12}J_4}{J_1^{11}J_8^2}+8q\dfrac{J_2^2J_4^3J_8^2}{J_1^7} &=-\dfrac{J_2^{12}J_4}{J_1^7J_8^2} {\bigg(\dfrac{J_4^{14}}{J_2^{14}J_8^4}+4q\dfrac{J_4^2J_8^4}{J_2^{10}}\bigg)}+8q\dfrac{J_2^2J_4^3J_8^2}{J_1^7}\nonumber\\ &=-\dfrac{J_4^{15}}{J_1^7J_2^2J_8^6}+4q\dfrac{J_2^2J_4^3J_8^2}{J_1^7}\nonumber\\ &=-\dfrac{J_4^5}{J_1^7J_8^2}{\bigg(\dfrac{J_4^{10}}{J_2^2J_8^4}-4q\dfrac{J_2^2J_8^4}{J_4^2}\bigg)}=-\dfrac{J_4^5}{J_1^3J_8^2}, \end{align} $$
$$ \begin{align} 2\dfrac{J_4^9}{J_1^7J_8^2}-\dfrac{J_2^{14}J_8^2}{J_1^{11}J_4^5} &=2\dfrac{J_4^9}{J_1^7J_8^2} -\dfrac{J_2^{14}J_8^2}{J_1^7J_4^5}{\bigg(\dfrac{J_4^{14}}{J_2^{14}J_8^4}+4q\dfrac{J_4^2J_8^4}{J_2^{10}}\bigg)}\nonumber\\ &=\dfrac{J_4^9}{J_1^7J_8^2}-4q\dfrac{J_2^4J_8^6}{J_1^7J_4^3}\nonumber\\ &=\dfrac{J_2^2J_8^2}{J_1^7J_4}{\bigg(\dfrac{J_4^{10}}{J_2^2J_8^4}-4q\dfrac{J_2^2J_8^4}{J_4^2}\bigg)} =\dfrac{J_2^2J_8^2}{J_1^3J_4}. \end{align} $$
Substituting (3.5) and (3.6) into (3.4), we obtain (3.2).
Moreover, Baruah et al. [Reference Baruah, Bhoria, Eyyunni and Maji5, (4.34)] proved that
$$ \begin{align*} 2\sum_{n=0}^\infty\sigma_e\textrm{mex}(4n)q^n &=\dfrac{J_2^6}{J_1^2J_4^4}\sum_{n=0}^\infty\sigma_o\textrm{mex}(4n+1)q^n\nonumber\\ &=\overline{J}_{3,8}\cdot\dfrac{J_2^6}{J_4^2}\cdot\dfrac{1}{J_1}\cdot\dfrac{1}{J_1^4} {\bigg(2\dfrac{J_4^5}{J_2J_8^2}\cdot\dfrac{1}{J_1^2}-\dfrac{1}{J_2^2}\cdot J_1^4-1\bigg)}. \end{align*} $$
Substituting (2.2), (2.3), (2.5), (2.6) and (3.3), upon simplification, we deduce that
$$ \begin{align} &2\sum_{n=0}^\infty\sigma_e\textrm{mex}(8n)q^n\nonumber\\ &\quad=(\overline{J}_{3,8}\overline{J}_{7,16}+q^2\overline{J}_{1,8}\overline{J}_{1,16})\nonumber\\ &\qquad\quad\times{\bigg({-}\dfrac{J_2^{22}}{J_1^{14}J_4^8}+16q\dfrac{J_4^8}{J_1^6J_2^2}-\dfrac{J_2^{12}}{J_1^{10}J_4^4} +2\dfrac{J_2^{17}}{J_1^{16}J_4J_8^2}+16q\dfrac{J_2^7J_4J_8^2}{J_1^{12}}\bigg)}\nonumber\\ &\qquad+4q(\overline{J}_{1,8}\overline{J}_{7,16}+q\overline{J}_{3,8}\overline{J}_{1,16}) {\bigg({-}\dfrac{J_4^4}{J_1^6}+2\dfrac{J_2^5J_4^7}{J_1^{12}J_8^2}+\dfrac{J_2^{19}J_8^2}{J_1^{16}J_4^7}\bigg)}. \end{align} $$
Substituting (2.7) into (3.7) yields
$$ \begin{align*} 2\sum_{n=0}^\infty\sigma_e\textrm{mex}(8n)q^n &=(\overline{J}_{3,8}\overline{J}_{7,16}+q^2\overline{J}_{1,8}\overline{J}_{1,16})\nonumber\\ &\quad\quad\times{\bigg({-}\dfrac{J_1^2}{J_2^2}-\dfrac{J_2^{12}}{J_1^{10}J_4^4} +2\dfrac{J_2^{17}}{J_1^{16}J_4J_8^2}+16q\dfrac{J_2^7J_4J_8^2}{J_1^{12}}\bigg)}\nonumber\\ &\quad+4q(\overline{J}_{1,8}\overline{J}_{7,16}+q\overline{J}_{3,8}\overline{J}_{1,16}) {\bigg({-}\dfrac{J_4^4}{J_1^6}+2\dfrac{J_2^5J_4^7}{J_1^{12}J_8^2}+\dfrac{J_2^{19}J_8^2}{J_1^{16}J_4^7}\bigg)}. \end{align*} $$
Replacing q by
$-q$
in (3.1) and using the identity
$$ \begin{align} (-q;-q)_\infty=\dfrac{J_2^3}{J_1J_4}, \end{align} $$
after simplification,
$$ \begin{align} \sum_{n=0}^\infty\sigma_o\textrm{mex}(8n+1)(-q)^n &=(J_{3,8}J_{7,16}+q^2J_{1,8}J_{1,16})\nonumber\\ &\quad\quad\times{\bigg({-}\dfrac{J_1^7J_4^{12}}{J_2^{19}J_8^2}-J_1^3\cdot\dfrac{J_4^8}{J_2^9J_8^2}+2\dfrac{J_1^{13}J_4^{21}}{J_2^{32}J_8^4} -8q\dfrac{J_1^{13}J_4^9J_8^4}{J_2^{28}}\bigg)}\nonumber\\ &\quad-2q(J_{1,8}J_{7,16}-qJ_{3,8}J_{1,16})\nonumber\\ &\quad\quad\times{\bigg({-}\dfrac{J_1^7J_4^6J_8^2}{J_2^{17}}+J_1^3\cdot\dfrac{J_4^2J_8^2}{J_2^7} +4\dfrac{J_1^{13}J_4^{15}}{J_2^{30}}\bigg)}. \end{align} $$
Also, we note that
$$ \begin{align} 2\sum_{n=0}^\infty\sigma_e\textrm{mex}(8n)q^n &=(\overline{J}_{3,8}\overline{J}_{7,16}+q^2\overline{J}_{1,8}\overline{J}_{1,16})\nonumber\\ &\quad\quad\times{\bigg({-}J_1^3\cdot\dfrac{1}{J_1J_2^2}-\dfrac{J_2^{12}}{J_1^{10}J_4^4} +2\dfrac{J_2^{17}}{J_1^{16}J_4J_8^2}+16q\dfrac{J_2^7J_4J_8^2}{J_1^{12}}\bigg)}\nonumber\\ &\quad+4q(\overline{J}_{1,8}\overline{J}_{7,16}+q\overline{J}_{3,8}\overline{J}_{1,16}) {\bigg({-}\dfrac{J_4^4}{J_1^6}+2\dfrac{J_2^5J_4^7}{J_1^{12}J_8^2}+\dfrac{J_2^{19}J_8^2}{J_1^{16}J_4^7}\bigg)}. \end{align} $$
Substituting (2.9) into (3.9) and (3.10) and using the identity
$$ \begin{align*} \overline{J}_{a,b}=\dfrac{J_{2a,2b}J_b^2}{J_{a,b}J_{2b}}, \end{align*} $$
we find that
$$ \begin{align} &\sum_{n=0}^\infty\sigma_o\textrm{mex}(8n+1)(-q)^n\nonumber\\ &\quad\equiv(J_{3,8}J_{7,16}+q^2J_{1,8}J_{1,16}) {\bigg({-}\dfrac{J_1^7J_4^{12}}{J_2^{19}J_8^2}+2\dfrac{J_1^{13}J_4^{21}}{J_2^{32}J_8^4} -8q\dfrac{J_1^{13}J_4^9J_8^4}{J_2^{28}}\bigg)}\nonumber\\ &\qquad-\dfrac{J_4^8}{J_2^9J_8^2}{\bigg(\dfrac{J_{56,128}J_{64}^2}{J_{28,64}J_{128}} -3q\dfrac{J_{40,128}J_{64}^2}{J_{20,64}J_{128}}+5q^3\dfrac{J_{24,128}J_{64}^2}{J_{12,64}J_{128}} -7q^6\dfrac{J_{8,128}J_{64}^2}{J_{4,64}J_{128}}\bigg)}\nonumber\\ &\qquad\quad\times(J_{3,8}J_{7,16}+q^2J_{1,8}J_{1,16})\nonumber\\ &\qquad-2q(J_{1,8}J_{7,16}-qJ_{3,8}J_{1,16}) {\bigg({-}\dfrac{J_1^7J_4^6J_8^2}{J_2^{17}}+4\dfrac{J_1^{13}J_4^{15}}{J_2^{30}}\bigg)}\nonumber\\ &\qquad-2q\dfrac{J_4^2J_8^2}{J_2^7}{\bigg(\dfrac{J_{56,128}J_{64}^2}{J_{28,64}J_{128}} -3q\dfrac{J_{40,128}J_{64}^2}{J_{20,64}J_{128}}+5q^3\dfrac{J_{24,128}J_{64}^2}{J_{12,64}J_{128}} -7q^6\dfrac{J_{8,128}J_{64}^2}{J_{4,64}J_{128}}\bigg)}\nonumber\\ &\qquad\quad\times(J_{1,8}J_{7,16}-qJ_{3,8}J_{1,16})\pmod{16} \end{align} $$
and
$$ \begin{align} &2\sum_{n=0}^\infty\sigma_e\textrm{mex}(8n)q^n\nonumber\\ &\quad\equiv{\bigg(\dfrac{J_{6,16}J_{14,32}J_8^2J_{16}}{J_{3,8}J_{7,16}J_{32}} +q^2\dfrac{J_{2,16}J_{2,32}J_8^2J_{16}}{J_{1,8}J_{1,16}J_{32}}\bigg)} {\bigg({-}\dfrac{J_2^{12}}{J_1^{10}J_4^4}+2\dfrac{J_2^{17}}{J_1^{16}J_4J_8^2}+16q\dfrac{J_2^7J_4J_8^2}{J_1^{12}}\bigg)}\nonumber\\ &\qquad-\dfrac{1}{J_1J_2^2}\bigg(\dfrac{J_{56,128}J_{64}^2}{J_{28,64}J_{128}}-3q\dfrac{J_{40,128}J_{64}^2}{J_{20,64}J_{128}} +5q^3\dfrac{J_{24,128}J_{64}^2}{J_{12,64}J_{128}}-7q^6\dfrac{J_{8,128}J_{64}^2}{J_{4,64}J_{128}}\bigg)\nonumber\\ &\quad\qquad\times{\bigg(\dfrac{J_{6,16}J_{14,32}J_8^2J_{16}}{J_{3,8}J_{7,16}J_{32}} +q^2\dfrac{J_{2,16}J_{2,32}J_8^2J_{16}}{J_{1,8}J_{1,16}J_{32}}\bigg)}\nonumber\\ &\qquad+4q{\bigg(\dfrac{J_{2,16}J_{14,32}J_8^2J_{16}}{J_{1,8}J_{7,16}J_{32}} +q\dfrac{J_{6,16}J_{2,32}J_8^2J_{16}}{J_{3,8}J_{1,16}J_{32}}\bigg)}\nonumber\\ &\quad\qquad\times{\bigg({-}\dfrac{J_4^4}{J_1^6}+2\dfrac{J_2^5J_4^7}{J_1^{12}J_8^2} +\dfrac{J_2^{19}J_8^2}{J_1^{16}J_4^7}\bigg)}\pmod{16}. \end{align} $$
Therefore, to prove (1.6) and (1.7), we need to prove that the coefficients on the right-hand sides of (3.11) and (3.12) vanish modulo
$16$
.
Let f and g denote the right-hand sides of (3.11) and (3.12), respectively. By Theorem 2.4,
$$ \begin{align} F(\tau)=q^{{13}/{96}} \frac{\eta^{122}(4\tau)\eta^2(8\tau)\eta^7_{16,8}(\tau)}{\eta^{48}(2\tau)\eta_{16,7}^{15}(\tau)}f \end{align} $$
and
$$ \begin{align} G(\tau)=q^{{1}/{96}} \frac{\eta^{168}(4\tau)\eta^{21}_{16,7}(\tau)\eta_{16,8}^{10}(\tau)}{\eta^{72}(2\tau)}g \end{align} $$
satisfy the transformation formulae
$$ \begin{align*} F(\gamma\tau)=(c\tau+d)^{38}F(\tau)\quad \text{and} \quad G(\gamma\tau)=(c\tau+d)^{48}G(\tau) \end{align*} $$
for any
$\gamma \in \Gamma _1(128)$
. From (2.11), the orders of
$F(\tau )$
and
$G(\tau )$
at every cusp of
$\Gamma _1(128)$
are nonnegative, and so they are modular forms for
$\Gamma _1(128)$
of weight 38 and 48, respectively. One can check the coefficients of the first 38912 terms of (3.13) are congruent to 0 modulo 16, and the coefficients of the first 49152 terms of (3.14) are congruent to 0 modulo 16. Therefore, by Theorem 2.6,
$f\equiv 0\pmod {16}$
and
$g\equiv 0\pmod {16}$
. This completes the proof of Theorem 1.2.
4 Concluding remarks
We conclude this paper with two remarks.
First, Baruah et al. [Reference Baruah, Bhoria, Eyyunni and Maji5] proved (1.4) and (1.5) by using several identities involving
$\varphi (q)$
and
$\psi (q)$
and the Lambert series representations of
$\varphi ^2(q)$
and
$\varphi (q)\varphi (q^2)$
, where
$\varphi (q)$
and
$\psi (q)$
are two of Ramanujan’s three classical theta functions. We provide a simplified proof of (1.4) based on (2.2), (2.9) and (3.3).
Baruah et al. [Reference Baruah, Bhoria, Eyyunni and Maji5, (4.17)] derived
$$ \begin{align} \sum_{n=0}^\infty\sigma_o\textrm{mex}(2n+1)q^n=\dfrac{J_2^2J_8^2}{J_1^3J_4}-\dfrac{J_1J_8^2}{J_4}. \end{align} $$
Replacing q by
$-q$
in (4.1) and using (3.8),
$$ \begin{align} \sum_{n=0}^\infty\sigma_o\textrm{mex}(2n+1)(-q)^n &=\dfrac{J_4^2J_8^2}{J_2^7}\cdot J_1^3-\dfrac{J_2^3J_8^2}{J_4^2}\cdot\dfrac{1}{J_1}. \end{align} $$
Substituting (2.2) and (2.9) into (4.2), taking all the terms of the form
$q^{2n}$
, after simplification,
$$ \begin{align*} \sum_{n=0}^\infty\sigma_o\textrm{mex}(4n+1)q^n &\equiv\dfrac{J_2^2J_4^2}{J_1^7}(\overline{J}_{14,32}-7q^3\overline{J}_{2,32}) -\dfrac{J_1J_4^2}{J_2^2}\overline{J}_{3,8}\pmod{16}\nonumber\\ &=\dfrac{J_2^2J_4^2}{J_1^7}(\overline{J}_{14,32}-7q^3\overline{J}_{2,32}) -\dfrac{J_1J_4^2}{J_2^2}(\overline{J}_{14,32}+q^3\overline{J}_{2,32})\nonumber\\ &\equiv\dfrac{J_1J_4^2}{J_2^2}(\overline{J}_{14,32}-7q^3\overline{J}_{2,32} -\overline{J}_{14,32}-q^3\overline{J}_{2,32})\pmod{8}\nonumber\\ &=-8q^3\dfrac{J_1J_4^2}{J_2^2}\overline{J}_{2,32}\equiv0\pmod{8}, \end{align*} $$
where the second identity follows from (3.3). The congruence (1.4) thus follows.
Second, the numerical evidence suggests the following conjecture.
Conjecture 4.1. We have
$$ \begin{align*} \lim_{X\rightarrow\infty}\dfrac{\#\{0\leq n<X\colon\sigma_o\textrm{mex}(31n+18)\equiv0\pmod{16}\}}{X}=1,\\ \end{align*} $$
$$ \begin{align*} \lim_{X\rightarrow\infty}\dfrac{\#\{0\leq n<X\colon\sigma_e\textrm{mex}(31n+18)\equiv0\pmod{16}\}}{X}=1. \end{align*} $$
Acknowledgements
The authors would like to express their sincere gratitude to two anonymous referees for their careful reading of the manuscript and many constructive suggestions, which improved the quality of the paper to a great extent.
