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\title{A note on the exact formulas for certain $2$-color partitions}

\alttitle{Note sur les formules exactes de certaines partitions à 2 couleurs}


\author{\firstname{Russelle} \lastname{Guadalupe}\CDRorcid{0009-0001-8974-4502}}

\address{Institute of Mathematics, University of the Philippines-Diliman, Quezon City, 1101, Philippines}

\email{rguadalupe@math.upd.edu.ph}


\subjclass{11P55, 11P82, 05A16}

\keywords{\kwd{Circle method} \kwd{$\eta$-quotients} \kwd{partitions} \kwd{asymptotic formula} }

\altkeywords{\kwd{Méthode des cercles} \kwd{$\eta$-quotients} 
\kwd{partitions} \kwd{formule asymptotique}}


\begin{abstract} 
Let $p\leq 23$ be a prime and $a_p(n)$ count the number of partitions of $n$ where parts that are multiple of $p$ come up with $2$ colors. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula for $\log a_p(n)$. Our results partially extend the work of Mauth, who proved the asymptotic formula for $\log a_2(n)$ conjectured by Banerjee et al.
\end{abstract}

\begin{altabstract} 
Soit $p\leq 23$ un nombre premier et $a_p(n)$ compte le nombre de partitions de n où les parties qui sont multiples de $p$ donnent $2$ couleurs. En utilisant un résultat de Sussman, nous dérivons la formule exacte pour $a_p(n)$ et obtenons une formule asymptotique pour $\log a_p(n)$.
Nos résultats étendent partiellement le travail de Mauth, qui a prouvé la formule asymptotique pour $\log a_2(n)$ conjecturée par Banerjee et al.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}\label{sec1}

Throughout this paper, we denote $(a;q)_\infty = \prod_{n\geq 0}(1-aq^n)$ for $a\in\mathbb{C}$. Recall that a partition of a positive integer $n$ is a nonincreasing finite sequence of positive integers, known as its parts, whose sum is $n$. We define $p(n)$ as the number of partitions of $n$, which can be seen as the coefficients of its generating function given by
\begin{align*}
	\dfrac{1}{(q;q)_\infty} = \sum_{n=0}^\infty p(n)q^n.
\end{align*} 
Hardy and Ramanujan~\cite{hadram} proved the following asymptotic formula for $p(n)$ given by
\begin{align*}
	p(n) \sim \dfrac{1}{4n\sqrt{3}}\exp\left(\pi\sqrt{\dfrac{2n}{3}}\right), \quad n\to\infty,
\end{align*}
using the celebrated Circle Method. Rademacher~\cite{rade} refined the Hardy--Ramanujan Circle Method and derived the exact formula
\begin{align*}
	p(n) = \dfrac{1}{\pi\sqrt{2}}\sum_{k=1}^\infty A_k(n)\sqrt{k}\dfrac{\dd }{\dd n}\left(\dfrac{\sinh\left(\frac{\pi}{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right)}{\sqrt{n-\frac{1}{24}}}\right)
\end{align*}
where 
\begin{align*}
	A_k(n) \coloneqq \sum_{\substack{0\leq h < k\\ \gcd(h,k)=1}} \exp\left[\pi \irm \left(s(h,k)-\dfrac{2nh}{k}\right)\right]
\end{align*}
and 
\begin{align*}
	s(h,k) \coloneqq \sum_{j=1}^{k-1}\dfrac{j}{k}\left(\left\{\dfrac{hj}{k}\right\}-\dfrac{1}{2}\right)
\end{align*}
is the Dedekind sum, where $\{t\}$ denotes the fractional part of $t$. Recently, Banerjee et al.~\cite{baner1} proved the following refined asymptotic formula for $p(n)$ given by
\begin{align*}
	\log p(n)\sim \pi\sqrt{\dfrac{2n}{3}}-\log n-\log 4\sqrt{3}-\dfrac{0.44\cdots}{\sqrt{n}}, \quad n\to\infty,
\end{align*}
from a family of inequalities for $p(n)$, which was used to prove a refinement of the inequality of DeSalvo and Pak~\cite{salvop}, and Chen, Wang and Xie~\cite{chenwx} that reads
\begin{align*}
	\left(1+\dfrac{\pi}{24n^{3/2}}-\dfrac{1}{n^2}\right)p(n-1)p(n+1) < p(n)^2 < \left(1+\dfrac{\pi}{24n^{3/2}}\right)p(n-1)p(n+1)
\end{align*}
for $n\geq 120$.

Let $a_k(n)$ be the number of partitions of $n$ where parts that are multiple of $k$ come up with $2$ colors. For $k=1$, we have $a_1(n)= p_2(n)$ where $p_2(n)$ is the total number of partitions of $n$ where parts come up with $2$ colors and for $k=2$, $a_2(n)$ also counts the number of cubic partitions (see~\cite{bkim}). The generating function for $a_k(n)$ is given by~\cite{ahmed}
\begin{align*}
	\dfrac{1}{(q;q)_\infty(q^k;q^k)_\infty} = \sum_{n=0}^\infty a_k(n)q^n.
\end{align*} 
Kotesovec~\cite{kotes} found the following asymptotic formula for $a_2(n)$ given by 
\begin{align*}
	a_2(n) \sim \dfrac{1}{8n^{5/4}}\exp(\pi\sqrt{n}), \quad n\to\infty.
\end{align*}
Banerjee et al.~\cite{baner1} conjectured that $a_2(n)$ satisfies the following asymptotic formula
\begin{align}
	\log a_2(n)\sim \pi\sqrt{n}-\dfrac{5}{4}\log n-\log 8-\dfrac{0.79\cdots}{\sqrt{n}}, \quad n\to\infty. \label{eq1}
\end{align}
Recently, Mauth~\cite{mauth} proved (\ref{eq1}) by finding the exact formula of Rademacher type for $a_2(n)$ using a result of Zuckerman~\cite{zucker}.

In this paper, we use a result of Sussman~\cite{suss} to derive Rademacher-type formulas for $a_p(n)$ when $p\leq 23$ is a prime, and deduce their asymptotic formulas in the spirit of (\ref{eq1}), which can be considered as a partial extension of Mauth's work. We give our main results as follows. 

\begin{theo}\label{thm11}
	Let $p\leq 23$ be a prime. Then for $n\geq 1$ we have
	\begin{multline*}
		a_p(n) = 2\pi\sqrt{p}\left(\dfrac{1+p^{-1}}{24n-p-1}\right)\sum_{j=1}^{p-1}\sum_{\substack{m=1\\ m\equiv j\bmod{p}}}^\infty I_2\left(\dfrac{\pi}{6m}\sqrt{(1+p^{-1})(24n-p-1)}\right)\dfrac{b_m(n)}{m}\\
		+ 2\pi\left(\dfrac{1+p}{24n-p-1}\right)\sum_{\substack{m=1\\ p\mid m}}^\infty I_2\left(\dfrac{\pi}{6m}\sqrt{(1+p)(24n-p-1)}\right)\dfrac{b_m(n)}{m},
	\end{multline*}
	where 
	\begin{align*}
		b_k(n) \coloneqq \sum_{\substack{0\leq h < k\\ \gcd(h,k)=1}} \exp\left[-\dfrac{2\pi nh\irm }{k}+\pi s(h,k)\irm +\pi s\left(\dfrac{ph}{\gcd(p,k)},\dfrac{k}{\gcd(p,k)}\right)\irm\right]
	\end{align*}
	and $I_2(s)$ is the second modified Bessel function of the first kind (see Section~\ref{sec2}).
\end{theo}

\begin{coro}\label{cor12}
	For $p\leq 23$ prime, we have 
	\begin{align*}
		\log a_p(n) \sim \pi\sqrt{\dfrac{2n(1+p^{-1})}{3}}-\dfrac{5}{4}\log n + \log \dfrac{2\sqrt{3p}(1+p^{-1})^{3/4}}{24^{5/4}}-\dfrac{c_p}{24\sqrt{6n}},\quad n\to\infty
	\end{align*}
	where 
	\begin{align*}
		c_p \coloneqq \dfrac{135}{\pi\sqrt{1+p^{-1}}}+\pi\sqrt{\dfrac{(1+p)^3}{p}}.
	\end{align*}
\end{coro}

Setting $p=2$ in Corollary~\ref{cor12}, we deduce the asymptotic formula due to Mauth~\cite{mauth} given~by
\begin{align*}
	\log a_2(n)\sim \pi\sqrt{n}-\dfrac{5}{4}\log n-\log 8-\left(\dfrac{15}{8\pi}+\dfrac{\pi}{16}\right)\dfrac{1}{\sqrt{n}}, \quad n\to\infty,
\end{align*}
which immediately yields (\ref{eq1}).

The paper is organized as follows. In Section~\ref{sec2}, we state the Sussman's result on the exact formulas for the Fourier coefficients of a class of $\eta$-quotients. In Section~\ref{sec3}, we apply this result to prove Theorem~\ref{thm11}, and using the asymptotic expansion of $I_2(s)$ due to Banerjee~\cite{baner2}, we deduce Corollary~\ref{cor12}.

\section{Fourier coefficients of a class of \texorpdfstring{$\eta$}{eta}-quotients}\label{sec2}

We consider in this section the exact formulas for the Fourier coefficients of the following class of holomorphic functions on the open disk given by
\begin{align*}
	G(q) \coloneqq \prod_{r=1}^R(q^{m_r};q^{m_r})_\infty^{\delta_r} =\sum_{n=0}^\infty g(n)q^n,
\end{align*}
where $\mathbf{m}=(m_1,\ldots,m_R)$ is a sequence of $R$ distinct positive integers and $\mathbf{\delta}=(\delta_1,\ldots,\delta_R)$ is a sequence of $R$ nonzero integers. The functions $G(q)$ can be seen as $\eta$-quotients since $(q;q)_\infty=q^{-1/24}\eta(\tau)$, where Dedekind's eta function, denoted
by $\eta(\tau)$, is defined by the infinite product $\eta(\tau)\coloneqq q^{1/24}\prod_{n\geq 1}(1-q^n)$ with $q=\expe^{2\pi \irm\tau}$and $\tau\in\mathbb{H}\coloneqq \{z\in\mathbb{C} : \mbox{Im}(z)>0\}$. For useful properties of the Dedekind's eta function and the definition of $\eta$-quotients, we refer the interested reader to the book~\cite{ono}. Given a particular $G(q)$ with $\sum_{r=1}^R \delta_r < 0$, Sussman~\cite{suss} gave a Rademacher-type exact formula for $g(n)$, which is a special case of the work of Bringmann and Ono~\cite{bringn} on the coefficients of harmonic Maass forms. Sussman's proof follows the original approach of Rademacher~\cite{rade} on the Hardy--Ramanujan Circle Method. In the case where $\sum_{r=1}^R \delta_r \geq 0$, Chern~\cite{chern} obtained an analogous formula with an error term using the method of O-Y. Chan~\cite{oych}.

Before we state Sussman's result, we need some definitions. For $(h, k)\in\mathbb{N}^2$ with $\gcd(h, k) = 1$, we set
\begin{align*}
	\Delta_1 &= -\dfrac{1}{2}\sum_{r=1}^R \delta_r, \quad \Delta_2 = \sum_{r=1}^R m_r\delta_r, \\
	\Delta_3(k) &= -\sum_{r=1}^R \dfrac{\gcd(m_r,k)^2}{m_r}\delta_r, \quad \Delta_4(k) = \prod_{r=1}^R \left(\dfrac{\gcd(m_r,k)}{m_r}\right)^{\delta_r/2},
	\intertext{and}
	\hat{A}_k(n) &= \sum_{\substack{0\leq h < k\\ \gcd(h,k)=1}}\exp\left(-\dfrac{2\pi nh\irm }{k}-\pi \irm \sum_{r=1}^R\delta_r s\left(\dfrac{m_rh}{\gcd(m_r,k)},\dfrac{k}{\gcd(m_r,k)}\right)\right),
\end{align*}
where $s(h,k)$ is the Dedekind sum defined in Section~\ref{sec1}. We also set $L$ as the least common multiple of $m_1,\ldots,m_R$, and partition the set $\{1,\ldots,L\}$ into two disjoint subsets
\begin{align*}
	\mathcal{L}_{>0} &\coloneqq \{1\leq l\leq L: \Delta_3(l) > 0\},\\
	\mathcal{L}_{\leq 0} &\coloneqq \{1\leq l\leq L: \Delta_3(l) \leq 0\}.
\end{align*}

\begin{theo}[\cite{suss}]\label{thm21}
	If $\Delta_1 > 0$ and the inequality
	\begin{align}
		\min_{1\leq r\leq R} \dfrac{\gcd(m_r,l)^2}{m_r}\geq \dfrac{\Delta_3(l)}{24}\label{eq2}
	\end{align}
	holds for $1\leq l \leq L$, then for positive integers $n > -\Delta_2/24$, we have
	\begin{align*}
		g(n) = 2\pi\sum_{l\in \mathcal{L}_{>0}}\Delta_4(l)\left(\dfrac{\Delta_3(l)}{24n+\Delta_2}\right)^{(\Delta_1+1)/2}\sum_{\substack{k=1\\ k\equiv l\bmod{L}}}^\infty I_{\Delta_1+1}\left(\dfrac{\pi}{6k}\sqrt{\Delta_3(l)(24n+\Delta_2)}\right)\dfrac{\hat{A}_k(n)}{k},
	\end{align*}
	where 
	\begin{align*}
		I_\nu(s) \coloneqq \sum_{m=0}^\infty \dfrac{(\frac{s}{2})^{\nu+2m}}{m!\Gamma(\nu+m+1)}
	\end{align*}
	is the $\nu$th modified Bessel function of the first kind, and $\Gamma(s)=\int_0^\infty \expe^{-t}t^{s-1}\,\dd t$ is the gamma function.
\end{theo}

\section{Proofs of Theorem~\ref{thm11} and Corollary~\ref{cor12}}\label{sec3}

In this section, we use Theorem~\ref{thm21} to prove Theorem~\ref{thm11}. As a consequence, we obtain Corollary~\ref{cor12} using the asymptotic expansion of the modified Bessel function of the first kind due to Banerjee~\cite{baner2}.

\begin{proof}[Proof of Theorem~\ref{thm11}]
	Writing
	\begin{align*}
		\dfrac{1}{(q;q)_\infty(q^p;q^p)_\infty} = \sum_{n=0}^\infty a_p(n)q^n,
	\end{align*}
	we have $\mathbf{m}=(1,p), \mathbf{\delta}=(-1,-1)$ and $L=p$. We compute $\Delta_1=1$ and $\Delta_2=-p-1$. Recalling that $p$ is a prime, we have that for $l\in\{1,\ldots, p\}$,
	\begin{align*}
		\Delta_3(l) =\begin{cases}
			1+\dfrac{1}{p}, & l\neq p\\
			1+p, & l=p
		\end{cases},\qquad
		\Delta_4(l) =\begin{cases}
			\sqrt{p}, & l\neq p\\
			1, & l=p
		\end{cases},
	\end{align*} 
	so that $\mathcal{L}_{>0}=\{1,\ldots, p\}$. Since $p\leq 23$, we see that condition (\ref{eq2}) holds for $l\in \{1,\ldots, p\}$. Applying Theorem~\ref{thm21} with $\hat{A}_k(n)=b_k(n)$, we obtain
	\begin{align*}
		a_p(n) = 2\pi\sum_{l\in \mathcal{L}_{>0}}\Delta_4(l)\left(\dfrac{\Delta_3(l)}{24n-p-1}\right)\sum_{\substack{m=1\\ m\equiv l\bmod{p}}}^\infty I_2\left(\dfrac{\pi}{6m}\sqrt{\Delta_3(l)(24n-p-1)}\right)\dfrac{b_m(n)}{m}
	\end{align*}
	for $n>(p+1)/24$. Plugging in the values of $\Delta_3(l)$ and $\Delta_4(l)$ yields the desired result.
\end{proof}

We see that the series for $a_p(n)$ in Theorem~\ref{thm11} converges rapidly and that the term $m=1$ contributes signficantly to the series. We thus obtain the asymptotic formula
\begin{align}
	a_p(n) \sim 2\pi\sqrt{p}\left(\dfrac{1+p^{-1}}{24n-p-1}\right)I_2\left(\dfrac{\pi}{6}\sqrt{(1+p^{-1})(24n-p-1)}\right),\quad n\to\infty.\label{eq3}
\end{align}
To prove Corollary~\ref{cor12}, we use the following asymptotic formula~\cite[Equation~(2.11)]{baner2}
\begin{align}
	I_\nu(s) \sim \dfrac{\expe^s}{\sqrt{2\pi s}}\sum_{m=0}^\infty \dfrac{(-1)^md_m(\nu)}{x^m}, \quad s\to\infty,\label{eq4}
\end{align}
where 
\begin{align*}
	d_m(\nu) = \dfrac{\binom{\nu-1/2}{m}(\nu+\frac{1}{2})_m}{2^m}
\end{align*}
with 
\begin{align*}
	\dbinom{a}{m}&=\begin{cases}
		\dfrac{a(a-1)\cdots(a-m+1)}{m!}, & m\in \mathbb{N}\\
		1, & m=0
	\end{cases},\\
	(a)_m&=\begin{cases}
		a(a+1)\cdots (a+m-1), & m\in \mathbb{N}\\
		1, & m=0
	\end{cases}
\end{align*}
for $a\in\mathbb{R}$.

\begin{proof}[Proof of Corollary~\ref{cor12}]
	From (\ref{eq4}) we get
	\begin{align*}
		I_2(s) = \dfrac{\expe^s}{\sqrt{2\pi s}}\left(1-\dfrac{15}{8s}+O\left(\dfrac{1}{s^2}\right)\right).
	\end{align*}
	In view of (\ref{eq3}), we have
	\begin{align}
		a_p(n)\Mk \sim \Mk 2\sqrt{3p}\cdot\dfrac{(1+p^{-1})^{3/4}}{(24n\Mk -\Mk p\Mk -\Mk 1)^{5/4}} %\nonumber\\
		\cdot\exp\left(\dfrac{\pi}{6}\sqrt{(1\Mk +\Mk p^{-1})(24n\Mk -\Mk p\Mk -\Mk 1)}\right)\left(1\Mk -\Mk \dfrac{45}{8\pi\sqrt{6n(1\Mk +\Mk p^{-1})}}\Mk +\Mk O\left(\dfrac{1}{n}\right)\right).\label{eq5}
	\end{align}
	Since 
	\begin{align*}
		\dfrac{1}{(24n-p-1)^{5/4}}&=\dfrac{1}{(24n)^{5/4}}\left(1+O\left(\dfrac{1}{n}\right)\right),\\
		\exp\left(\dfrac{\pi}{6}\sqrt{(1+p^{-1})(24n-p-1)}\right) &=\exp\left(\pi\sqrt{\dfrac{2n(1+p^{-1})}{3}}\right)\left(1-\dfrac{p^{-1/2}(1+p)^{3/2}\pi}{24\sqrt{6n}}+O\left(\dfrac{1}{n}\right)\right),
	\end{align*}
	we infer from (\ref{eq5}) that
	\begin{align*}
		a_p(n)\sim 2\sqrt{3p}&\cdot\dfrac{(1+p^{-1})^{3/4}}{(24n)^{5/4}}\left(1+O\left(\dfrac{1}{n}\right)\right)\cdot\exp\left(\pi\sqrt{\dfrac{2n(1+p^{-1})}{3}}\right)\\
		&\cdot\left(1-\dfrac{p^{-1/2}(1+p)^{3/2}\pi}{24\sqrt{6n}}+O\left(\dfrac{1}{n}\right)\right)\left(1-\dfrac{45}{8\pi\sqrt{6n(1+p^{-1})}}+O\left(\dfrac{1}{n}\right)\right),
	\end{align*}
	and taking logarithms yields the desired conclusion.
\end{proof}
We remark that when $p > 23$ is a prime, condition (\ref{eq2}) of Theorem~\ref{thm21} fails. Nonetheless, by numerical experiments via \textit{Mathematica}, we propose the following conjecture. 

\begin{conj}
Corollary~\ref{cor12} also holds for primes $p > 23$.
\end{conj}


\section*{Acknowledgments}
The author would like to thank the anonymous referee for helpful comments that improved the contents of the paper.

\section*{Declaration of Interest}

The author do not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and have declared no affiliation other than their research organisations.

\printbibliography
\end{document}
