\section{Introduction and statement of results}

A partition $\lambda = (\lambda_1,\dots,\lambda_{s})$ of a positive integer $n$ is a list of non-increasing parts $\lambda_j$ such that $\lvert \lambda \rvert \coloneqq \sum_{j=1}^{s} \lambda_j=n$. Letting $p(n)$ denote the number of partitions of $n$, Hardy and Ramanujan \cite{HardyRamanujan} proved their asymptotic formula
\begin{equation}\label{HR}
	p(n)\sim \frac{1}{4\sqrt{3}n} e^{\pi \sqrt{\frac{2n}{3}}},
\end{equation}
as $n\rightarrow \infty.$ This work marked the birth of the powerful Circle Method that has seen a wide range of applications in the last century, in particular in obtaining asymptotic and exact formulae for partitions and their statistics. For example, the exact formula for $p(n)$ given by Rademacher \cite{RademacherExact}, and formulae and inequalities for certain partition statistics \cite{Bring,BM} among many others.

We use Wright's variant of the Circle Method to obtain asymptotic formulae and distribution results or a certain family of partitions, first introduced by Berkovich and Garvan in \cite{BG}. For this we need two partition statistics. Recall that a partition has a Ferrers--Young diagram given by
$$
\begin{matrix}
	\bullet & \bullet & \bullet & \dots & \bullet & \leftarrow & \lambda_1 \text{ many nodes}\\
	\bullet & \bullet & \dots & \bullet & & \leftarrow & \lambda_2 \text{ many nodes}\\
	\vdots & \vdots & \vdots & &  &  \\
	\bullet & \dots & \bullet & & & \leftarrow & \lambda_s \text{ many nodes},
\end{matrix}
$$
and each node has a hook length. The node in row $k$ and column $j$ has hook length
$h(k,j):=(\lambda_k-k)+(\lambda'_j-j)+1,$ where $\lambda'_j$ is the number of nodes in column $j$. A $t$-core partition is a partition that has no hook lengths divisible by $t$. Let $P$ denote the set of all partitions, $P_t$ the set of $t$-core partitions, and consider the well-known map of Littlewood \cite{Littlewood} $\phi \colon P_t \times P \times \dots \times P$ defined via
\begin{align*}
	\phi(\lambda) = (\lambda_{\text{$t$-core}}, \hat{\lambda}_t),\qquad \hat{\lambda} = (\hat{\lambda}_0,\hat{\lambda}_1,\dots,\hat{\lambda}_{t-1})
\end{align*}
such that
\begin{align*}
	|\lambda| = |\lambda_{\text{$t$-core}}| +t \sum_{j=0}^{t-1} |\hat{\lambda}_j|.
\end{align*}
Then the $2$-quotient rank of $\lambda$ is defined by \cite{BG}
\begin{align*}
	\text{$2$-quotient rank}(\lambda) \coloneqq \nu(\lambda_0) - \nu(\lambda_1),
\end{align*}
where $\nu(\lambda)$ is the number of parts of $\lambda$ and
\begin{align*}
	\phi(\lambda) = (\lambda_{\text{$2$-core}},\hat{\lambda}_0,\hat{\lambda}_1).
\end{align*}
It was shown in \cite[Theorem 3.1]{BG} that the $2$-quotient rank naturally divides partitions of $5n+4$ with so-called s-rank $i=0,2$ into five equal classes, thereby proving an analogue of Ramanujan's congruence $p(5n+4) \equiv 0 \pmod{5}$ in these cases.
The BG-rank was also defined in \cite{BG}
\begin{align*}
	\text{BG-rank}(\lambda) \coloneqq \sum_{j=1}^{\nu} (-1)^{j+1} \operatorname{par}(\lambda_j),
\end{align*}
where $\operatorname{par}$ denotes the parity of a positive integer. It was used to prove a further refinement of Ramanujan's congruence modulo $5$ as well as to prove new congruences modulo $5$ of $p(n)$.

Let $\overline{p}_j(m,n)$ denote the number of partitions of a positive integer $n$ with BG-rank $j$ and $2$-quotient rank $m$. Then Berkovich and Garvan \cite{BG} proved that the generating function is
\begin{align*}
	H(\zeta;q) \coloneqq \sum_{\substack{n \geq 0 \\ m \in \mathbb{Z}}} \overline{p}_j(m,n) \zeta^m q^n = \frac{q^{(2j-1)j}}{(q^2 \zeta,q^2 \zeta^{-1};q^2)_\infty},
\end{align*}
where $(a;q)_\infty \coloneqq \prod_{n\geq0} (1-aq^n)$ is the usual $q$-Pochhammer symbol, and we define $(a,b;q)_\infty \coloneqq (a;q)_\infty(b;q)_\infty$.

Let $\overline{p}_j(n)$ count the number of partitions of $n$ with BG-rank $j$, and $\overline{p}_j(a,b;n)$ denote the number of partitions with BG-rank $j$ and $2$-quotient rank congruent to $a \pmod{b}$. Then a standard computation using orthogonality of roots of unity gives
\begin{align}\label{eqn: splitting}
H(a,b;q) \coloneqq	\sum_{n \geq 0} \overline{p}_j(a,b;n) q^n = \frac{1}{b} \sum_{n \geq 0} \overline{p}_j(n) q^n + \frac{1}{b} \sum_{k=1}^{b-1} \zeta_b^{-ak} 	H(\zeta_b^k;q).
\end{align}

We follow the framework provided by \cite{CCM}, which builds on work of \cite{BCMO}, and prove the following equidistribution result.

\begin{theorem}\label{thm: main}
	Let $0\leq a <b$ and $b>1$. Then as $n \to \infty$ we have
	\begin{align*}
		\overline{p}_j(a,b;n) = \frac{1}{3^{\frac{3}{4}} (2n)^{\frac{5}{4}} b } e^{\pi \sqrt{\frac{2n}{3}}} \left(1+ O \left(n^{-\frac{1}{2}}\right)\right).
	\end{align*}

\end{theorem}

A direct corollary yields the asymptotic behaviour of $\overline{p}_j(n)$.
\begin{corollary}\label{cor: asymps of p_j}
	As $n \to \infty$ we have
	\begin{align*}
		\overline{p}_j(n) = \frac{1}{3^{\frac{3}{4}} (2n)^{\frac{5}{4}} } e^{\pi \sqrt{\frac{2n}{3}}} \left(1+ O \left(n^{-\frac{1}{2}}\right)\right).
	\end{align*}

\end{corollary}

\begin{remarks}
	\begin{enumerate}[label=(\roman*)]
		\item We remark that neither Theorem \ref{thm: main} or \ref{cor: asymps of p_j} depend on $j$, implying that asymptotically the BG-rank is also equidistributed over partitions. However, the dependence on $j$ can be seen if one applies further terms arising from the application of Wright's Circle Method in the proof of Theorem \ref{thm: main}.

		\item For $b=5$ we recover an asymptotic form of \cite[Theorem 7.1]{BG}.
	\end{enumerate}
\end{remarks}

A direct application of \cite[Corollary 3.2]{CCM}  to our results gives that both $\overline{p}_j(n)$ and $\overline{p}_j(a,b;n)$ asymptotically satisfy the following convexity results. Similar results for other partition-theoretic objects have been obtained by many authors, see e.g. \cite{BO,HJ,Males}.
\begin{corollary}\label{cor: convexity}
	For large enough $n_1$ and $n_2$, we have that
	\begin{align*}
		\overline{p}_j(a,b;n_1)\overline{p}_j(a,b;n_2) > \overline{p}_j(a,b;n_1+n_2),
	\end{align*}
and
\begin{align*}
	\overline{p}_j(n_1) \overline{p}_j(n_2) > \overline{p}_j(n_1+n_2).
	\end{align*}
\end{corollary}

The Tur\'{a}n inequalities for functions in the real entire Laguerre--P\'{o}lya class are intricately linked to the Riemann hypothesis \cite{Dimitrov, Sz} and have seen renewed interest in recent years. In fact, the Riemann hypothesis is true if and only if the Riemann Xi function lies in the Laguerre--P\'{o}lya class, and a necessary condition is that the Maclaurin coefficients satisfy Tur\'{a}n inequalities of all orders \cite{Dimitrov,SchurPolya}.

The second order Tur\'{a}n inequality for a sequence $\{a_n\}_{n \geq 0}$ is known as log-concavity, and is satisfied if
\begin{align*}
	a_n^2 \geq a_{n-1}a_{n+1}
\end{align*}
for all $n \geq 1$. Log-concavity for modular type objects is well-studied in the literature, for example \cite{BJMR,CCM,CraigPun,DawMas,DeSalvo}. For an explicit description of the higher order inequalities, we refer the reader to e.g. the introduction of \cite{CraigPun}.
The higher-order Tur\'{a}n inequalities are linked to the Jensen polynomial associated to a sequence $\{\alpha(n)\}$, given by
\begin{align*}
	J_{\alpha}^{d,n}(X) \coloneqq \sum_{k=0}^{d-1} \binom{d}{k} \alpha(n+k) X^k.
\end{align*}
In particular, all higher-order Tur\'{a}n inequalities are satisfied by the sequence $\alpha(n)$ if the associated Jensen polynomial is hyperbolic. Similar situations to ours have been studied in e.g. \cite{GORZ,LW,OPR}.

\begin{theorem}\label{thm: turan ineqs}
	Let  $\widehat{J}_\alpha^{d,n}$ be the renormalised Jensen polynomials defined in Theorem \ref{Thm: Gorz} and $H_d(X)$ the Hermite polynomials. Then we have
	\begin{align*}
	 \lim_{n \to \infty} \widehat{J}_{\overline{p}_j}^{d,n}(X) = H_d(X)
	\end{align*}
uniformly for $X$ on compact subsets of $\mathbb{R}$. This implies that both $\overline{p}_j(a,b;n)$ and $\overline{p}_j(n)$ asymptotically satisfy all higher order Tur\'{a}n inequalities.
\end{theorem}

\subsection*{Outline}
We begin in Section \ref{Sec: prelims} by recalling preliminary results needed through the rest of the paper. Section \ref{Sec: proofs} is dedicated to proving Theorem \ref{thm: main} and Corollary \ref{cor: asymps of p_j}. In Section \ref{Sec: Turan} we prove Theorem \ref{thm: turan ineqs}.

\subsection*{Acknowledgements}
The authors thank the Manitoba eXperimental Mathematics Laboratory for their support of the project.

\section{Preliminaries}\label{Sec: prelims}

\subsection{Asymptotics of infinite $q$-products}
Following \cite{BCMO}, we let
\begin{align*}
	F_1(\zeta; q)\coloneqq \prod_{n=1}^{\infty}\left(1-\zeta q^n\right), \qquad
\end{align*}
along with Lerch's transcendent
\begin{align*}
	\Phi(z,s,a)\coloneqq \sum_{n=0}^\infty \frac{z^n}{(n+a)^s},
\end{align*}
and for $0\leq \theta < \frac{\pi}{2}$ define the domain $D_{\theta} \coloneqq \left\{ z=re^{i\alpha} \colon r \geq 0 \text{ and } |\alpha| \leq \theta \right\}$. Throughout, the Gamma function is defined as usual by $\Gamma(x)\coloneqq \int_0^\infty t^{x-1} e^{-t} dt $, for $\operatorname{Re}(x)>0$.
Then \cite[Theorem 2.1]{BCMO} in our context reads as follows.
\begin{theorem} \label{Thm: asymptotics F}
	For $b\geq 2$, let $\zeta$ be a primitive $b$-th root of unity. Then as $z \to 0$ in $D_\theta$, we have
		\begin{align*}
			F_{1}\left(\zeta;e^{-z}\right)  =\frac{1}{\sqrt{1-\zeta}} \, e^{-\frac{\zeta\Phi(\zeta,2,1)}{z}}\left( 1+O\left(|z|\right) \right).
		\end{align*}
\end{theorem}
In particular, this allows us to obtain estimates of $F_1$ on the major arcs arising from Wright's Circle Method.

\subsection{Wright's Circle Method}
Our central tool is a variant of Wright's Circle Method, which was proved by Bringmann, Craig, Ono, and one of the authors \cite[Proposition 4.4]{BCMO}, following work of Wright \cite{Wright2} - see also Ngo and Rhoades \cite{NgoRhoades}. The essence of Wright's Circle Method is that one uses Cauchy's integral formula to recover the the Fourier coefficients as an integral of the generating function over a complex circle transversed exactly once in the counterclockwise direction. One then splits the circle into major (resp.\@ minor) arcs, where the generating function has relatively large (resp.\@ small) growth. In turn, this provides one with an asymptotic estimate for the coefficients. Often, the major arc is close to $q=1$, as is the case in this paper.

\begin{proposition} \label{WrightCircleMethod}
	Suppose that $F(q)$ is analytic for $q = e^{-z}$ where $z=x+iy \in \mathbb{C}$ satisfies $x > 0$ and $|y| < \pi$, and suppose that $F(q)$ has an expansion $F(q) = \sum_{n=0}^\infty c(n) q^n$ near 1. Let $N,M>0$ be fixed constants. Consider the following hypotheses:

	\begin{enumerate}[leftmargin=*]
		\item[\rm(1)] As $z\to 0$ in the bounded cone $|y|\le Mx$ (major arc), we have
		\begin{align*}
			F(e^{-z}) = z^{B} e^{\frac{A}{z}} \left( \sum_{j=0}^{N-1} \alpha_j z^j + O_M\left(|z|^N\right) \right),
		\end{align*}
		where $\alpha_j \in \mathbb{C}$, $A\in \mathbb{R}^+$, and $B \in \mathbb{R}$.

		\item[\rm(2)] As $z\to0$ in the bounded cone $Mx\le|y| < \pi$ (minor arc), we have
		\begin{align*}
			\lvert	F(e^{-z}) \rvert \ll_M e^{\frac{1}{\mathrm{Re}(z)}(A - \kappa)},
		\end{align*}
		for some $\kappa\in \mathbb{R}^+$.
	\end{enumerate}
	If  {\rm(1)} and {\rm(2)} hold, then as $n \to \infty$ we have for any $N\in \mathbb{R}^+$
	\begin{align*}
		c(n) = n^{\frac{1}{4}(- 2B -3)}e^{2\sqrt{An}} \left( \sum\limits_{r=0}^{N-1} p_r n^{-\frac{r}{2}} + O\left(n^{-\frac N2}\right) \right),
	\end{align*}
	where $p_r \coloneqq  \sum\limits_{j=0}^r \alpha_j c_{j,r-j}$ and $c_{j,r} \coloneqq  \dfrac{(-\frac{1}{4\sqrt{A}})^r \sqrt{A}^{j + B + \frac 12}}{2\sqrt{\pi}} \dfrac{\Gamma(j + B + \frac 32 + r)}{r! \Gamma(j + B + \frac 32 - r)}$.
\end{proposition}

\subsection{Hyperbolicity of Jensen polynomials}

We use a result of \cite{GORZ}, where the authors showed that for certain classes of functions, the Jensen polynomials of the coefficients tend to the order $d$ Hermite polynomials $H_d(X)$, in turn implying the asymptotic hyperbolicity of the Jensen polynomials.

\begin{theorem}[Theorem 3 of \cite{GORZ}]\label{Thm: Gorz}
	Let $\{ \alpha(n)\}$, $\{A(n)\}$, and $\{\delta(n)\}$ be three sequences of positive real numbers with $\delta(n)$ tending to $0$ and satisfying
	\begin{align*}
		\log\left( \frac{\alpha(n+k)}{\alpha(n)} \right) = A(n)k - \delta(n)^2k^2+\sum_{i=3}^{d} g_i(n)k^i +o(\delta(n)^d)
	\end{align*}
	as $n \to \infty$, where each $g_i(n) = o(\delta(n)^i)$ for each $3\leq i \leq d$. Then the renormalized Jensen polynomials
	\begin{align*}
		\widehat{J}_{\alpha}^{d,n}(X) \coloneqq \frac{\delta(n)^{-d}}{\alpha(n)} J_{\alpha}^{d,n}\left( \frac{\delta(n)X-1}{\exp(A(n))}\right)
	\end{align*}
	satisfy
	\begin{align*}
		\lim_{n \to \infty} \widehat{J}_{\alpha}^{d,n}(X) = H_d(X),
	\end{align*}
	uniformly for $X$ in any compact subset of $\mathbb{R}$. Moreover, this implies that the Jensen polynomials $J_{\alpha}^{d,n}$ are each hyperbolic for all but finitely many $n$.

\end{theorem}

\section{Proofs of Theorem \ref{Thm: asymptotics F} and Corollary \ref{cor: asymps of p_j}}\label{Sec: proofs}
In this section we prove the main results on the asymptotic behaviour of $\overline{p}_j(a,b;n)$ and $\overline{p}_j(n)$.

\begin{proof}[Proof of Theorem \ref{Thm: asymptotics F}]
	We begin by rewriting
	\begin{align*}
		H(\zeta;q) = \frac{q^{(2j-1)j}}{F_1(\zeta;q^2) F_1(\zeta^{-1};q^2)},
	\end{align*}
noting that $H(1;q) = \frac{q^{(2j-1)j}}{(q^2;q^2)_\infty^2}$. Then by \eqref{eqn: splitting} we obtain
\begin{align*}
	H(a,b;q)  = \frac{q^{(2j-1)j}}{b (q^2;q^2)_\infty^2} +  \frac{1}{b} \sum_{k=1}^{b-1} \zeta_b^{-ak} 	 \frac{q^{(2j-1)j}}{F_1(\zeta_b^j;q^2) F_1(\zeta_b^{-j};q^2)}.
\end{align*}
We aim to prove that the first term on the right-hand side dominates on both the major and minor arcs, and that the resulting asymptotic expansion for $q=e^{-z}$ as $z \to 0$ satisfies the conditions of Proposition \ref{WrightCircleMethod}.

First consider the major arc, so $|y|\le Mx$ for arbitrary $M>0$. Letting $P(q) \coloneqq \sum_{n \geq 0} p(n)q^n = (q;q)_\infty^{-1}$, we have the classical asymptotic (for $|y|\le Mx$, as $z\to0$)
\[
P\left(e^{-z}\right) = \sqrt{\frac{z}{2\pi}} e^{\frac{\pi^2}{6z}} (1+O(|z|))
\]
and so
\begin{align*}
 e^{-(2j-1)jz} \left(e^{-2z}; e^{-2z}\right)_\infty^{-2} =  \frac{z}{\pi} e^{\frac{\pi^2}{6z}} (1+O(|z|)).
\end{align*}
where we expanded the first exponential using its Taylor expansion.
By Theorem \ref{Thm: asymptotics F}, for $\zeta_b^j \neq 1$ we have that
\begin{align*}
		& F_{1}\left(\zeta_b^j;e^{-2z}\right)  = \frac{1}{\sqrt{1-\zeta_b^j}} \, e^{-\frac{\zeta_b^j\Phi(\zeta_b^j,2,1)}{2z}}\left( 1+O\left(|z|\right) \right),\\
		&F_{1}\left(\zeta_b^{-j};e^{-2z}\right)  =\frac{1}{\sqrt{1-\zeta_b^{-j}}} \, e^{-\frac{\zeta_b^{-j}\Phi(\zeta_b^{-j},2,1)}{2z}}\left( 1+O\left(|z|\right) \right).
\end{align*}
So the major arc bound follows if
	\begin{align*}
		\frac{1}{\sqrt{(1-\zeta_b^j)(1-\zeta_b^{-j}) }} \, e^{-\frac{\zeta_b^j\Phi(\zeta_b^j,2,1) +\zeta_b^{-j}\Phi(\zeta_b^{-j},2,1)}{2z}}  \left( 1+O\left(|z|\right) \right) \ll \sqrt{\frac{z}{2\pi}} e^{\frac{\pi^2}{6z}} (1+O(|z|)).
	\end{align*}
Since $\zeta \Phi(\zeta,2,1) = \operatorname{Li}_2(\zeta)$ is the dilogarithm function, and $\operatorname{Li}_2(z) + \operatorname{Li}_2(z^{-1}) = -\frac{\pi^2}{6} -\frac{1}{2}\log(-z)^2$, the bound follows if
\begin{align*}
\textnormal{Re} \left(	\frac{\pi^2}{12z} + \frac{\log(-\zeta_b^j)^2}{4z} \right) \ll \textnormal{Re} \left( \frac{\pi^2}{6z} \right).
\end{align*}
Note that $\log(-\zeta_b^j) = \log(1)+i \arg(-\zeta_b^j) = i \arg(-\zeta_b^j)$, so the condition becomes
\begin{align*}
	\pi^2 - 3\arg(-\zeta_b^j)^2 < 2\pi^2,
\end{align*}
which is clearly satisfied. Thus on the major arc as $z \to 0$, we obtain that
\begin{align*}
	H(a,b;e^{-z}) =  \frac{z}{b \pi} e^{\frac{\pi^2}{6z}} (1+O(|z|)).
\end{align*}

We now turn to the minor arcs, where $y \geq Mx$. It is well-known that for some $\mathcal{C}>0$
\[
\left|P\left(e^{-z}\right)\right| \le x^{\frac12} e^{\frac{\pi}{6x}-\frac{\mathcal{C}}{x}},
\]
and so
\begin{align*}
	\left\lvert e^{-(2j-1)jz} \left(e^{-2z}; e^{-2z}\right)_\infty^{-2} \right\rvert \le 2x e^{\frac{\pi}{6x}-\frac{\mathcal{C}}{x}}.
\end{align*}
Next we turn to the function $\frac{1}{F_1(\zeta;q)}$ for a root of unity $\zeta \neq 1$. We have
\begin{align*}
	\log \left(F_1(\zeta;q)^{-1} \right) = \sum_{n \geq 1} -\log\left(1-\zeta q^n\right) = \sum_{n,m \geq 1} \frac{\zeta^m q^{mn}}{m} = \sum_{m \geq 1} \frac{\zeta^m q^m}{(1-q^m)}.
\end{align*}
We then bound
\begin{align*}
	\log \left(F_1(\zeta;q)^{-1} \right) \leq \left\lvert \frac{\zeta q}{1-q} \right\rvert - \frac{\lvert q \rvert}{1-\lvert q \rvert} + \log P(\lvert q\rvert)
\end{align*}
As in the proof of Theorem 1.4 of \cite{BCMO}, we obtain that
\begin{align*}
	\left\lvert \frac{\zeta q}{1-q} \right\rvert - \frac{\lvert q \rvert}{1-\lvert q \rvert} = -\frac{1}{x} +O(1).
\end{align*}
Thus on the minor arc we have
\begin{align*}
	F_1(\zeta_b^j;e^{-2z})^{-1} F_1(\zeta_b^{-j};e^{-2z})^{-1} \ll \left\lvert e^{-(2j-1)jz} \left(e^{-2z}; e^{-2z}\right)_\infty^{-2} \right\rvert \le 2x e^{\frac{\pi}{6x}-\frac{\mathcal{C}}{x}}.
\end{align*}
Finally, we apply Proposition \ref{WrightCircleMethod} with $B=1$, $N=1$, $A = \frac{\pi^2}{6}$ and $\alpha_0 = \frac{1}{b \pi}$ to obtain the result.
\end{proof}

We now prove Corollary \ref{cor: asymps of p_j}, relying on the proof of Theorem \ref{thm: main}.
\begin{proof}[Proof of Corollary \ref{cor: asymps of p_j}]
	The result follows by noting that the generating function for $\overline{p}_j(n)$ dominated on both the major and minor arc in the proof of Theorem \ref{thm: main}. Thus, up to a factor of $b$, the asymptotic from Theorem \ref{thm: main} holds, giving the result.
\end{proof}

\section{Tur\'{a}n Inequalities}\label{Sec: Turan}
This section is dedicated to proving Theorem \ref{thm: turan ineqs}. We directly apply Theorem \ref{Thm: Gorz} to the asymptotic formulae obtained in the previous section.

\begin{proof}[Proof of Theorem \ref{thm: turan ineqs}]
	We present the proof for $\overline{p}_j(n)$, as the proof for $\overline{p}_j(a,b;n)$ differs only by a factor of $b$. Applying the general $N$ version of Proposition \ref{WrightCircleMethod} instead of $N=1$ in the proof of Theorem \ref{thm: main} and Corollary \ref{cor: asymps of p_j}, it is easy to show that there are constants $c_\nu$ such that as $n \to \infty$
	\begin{align*}
				\overline{p}_j(n) = \frac{1}{3^{\frac{3}{4}} (2n)^{\frac{5}{4}} } \exp\left({\pi \sqrt{\frac{2n}{3}}}\right) \exp\left( c_0 + \frac{c_1}{n^{\frac{1}{2}}} + \frac{c_2}{n} +\dots \right)
	\end{align*}
to all orders of $n$.

	It follows in a similar way to \cite[Proof of Theorem 7]{GORZ} that
	\begin{align*}
		\log\left( \frac{\overline{p}_j(n+k)}{\overline{p}_j(n)}\right) \sim \pi \sqrt{\frac{2}{3}} \sum_{i \geq 1} \binom{1/2}{i} \frac{k^i}{n^{i-\frac{1}{2}}} - \frac{5}{4} \sum_{i \geq 1} \frac{(-1)^{i-1}k^i}{in^i} + \sum_{s,t\geq 1} c_t \binom{-t}{s} \frac{k^s}{n^{\frac{s+t}{2}}}.
	\end{align*}
It is not difficult to see that one can then apply Theorem \ref{Thm: Gorz} with the sequences $A(n)$, $\delta(n)$ defined via
\begin{align*}
	A(n) = \pi \sqrt{\frac{1}{6n}} + O\left(\frac{1}{n} \right), \qquad \delta(n)^2 = \pi \sqrt{\frac{2}{3}} \binom{1/2}{2} n^{-\frac{3}{2}} + O\left(n^{-2}\right).
\end{align*}to obtain the result.
\end{proof}
We remark that it is not difficult to see that any function satisfying Proposition \ref{WrightCircleMethod} will have an associated Jensen polynomial that is asymptotically hyperbolic. We pose the following question for future research: can one prove lower bounds $N_{A,B}(d)$ such that the Jensen polynomial associated to such a function is hyperbolic for all $n>N_{A,B}(d)$? A resolution would immediately yield effective results for large swathes of partition theoretic objects, coefficients of modular-type objects, and topological invariants.