# Zero Attractors of Partition Polynomials

Robert P. Boyer<EMAIL_ADDRESS>Daniel T. Parry<EMAIL_ADDRESS>Department of Mathematics, Drexel University
Philadelphia, PA 19104  Model Risk Governance and Review, JP Morgan Chase,
4 Chase, MetroTech Center 11245 New York, New York

###### Abstract

A partition polynomial is a refinement of the partition number $p(n)$ whose
coefficients count some special partition statistic. Just as partition numbers
have useful asymptotics so do partition polynomials. In fact, their
asymptotics determine the limiting behavior of their zeros which form a
network of curves inside the unit disk. An important new feature in their
study requires a detailed analysis of. the “root dilogarithm” given as the
real part of the square root of the usual dilogarithm.

###### keywords:

Partition, Polynomials , Asymptotics , Dilogarithm , Lerch zeta function

###### MSC:

11M35, 11P82, 11P55, 33B30, 30E15

††journal: JMAA

##  1 Introduction

More than twenty years ago, Richard Stanley gave a talk on the geometry of the
roots of various polynomial sequences where the roots appear to lie on
complicated unknown curves in the complex plane. The examples he emphasized
are influential within combinatorics and include the Bernoulli polynomials
[17], chromatic polynomials [31], and $q$-Catalan numbers. The list contains a
single example from the theory of integer partitions [3, Chapter 2]

$F_{n}(z)=\sum_{k=1}^{n}p_{k}(n)z^{k},$

which we now call the partition polynomials. Here the number $p_{k}(n)$ counts
the total number of integer partitions of $n$ with $k$ parts. This polynomial
sequence appeared already in several areas of mathematics and physics. First,
E.M. Wright studied $F_{n}(z)$ to develop the “Wright circle method” to
produce estimates for $F_{n}(z)$ when $z>0$ [34] under the notion of “weighted
partitions.” This paper has been an inspiration to many works in combinatorics
and number theory (see [15] for example). Second, Navez et al. discovered that
$F_{n}(z)$ can define a probability measure called the “Maxwell’s Demon
Ensemble” which approximates a Bose gas near certain temperatures [33, 20,
26].

With Richard Stanley’s talk in mind, Boyer and Goh [9, 10] initiated a study
of the roots of $F_{n}(z).$ They announced the results of their study but they
never published proofs. In fact they confirmed Richard Stanley’s suspicion
that there are indeed curves that attract the roots of the partition
polynomials.

For background, the Wright circle method produces an asymptotic estimate of
$F_{n}(z)$ as $n\to\infty,$ which requires the choice of a “heaviest” or
“dominant” singularity of a bivariate generating function $P(z,q).$ It turns
out that for $z>0,$ this heaviest singularity always occurs at $q=1$ but for
general $z$ in the unit disk $\mathbb{D}$ the heaviest singularity depends on
the location of $z.$ Since the asymptotics for $F_{n}(z)$ depends primarily on
the heaviest singularity, a Stokes phenomenon [24] appears forcing the roots
fall near the Stokes/anti-Stokes lines. This is analogous to the phenomenon
observed by the well studied Szego curve [30] and the Appell polynomials [8,
16].

Boyer and Goh also observed that the dilogarithm
$Li_{2}(z)=\sum_{n=1}^{\infty}z^{n}/n^{2}$ plays an important role in
determination of the heaviest singularity. In itself, this is not unusual in
integer partition theory because the dilogarithm is intricately connected to
partitions ([29, 35, 27] for example) but what is unusual is that overcoming
the additional challenge of allowing $z$ be a complex number required
univalent function theory including conformal properties of the dilogarithm
[5, 25, 23].

With that, Boyer and Parry began the study of a similar polynomial sequence
called the plane partition polynomials [14, 13, 12] whose study was suggested
by Stanley in conversation in 2005. This polynomial sequence acts in the same
way as Boyer and Goh predicted with the different “phases” of the asymptotic
structure of $F_{n}(z)$ driving the location of the roots of $F_{n}(z).$ It is
now clear that the partition polynomials are not an isolated case but part of
an entire family of polynomial sequences in combinatorics that have this
Stokes phenomenon.

Construction of a general framework for partition polynomials began with [28]
when Parry found a polynomial version of the classical Meinardus Theorem for
the asymptotics of sequences whose generating functions have a special (Euler)
infinite product structure. This structure is common in the classical study of
integer partitions and given one knows the phase structure, one can determine
the asymptotics of the polynomial sequence.

This paper finishes the framework of Boyer and Parry and confirms the
conjectures of Boyer and Goh. Specifically, we will show that the same
phenomenon observed by the original partition polynomials and established for
the plane partition polynomials occurs for the partition polynomials for
arithmetic progressions.

##  2 Mathematical Preliminaries

Partition polynomials have been studied in a few contexts [4, 6] and loosely
define a family of polynomial generating functions that have an interpretation
in integer partition theory. For our purposes, we will use the following
definition:

######  Definition 1.

The partition polynomials, $F_{n}(z)$, are defined as the Fourier coefficients
of

$1+\sum_{n=1}^{\infty}F_{n}(z)q^{n}=\prod_{m=1}^{\infty}\frac{1}{(1-zq^{m})^{a_{m}}}$
(1)

where the integer exponents $a_{m}$ are either $0$ or $1$.

For $S\subset{\mathbb{Z}}^{+}$, let $a_{m}=\chi_{S}(m)$, then the coefficient
$[z^{k}]F_{n}(z)$ counts the number of partitions of $n$ with $k$ parts that
all lie in $S$. The principal cases we study are $S={\mathbb{Z}}^{+}$; that
is, the parts have no restriction; and where the elements of $S$ satisfy a
linear congruent equation modulo $p$.

We recall the notion of phase [3, Definition 1] for a sequence
$\\{L_{n}(z)\\}$ of functions on the open subset $U$ of the complex plane
where $L_{n}(z)$ is analytic on $U$ except perhaps for a branch cut.

######  Definition 2.

Assume $U$ has a decomposition as a finite disjoint union of open nonempty
sets $R(m_{1}),\allowbreak\cdots,\allowbreak R(m_{k})$ together with their
boundaries such that

$\Re L_{m_{j}}(z)>\max\\{L_{m}(z):m\neq m_{j}\\}$ (2)

for $z\in R(m_{j})$. If the family of sets $R(m)$ is a maximal family with
these properties, we call $R(m_{j})$ a phase with phase function
$L_{m_{j}}(z)$.

The term phase comes from statistical mechanics; in that context a phase
function represents a metastable free energy. In our examples, $U$ will either
be the complex plane $\mathbb{C}$ or the open unit disk $\mathbb{D}$. By
maximality, the phase functions are not analytic continuations of each other.

Our description of the asymptotics for partition polynomials $\\{F_{n}(z)\\}$
inside the open unit disk $\mathbb{D}$ uses the notion of phase and phase
functions. On $\mathbb{D}$, we find that there is a sequence $\\{L_{m}(z)\\}$
of functions and finitely many indices $m_{1},m_{2},\dots,m_{\ell}$, say, so
on each $R(m_{j})$

$F_{n}(z)\sim_{X}A_{n,m}(z)\exp(n^{-\beta}L_{m_{j}}(z)),\quad\beta>0,$

where $A_{n,m}(z)$ are nonzero functions on $R(m_{j})$ such that

$\lim_{n\to\infty}n^{-\beta}\ln|A_{n,m}(z)|=0$

uniformly on the compact subsets of $R(m_{j})$, $L_{m_{j}}(z)$ is analytic on
$R(m_{j})$ except perhaps for a branch cut, and $\beta=1/2$.

Next we recall the definition of principal object of this paper.

######  Definition 3.

Let $Z(P(z))$ denote the finite set of zeros of the polynomial $P(z)$. For a
polynomial sequence $\\{F_{n}(z)\\}$ whose degrees go to infinity, its zero
attractor $A$ is the limit of $Z(F_{n}(z))$ in the Hausdorff metric on the
non-empty compact subsets $K$ of $\mathbb{C}\cup\\{\infty\\}$ [8].

We now discuss fully below the connection between phases and the zero
attractor For a partition polynomial sequence $\\{F_{n}(z)\\}$ on the unit
disk, if its phases there are, $R(m_{1}),\allowbreak\dots,\allowbreak
R(m_{\ell})$ say, then its zero attractor $A$ is simply the union of the
boundaries of the phases

$A=\partial R(m_{1})\cup\cdots\cup\partial R(m_{\ell})$

with perhaps further contributions from branch cuts if necessary.

Finally we mention that we continue using our notational conventions in [3].
So $\sqrt[s]{x}$ is the $s$ root $\exp(\frac{\log{x}}{s})$ with the imaginary
part of the logarithm defined on $(-\pi,\pi].$ Both $[x]^{-}$ and
$\overline{x}$ denote the complex conjugate of $x.$ Let $\\{g_{n}(x)\\}$ a
sequence of functions, we say $g_{n}(x)=O_{V}(a_{n})$ where $a_{n}$ a sequence
of complex numbers, if there exists a constant $C_{V}$ dependent solely on a
collection of parameters $V,$ such that $|g_{n}(x)|\leq C_{V}|a_{n}|$ as
$n\to\infty.$ Similarly, we say $g_{n}(x)=o_{V}(a_{n})$ for every $C_{V}$
dependent solely on a collection of parameters $V,$ $|g_{n}(x)|\leq
C_{V}|a_{n}|$ as $n\to\infty.$ Absence of any $V$ indicates that the constant
is uniform.

For convenience, we use the notation $e_{k}(z)=\exp(2\pi iz/k)$ and
$e(z)=e_{1}(z).$

###  2.1 Summary of the Meinardus Polynomial Setup [28]

Let $\\{a_{m}\\}$ be the exponent sequence for the generating function for the
partition polynomials $\\{F_{n}(z)\\}$. For each positive integer $k$ and
character of ${\mathbb{Z}}_{k}$, we associate the Dirichlet series
$D_{t,k}(s)$ where $t$ is an integer $1\leq t\leq k$ that indexes the
character of the finite cyclic group ${\mathbb{Z}}_{k}$ given by $j\mapsto
e_{k}(tj)$. We define the Dirichlet series as

$D_{t,k}(s)=\sum_{m=1}^{\infty}a_{m}\frac{e_{k}(mt)}{m^{s}}.$

Observe that $D_{k,k}(s)$ is the usual Dirchlet generating function for the
sequence $\\{a_{m}\\}$. In the examples in this paper, these Dirichlet series
either have an analytic continuation to the complex plane as an entire
function or as a meromorphic function with a unique singularity at $s=1$,
which will be a simple pole. Further, for each positive integer $k$, we need
to introduce two functions on the character group $\widehat{{\mathbb{Z}}_{k}}$
as well as their finite Fourier transforms given in terms of the Dirichlet
series family.

######  Definition 4.

Let $b_{k}$ be the $\widehat{{\mathbb{Z}}_{k}}$-Fourier transform of $t\mapsto
D_{t,k}(0)$ while $c_{k}$ be the $\widehat{{\mathbb{Z}}_{k}}$-Fourier
transform of $t\mapsto Res(D_{t,k},s=1)$; explicity, for $1\leq j\leq k$,

$b_{k}(j)=\frac{1}{k}\sum_{t=1}^{k}e_{k}(-tj)D_{t,k}(0),\quad
c_{k}(j)=\frac{1}{k}\sum_{t=1}^{k}e_{k}(-tj)\,Res(D_{t,k}(s),s=1).$

The actual form of the asymptotics of $\\{F_{n}(z)\\}$ is given in terms of
these functions on ${\mathbb{Z}}_{k}$ and its dual. For
$k,n\in{\mathbb{Z}}^{+}$ and $1\leq h<k$ relatively prime to $k$, define
$L_{h,k}(z)$ and $\omega_{h,k,n}(z)$:

$\displaystyle L_{h,k}(z)^{2}$
$\displaystyle=\frac{1}{k^{2}}\sum_{r=1}^{k}z^{r}L(z^{k},2,r/k)\,Res(D_{rh,k},s=1)=\sum_{j=1}^{k}c_{k}(j)Li_{2}(e_{k}(jh)z),$
$\displaystyle\omega_{h,k,n}(z)$
$\displaystyle=e_{k}(-hn)\prod_{j=1}^{k}(1-e_{k}(hj)z)^{-b_{k}(j)},\quad\Omega_{n,k}(z)=\sum_{(h,k)=1}\omega_{h,k,n}(z)$

where $L(z,s,\nu)$ is the Lerch zeta function. When the functions $L_{h,k}(z)$
are independent of $h$, we will suppress the $h$ dependence in the indexing
when convenient.

The asymptotic results in [28] depend on knowing the phases for the function
sequence $\\{L_{h,k}(z)\\}$. For simplicity of exposition, we assume that
$L_{h,k}(z)$ are independent of $h$ and that the phases are
$R(m_{1}),\dots,R(m_{j})$. On each phase $R(m)$ there are possibly two
different regimes of the asymptotics on compact subsets $X$:

$\displaystyle
F_{n}(z)\sim_{X}\frac{1}{2\sqrt{\pi}n^{3/4}}\Omega_{n,m}(z)\sqrt{L_{m}(z)}\exp(2n^{1/2}L_{m}(z)),\,z\in
X\setminus\\{z:L_{m}(z)\leq 0\\},$ (3)
$\displaystyle\sim_{X}2\Re\left[\frac{1}{2\sqrt{\pi}n^{3/4}}\Omega_{n,m}(z)\sqrt{L_{m}(z)}\exp(2n^{1/2}L_{m}(z))\right],\,z\in
X\cap\\{z:L_{m}(z)\leq 0\\},$ (4)

where the second asymptotics represent the contribution of a branch cut, if
needed.

###  2.2 General Results about Zero Attractor

We show that the zero attractor lies inside the closed unit disk
$\overline{\mathbb{D}}$ and always contains the unit circle for any family of
partition polynomials if their exponent sequence $\\{a_{n}\\}$ satisfies the
initial condition $a_{1}=1$ and contains infinitely many $1$’s. Of course, the
more detailed structure of the zero attractor depends on the moduli of
$\\{F_{n}(z)\\}$ so the chief contribution to consider is $\exp(2n^{1/2}\Re
L_{m}(z))$ in its asymptotics.

We restate Sokal’s result in language convenient for partition polynomials.

######  Proposition 1.

[31] Let $\\{\phi_{n}(z)\\}$ be a sequence of analytic functions on an open
connected set $U$. For a fixed $\beta>0$, we assume that the sequence
$\\{|\phi(z)|^{{n^{\beta}}}\\}$ is uniformly bounded on the compact subsets of
$U$. A point $z_{0}\in{\mathbb{C}}$ is in their zero attractor if there exists
a neighborhood $V(z_{0})$ of $z_{0}$ with the property that there cannot exist
a harmonic function $v(z)$ on $V(z_{0})$ satisfying

$\liminf_{n\to\infty}n^{-\beta}\ln|\phi_{n}(z)|\leq
v(z)\leq\limsup_{n\to\infty}n^{-\beta}\ln|\phi_{n}(z)|.$

For partition polynomials, there are two choices for the scales $\beta=1/2$ or
$1$. In next Proposition, we make no assumptions on the phase structions
inside the unit disk.

######  Proposition 2.

Let $\\{F_{n}(z)\\}$ be a sequence of partition polynomials. Then
(a) $|F_{n}(z)|^{1/\sqrt{n}}$ is uniformly bounded on the unit disk.
(b) Uniformly on compact subsets of
$\mathbb{C}\setminus\overline{\mathbb{D}}$:

$\lim_{n\to\infty}\frac{1}{n}\ln|F_{n}(z)|=\ln|z|.$

(c) Let $\\{a_{m}\\}_{m=1}^{\infty}$ be an exponent sequence for the partition
polynomial sequence $\\{F_{n}(z)\\}$ with $a_{1}=1$. Then its zero attractor
$A$ lies inside the closed unit disk and $\infty\notin A$.

###### Proof.

(a) Since the exponents $\\{a_{m}\\}$ form a binary $0$-$1$ sequence,
$F_{n}(1)$ is bounded above by the total number $p(n)$ of the standard
partitions of $n$. But $p(n)$ has order $\exp(\pi\sqrt{2n/3})$.
(b) Assume $|z|>1$. Then

$\displaystyle\frac{1}{n}\ln|F_{n}(z)|$
$\displaystyle=\frac{1}{n}n\ln|z|+\frac{1}{n}\ln\left|1+([z^{n-1}]F_{n}(z))/z+\cdots+([z^{1}]F_{n}(z))/z^{n-1}\right|$

But

$\displaystyle\frac{1}{n}\ln\left|1+([z^{n-1}]F_{n}(z))/z+\cdots+([z^{1}]F_{n}(z))/z^{n-1}\right|\leq\frac{1}{n}\ln
F_{n}(1)$ $\displaystyle\qquad\leq\frac{1}{n}\ln p(n)\to 0$

as $n\to\infty$. Hence $\ln|F_{n}(z)|/n\to\ln|z|$.

(c) Let $H(z)=\prod_{m=1}^{\infty}{(1-z^{m})^{-a_{m+1}}}$ so $H(z)$ is
analytic, nonvanishing in the open unit disk $\mathbb{D}$, and $H(0)=1$. By
the Stabilization result in [11], we find that

$[z^{n-k}]F_{n}(z)=[z^{k}]H(z),\quad 0\leq k<\lfloor n/2\rfloor.$

Write $H(z)=1+\sum_{m=1}^{\infty}h_{m}z^{m}$. Let $1<r_{0}$. For $|z|\geq
r_{0}$, we find there is a cancellation of the first $\lfloor n/2\rfloor$
terms below to give the bound:

$\displaystyle\left|\frac{F_{n}(z)}{z^{n}}-\prod_{m=1}^{\infty}(1-1/z)^{-a_{m+1}}\right|$
$\displaystyle\leq\sum_{\ell=\lfloor
n/2\rfloor}^{n}[z^{n-\ell}]F_{n}(z)\frac{1}{|z|^{\ell}}+\sum_{\ell=\lfloor
n/2\rfloor+1}^{\infty}h_{\ell}|z|^{-\ell}$ $\displaystyle\leq r_{0}^{\lfloor
n/2\rfloor}F_{n}(1)+\sum_{\ell=\lfloor
n/2\rfloor+1}^{\infty}h_{\ell}r_{0}^{-\ell}.$

Hence, on $|z|\geq r_{0}>1$, $F_{n}(z)/z^{n}$ converges uniformly to $H(1/z)$.
Let $\\{z_{n_{k}}\\}$ be a convergent sequence of zeros of
$F_{n_{k}}(z_{n_{k}})=0$ such that $|z_{n_{k}}|\geq r_{0}$ with limit $z^{*}$.
Then $F_{n_{k}}(z_{n_{k}})/z_{n_{k}}^{n_{k}}=0\to H(1/z^{*})\neq 0$ which
ontradicts that $H(1/z)$ does not vanish on $\mathbb{D}$. On the other hand,
$|z_{n_{k}}|\to\infty$ forces $H(0)=0$ but this contradicts $H(1)=0$.

∎

######  Theorem 1.

Let $\\{F_{n}(z)\\}$ be a partition polynomial sequence whose phases are the
disjoint nonempty open subsets are $R(m_{1}),\cdots,R(m_{\ell})$ inside
$\mathbb{D}$. Assume the functions $\Omega_{n,m_{j}}(z)\neq 0$ for $0\neq z\in
R(m_{j})$.
(a) Then any $z_{0}\in\partial R(m_{j})\cap{\mathbb{D}}$ is in the zero
attractor of $\\{F_{n}(z)\\}$.
(b) No nonzero point in $R(m)$ lies in the zero attractor.
(c) The unit circle lies in the zero attractor.

###### Proof.

(a) To apply Sokal’s result (see Proposition 1), we need to consider the limit

$\lim_{n\to\infty}\frac{1}{\sqrt{n}}\ln|F_{n}(z)|=2\Re L_{m}(z),\quad z\in
R(m),$

that holds uniformly on compact subsets of the phase $R(m)$. By assumption,
$\Re L_{m}(z)$ is harmonic on $R(m)$ and for $\ell\neq k$, $\Re L_{\ell}(z)$
and $\Re L_{k}(z)$ are not harmonic continuations of each other. Hence, for
$z_{0}\in{\mathbb{D}}\cap\partial R(m)$, there cannot be a harmonic function
$v(z)$ in a neighborhood of $z_{0}$ such that $\liminf\ln|F_{n}(z)|\leq
v(z)\leq\limsup\ln|F_{n}(z)|$. We conclude that $z_{0}$ lies in the zero
attractor.

(b) By assumption, we know that

$\lim_{n\to\infty}n^{-3/4}F_{n}(z)\exp(-2n^{1/2}L_{m}(z))=\frac{1}{2\sqrt{\pi}}\Omega_{n,m}(z)\sqrt{L_{m}(z)}$

uniformly on the compact subsets of $R(m)$. Further, this limit is nonzero by
construction. Hence $z$ cannot lie in the zero attractor.

(c) Let $z^{*}$ be a point on the unit circle. Consider the open disk $V$ with
center $z^{*}$ and radius $r>0$. We consider the limit

$\lim_{n\to\infty}\frac{1}{n}\ln|F_{n}(z)|,\quad z\in V.$

∎

The above theorem leads to an algorithm for drawing zero attractors. Locally,
the boundary of a phase is a portion of an integral curve of the differential
equation:

$\displaystyle\frac{dy}{dx}$
$\displaystyle=\frac{\Re[L_{k}^{\prime}(x+iy)-L_{\ell}^{\prime}(x+iy)]}{\Im[L_{k}^{\prime}(x+iy)-L_{\ell}^{\prime}(x+iy)]},$

where its initial condition is found by solving the appropriate single non-
linear equation with a specified radius or angle; typically on the unit
circle, say $\Re L_{k}(e^{it})=\Re L_{\ell}(e^{it})$.

For chromatic polynomials, the analogue of the phases is the equimodular set
[7] which can also be described by means of integral curves but of a single
differential equation.

##  3 A New Special Function: The Root Dilogarithm

The determination of the phases for the families of partition polynomials in
this paper rests on finding the bounds among the family of functions given
through the dilogarithm.

######  Definition 5.

The root dilogarithms are the functions given by

$f_{k}(z)=\frac{1}{k}\Re\left[\sqrt{{\mathrm{Li}}_{2}(z^{k})}\right]$ (5)

where the square root is chosen as nonnegative and $k$ is a positive integer.

In general, the study of $f_{k}(z)$ is a lengthy tangent which we leave to the
appendix. We will state here what we need and refer the reader to the appendix
for their proofs. These results come in two kinds. First will be the calculus
of $f_{1}(z)$ on radial lines and circles.

######  Proposition 3.

The function $t\to f_{1}(it)$ is a positive, increasing, concave function of
$t\in(0,1).$

On circles, $f_{k}(re^{it})$ behaves in a cosine like fashion.

######  Proposition 4.

For a fixed value of $r\in(0,1],$ the function $t\mapsto f_{1}(re^{it})$ is
decreasing on $[0,\pi]$.

Second are root dilogarithm dominance facts that we will use extensively
throughout our examples.

######  Theorem 2.

For $0<|z|\leq 1$,

  1. 1.

$f_{k}(z)\leq f_{k}(|z|)\leq f_{2}(|z|)<f_{1}(z),$ $k\geq 2,$ $|\arg
z|\leq\pi/3.$

  2. 2.

$f_{k}(z)\leq f_{k}(|z|)<f_{1}(z),$ $k\geq 3,$ $|\arg z|\leq\pi/2$,

  3. 3.

$f_{k}(z)<f_{1}(z)$, $k\geq 2$, $0\leq\arg z\leq\pi/2$,

  4. 4.

$f_{k}(z)<\max[f_{1}(z),f_{2}(z),f_{3}(z)],$ $k\geq 4$, $\pi/2\leq\arg
z\leq\pi$,

##  4 Partition Polynomials

###  4.1 Partition Polynomials Weighted by Number of Parts

Consider the constant exponent sequence $\\{a_{m}\\}$ where $a_{m}=1$ for all
$m$ so there is no restriction on the parts of a partition. For
$k\in{\mathbb{Z}}^{+}$ and $t\in\widehat{{\mathbb{Z}}_{k}}$, the Dirichlet
series $D_{t,k}(s)$ are given by

$D_{t,k}(s)=\sum_{m=1}^{\infty}\frac{e_{k}(mt)}{m^{s}}=\frac{1}{k^{s}}\,\sum_{r=1}^{k}e_{k}(rt)\zeta(s,r/k)=F(t/k,s)$

where $F(\lambda,s)$ is the periodic zeta function

$F(\lambda,s)=\sum_{m=1}^{\infty}\frac{e(m\lambda)}{m^{s}}.$

$F(\lambda,s)$ is periodic in $\lambda$ with period 1. When $\lambda$ is not
an integer, then $F(\lambda,s)$ is an entire function of $s$; otherwise,
$F(\lambda,s)$ reduces to the Riemann zeta function.

######  Remark 1.

In this example $D_{t,k}(s)$ reduces to the usual Dirichlet $L$-function
relative to a character of ${\mathbb{Z}}_{k}$ when $(t,k)=1$.

Since $\zeta(0,r/k)=1/2-r/k$, for $1\leq r<k$,

$D_{t,k}(0)=\sum_{r=1}^{k}\,(\tfrac{1}{2}-\tfrac{r}{k})\,e_{k}(tr)=-\frac{1}{k}\sum_{r=1}^{k}re_{k}(tr),\quad
1\leq t<k,$

while $D_{k,k}(0)=-1/2$. Since $F(s,\lambda)$ has a pole at $s=1$ when
$\lambda$ is an integer, we find

$Res(D_{t,k},s=1)=Res(\zeta(s),s=1)\,\delta_{t,k}=\delta_{t,k}.$

Hence its $\widehat{{\mathbb{Z}}_{k}}$ Fourier transform is

$c_{k}(j)=\frac{1}{k},\quad 1\leq j\leq k.$

For $h$ relatively prime to $k$ such that $1\leq h<k$, we find an explicit
form for $L_{h,k}(z)$:

$\displaystyle L_{h,k}(z)^{2}$
$\displaystyle=\sum_{j=1}^{k}c_{k}(j)Li_{2}(e_{k}(j)z)=\frac{1}{k}\sum_{j=1}^{k}Li_{2}(e_{k}(j)z)=\frac{1}{k^{2}}Li_{2}(z^{k});$

in particular, the functions $L_{h,k}(z)$ are independent of the choice of $h$
so we write $L_{k}(z)$ for them. Furthermore, each $L_{k}(z)$ is analytic on
the unit disk except for branch cuts at the rays $\arg z=\pi j/k,$ $0\leq
j<k$.

For small values of $k$, we write out the values of $b_{k}$:

$b_{1}(1)=-1/2;\,b_{2}(1)=(-1)^{n}/2;\,b_{3}(1)=1/6,b_{3}(2)=-1/6,b_{3}(3)=-1/2.$

Hence we find that

$\displaystyle\omega_{1,1,n}(z)$
$\displaystyle=\sqrt{1-z},\quad\omega_{1,2,n}(z)=(-1)^{n}\sqrt{1-z},$
$\displaystyle\omega_{1,3,n}(z)$
$\displaystyle=e_{3}(-n)\,\frac{(1-e_{3}(2)z)^{1/6}(1-z)^{1/2}}{(1-e_{3}(1)z))^{1/6}},$
$\displaystyle\omega_{2,3,n}(z)$
$\displaystyle=e_{3}(-2n)\,\frac{(1-e_{3}(1)z)^{1/6}(1-z)^{1/2}}{(1-e_{3}(2)z))^{1/6}},$

We can verify directly that $\omega_{1,3,n}(z)\neq-\omega_{2,3,n}(z)$ for
$|z|<1$. Using Theorem 2, we obtain that there are only three phases $R(1),$
$R(2),$ and $R(3).$ These phases can be more concretely written down as

$\displaystyle R(1)$
$\displaystyle=\left\\{z\in\mathbb{D}:\Re\sqrt{Li_{2}(z)}>\frac{1}{2}\Re\sqrt{Li_{2}(z^{2})},\frac{1}{3}\Re\sqrt{Li_{2}(z^{3})}\right\\},$
$\displaystyle R(2)$
$\displaystyle=\left\\{z\in\mathbb{D}:\frac{1}{2}\Re\sqrt{Li_{2}(z^{2})}>\frac{1}{2}\Re\sqrt{Li_{2}(z)},\frac{1}{3}\Re\sqrt{Li_{2}(z^{3})}\right\\},$
$\displaystyle R(3)$
$\displaystyle=\left\\{z\in\mathbb{D}:\frac{1}{3}\Re\sqrt{Li_{2}(z^{3})}>\frac{1}{2}\Re\sqrt{Li_{2}(z)},\Re\sqrt{Li_{2}(z)}\right\\}.$

Using Theorem 1 we record the following proposition.

######  Proposition 5.

The zero attractor for the partition polynomials is

$S^{1}\cup\partial R(1)\cup\partial R(2)\cup\partial R(3).$

Figure 1:  Phases for weighted partition polynomials: (a) Region 1 (b) Region
2, (c) Region 3 in upper half plane.

Figure 2:  (a) Level sets $f_{1}(z)=f_{2}(z)$ (solid line),
$f_{2}(z)=f_{3}(z)$ (dotted line), $f_{1}(z)=f_{3}(z)$ (dash-dot lines) inside
the unit disk, (b) Closeup near the triple point

######  Remark 2.

The appendix shows that there are at most three phases; in fact, all three do
occur. In the right-half plane, including the imaginary axis,
$f_{2}(z),f_{3}(z)<f_{1}(z)$. In the second quadrant, consider
$f_{1}(re^{it})$ which is decreasing to $0$ while $f_{2}(re^{it})$ is
increasing from $0$, $t\in[\pi/2,\pi]$. Hence, for each $0<r\leq 1$, there is
a unique value $t_{12}(r)$ such that
$f_{1}(re^{it_{12}(r)})=f_{2}(re^{it_{12}(r)})$. Similar reasoning shows
$f_{2}(re^{it})=f_{3}(r^{it})$ has at most one solution for $2\pi/3\leq
t\leq\pi$; in fact, $f_{3}(re^{i5\pi/6})<f_{2}(re^{i5\pi/6})$ on $(0,1]$.

The phase $R(1)$ includes the unit disk that lies in the right-half plane. The
phases $R(2)$ and $R(3)$ lie in the left-hand plane since $f_{1}(z)>f_{k}(z)$,
for $k\geq 2$, $0<|z|\leq 1$, and $\Re z\geq 0$. For fixed $r>0$,
$f_{1}(r^{it})$ is decreasing to $0$ while $f_{2}(e^{it})$ is increasing from
$0$ on $[\pi/2,\pi]$. So $f_{1}(z)=f_{2}(z)$ determines a level set that
connects $0$ to a unique point on the unit circle. See Figure 1.

###  4.2 Partition Polynomials Whose Parts Satisfy a Linear Congruence

For a fixed positive integer $p\geq 2$, consider the exponent sequence
$\\{a_{m}\\}$ whose nonzero entries satisfy $m\equiv 1\,{\rm mod}\,p$. The
corresponding partition polynomials count partitions all of whose parts
satisfy the congruence $1\mod p$. For $k\in{\mathbb{Z}}^{+}$ and
$t\in\widehat{{\mathbb{Z}}_{k}}$, let

$D_{t,k}(s)=\sum_{m=1}^{\infty}\frac{a_{m}e_{k}(tm)}{m^{s}}=\sum_{\ell=0}^{\infty}\frac{e_{k}(t(p\ell+1))}{(p\ell+1)^{s}}=\frac{e_{k}(t)}{p^{s}}L(pt/k,s,1/p)$

where $L(\lambda,s,\nu)$ is the Lerch zeta function. We observe that if $k\mid
pt$ then $D_{t,k}(s)$ reduces to $e_{k}(t)p^{-s}\zeta(s,1/p)$. Since
$\zeta(0,\alpha)=1/2-\alpha$ and

$L(\lambda,0,\nu)=\frac{1}{1-\exp(2\pi
i\lambda)},\quad\lambda\notin{\mathbb{Z}}.$

we have for $1\leq t\leq k$

$D_{t,k}(0)=\begin{cases}e_{k}(t)(1/2-1/p),&k\mid pt,\\\
\frac{e_{k}(t)}{1-e_{k}(pt)},&k\nmid pt\end{cases}$

In the special case for $p=2$ with $1\leq t<k$ and $t$ is relatively prime to
$k$

$D_{t,k}(0)=\frac{i}{2\sin(2\pi t/k)}$

and, with $p=2$, $D_{k,k}(0)=0$.

Recall that $L(\lambda,s,\nu)$ is an entire function of $s$ if and only if
$\lambda\notin{\mathbb{Z}}$. When $\lambda$ is an integer, $L(\lambda,s,\nu)$
is a mereomorphic function of $s$ with a unique singularity at $s=1$ which is
a simple pole with residue $1$. Hence $D_{t,k}(s)$ has a singularity if and
only if $k\mid pt$. If this is case, then the singularity is a simple pole at
$s=1$ with residue

$Res(D_{t,k}(s),s=1)=\frac{1}{p}e_{k}(t).$

######  Lemma 1.

For $p\geq 3$, the finite Fourier transform $b_{k}(t)$ of $t\mapsto
D_{t,p}(0)$ is

$b_{k}(t)=\begin{cases}\frac{p/2-1}{p},&t=1,2\mid p\textrm{ and }2\mid p\\\
\frac{p-2}{2p},&t=1,2\nmid p\\\ 0,&2\leq t\leq p\end{cases}$

######  Example 1.

For the special case: $p=2$ with small values of $k$, we record the values of
the finite Fourier transform $b_{k}$:

$\displaystyle b_{3}(1)$ $\displaystyle=-b_{3}(2)=1/3,\quad b_{3}(3)=0$
$\displaystyle b_{4}(1)$ $\displaystyle=b_{4}(3)=1/4,\quad
b_{4}(2)=b_{4}(4)=0,$ $\displaystyle b_{6}(1)$
$\displaystyle=-b_{6}(5)=1/3,\quad b_{6}(2)=b_{6}(3)=b_{6}(4)=b_{6}(6)=0.$

This allows us to write out the explicit forms for the corresponding
$\omega_{h,k,n}(z)$ functions:

$\displaystyle\omega_{1,1,n}(z)$
$\displaystyle=1,\quad\omega_{1,2,n}(z)=(-1)^{n},\quad$
$\displaystyle\omega_{1,4,n}(z)$
$\displaystyle=i^{-n}\sqrt[4]{\frac{z-i}{z+i}},\quad\omega_{3,4,n}(z)=i^{n}\sqrt[4]{\frac{z+i}{z-i}}$

Further we can check directly that $\omega_{1,4,n}(z)=-\omega_{3,4,n}(z)$ only
if $z=0$.

######  Example 2.

For $p=3$, we have

$\displaystyle b_{1}(1)$ $\displaystyle=-1/2,\quad b_{2}(1)=1/3,b_{2}(2)=-1/6$
$\displaystyle b_{6}(1)$ $\displaystyle=1/3,b_{6}(4)=-1/6,\quad
b_{6}(2)=b_{6}(3)=b_{6}(5)=b_{6}(6)=0$

######  Lemma 2.

If $(k,p)=m>1$, then $c_{k}(j)=\begin{cases}\frac{(k,p)}{kp},&j\equiv
1\mod(k,p),\\\ 0,&j\not\equiv 1\mod(k,p)\end{cases}$

###### Proof.

Let $1\leq t\leq k$. Then $k\mid pt$ if and only if $k_{0}\mid p_{0}t$ where
$k_{0}=k/(k,p)$ and $p_{0}=p/(k,p)$. In other words, $k|\mid pt$ if and only
if $p_{0}t\in k_{0}{\mathbb{Z}}$. In our case, $t=vk_{0}$, $1\leq v\leq(k,p)$.
Consider

$\displaystyle c_{k}(j)$
$\displaystyle=\frac{1}{k}\sum_{t=1}^{k}e_{k}(-jt)\,Res(D_{t,k}(s),s=1)=\frac{1}{k}\sum_{t=1}^{k}e_{k}(-jt)\,\delta_{k\mid
pt}\frac{e_{k}(t)}{p}$
$\displaystyle=\frac{1}{k}\sum_{t=1}^{k}e_{k}(-jt)\,\delta_{k_{0}\mid
p_{0}t}\frac{e_{k}(t)}{p}=\frac{1}{k}\sum_{v=1}^{(k,p)}e_{k}(-jvk_{0})\,\frac{e_{k}(vk_{0})}{p}$
$\displaystyle=\frac{1}{kp}\sum_{v=1}^{(k,p)}e_{k}(vk_{0}(1-j))=\frac{1}{kp}\sum_{v=1}^{(k,p)}e_{(k,p)}(v(1-j))$
$\displaystyle=\begin{cases}\frac{(k,p)}{pk},&j\equiv 1\mod(k,p)\\\
0,&j\not\equiv 1\mod(k,p)\end{cases}$

∎

We use this Kubert-type identity in the next lemma:

$\sum_{m=1}^{k}Li_{2}(ze_{k}(m))=\frac{1}{k}Li_{2}(z^{k}).$

######  Lemma 3.

Let $1\leq h<k$ and $(h,k)=1$. Then

$\displaystyle
L_{h,k}(z)^{2}=\frac{(k,p)^{2}}{k^{2}p}Li_{2}\left(\,e_{(k,p)}(h)\,z^{k/(k,p)}\right).$

###### Proof.

Consider the following expansions

$\displaystyle
L_{h,k}(z)^{2}=\sum_{j=1}^{k}c_{k}(j)Li_{2}\left(e_{k}(jh)z\right)=\sum_{k\mid(j-1)}\frac{(k,p)}{kp}Li_{2}\left(e_{k}(jh)z\right)$
$\displaystyle=\sum_{v=1}^{k/(k,p)}\frac{(k,p)}{kp}Li_{2}\left(e_{k}((v(k,p)+1)h)z\right)$
$\displaystyle=\frac{(k,p)}{kp}\sum_{v=1}^{k/(k,p)}Li_{2}\left(e_{k/(k,p)}(v)\,[e_{k}(h)z]\right)$
$\displaystyle=\frac{(k,p)}{kp}\,\frac{1}{k/(k,p)}Li_{2}\left([e_{k}(h)z]^{k/(k,p)}\right)$
$\displaystyle=\frac{(k,p)}{kp}\,\frac{1}{k/(k,p)}Li_{2}\left([e_{k}(hk/(k,p))]\,z^{k/(k,p)}\right)$
$\displaystyle=\frac{(k,p)^{2}}{k^{2}p}Li_{2}\left(\,e_{(k,p)}(h)\,z^{k/(k,p)}\right).$

∎

######  Remark 3.

For polynomials counting partitions whose parts are congruent to $a$ modulo
$p$ with $(a,p)=1$, then

$L_{h,k}(z)^{2}=\frac{(k,p)^{2}}{k^{2}p}Li_{2}\left(\,e_{(k,p)}(ha)\,z^{k/(k,p)}\right).$

For $p\geq 2$, assume that the only nonzero entries of the exponent sequence
$\\{a_{m}\\}$ satisfy $m\equiv 1\,{\rm mod}\,p$. Calculating directly from the
generating function $G(z,q)$, we get

$F_{\ell}(e^{2\pi i/p}z)=e^{2\pi i\ell/p}F_{\ell}(z)$

Since the polynomial $F_{\ell}(z)$ has real coefficients,
$\overline{F_{\ell}(z)}=F_{\ell}(\overline{z})$ as well. So their moduli
$|F_{\ell}(z)|$ are invariant under the action of the dihedral group generated
by the rotation by angle $2\pi/p$ and reflection about the real axis. Note the
choice of $1\mod p$ is done for simplicity and a similar analysis can be done
for $a\mod p$ together with the requriement $a_{1}=1$ although the
computations are needlessly lengthy. Nonetheless, for clarity for the reminder
of this section, we will still use the two variable notation for phases

$R(h,k)=\\{z\in\mathbb{D}\setminus 0:\Re L_{h,k}(z)\ \textrm{is maximal}\\}.$

with $R(p,p)=R(1,1)=R(0,p)$ for notational uniformity. The phase functions
connect to the root dilogarithm function by

$\Re L_{h,k}(z)=\frac{1}{\sqrt{p}}f_{k/(k,p)}(e_{k}(h)z).$ (6)

We need an elementary number theory lemma to handle the cyclic nature of $\Re
L_{h,k}(z).$

######  Lemma 4.

For every $k,j\in\mathbb{N},$ $k/(k,j)=1$ if and only if $k\mid j.$

###### Proof.

Suppose that $k\mid j$ and of course $k\mid k.$ By the definition of greatest
common divisor, $k\mid(k,j).$ But $k\leq j$ so $k/(k,j)=1.$ If $k/(k,j)=1$
then $k\mid(k,j).$ Therefore by the definition of greatest common divisor,
$k\mid j.$ ∎

The next lemma is a consequence of Theorem 2 part (1).

######  Lemma 5.

We can reduce the set of phase functions for every $z\in\mathbb{D},$ $j\geq
3,$

$\max\\{\Re L_{h,k}(z):k\mid p,(k,p)=1\\}>\max\\{\Re L_{h,k}(z):k\nmid p\\}$

In terms of phases we have

$\overline{\mathbb{D}}=\bigcup_{h\in\mathbb{Z}_{p}}\overline{R(h,p)}.$

###### Proof.

If $p$ is a prime, we observe, using Lemma 4, that if $p/(p,k)=1$ then $k\mid
p$ and either $k=1$ or $k=p.$ In the case $k=1$, then $h=1$ since $(h,k)=1$;
otherwise, $k=p$ and so $h\in\mathbb{Z}_{p}^{\times}.$ Using the definition of
$\Re L_{h,k}(z)$ in Equation 6 and applying Proposition 4, we obtain

$\displaystyle\max\\{\Re L_{h,k}(z):k\mid p,(k,h)=1\\}$
$\displaystyle=\max\\{\frac{1}{\sqrt{p}}f_{1}\left(e_{p}(h)z\right):1\leq
h\leq p\\}$ $\displaystyle=\frac{1}{\sqrt{p}}f_{1}\left(e_{p}(h^{*})z\right).$

where $h^{*}$ is chosen as the minimizer of
$\left|\arg\left(e_{p}(h^{*})z\right)\right|$ over $h\in\mathbb{Z}_{p}$. By
Theorem 2 part (1) and the Equation 6, we next find that if
$\left|\arg\left(e_{p}(h^{*})z\right)\right|\leq\pi/3$ and $k\nmid p$ then

$\Re
L_{h,k}(z)=\frac{1}{\sqrt{p}}f_{k/(k,p)}\left(e_{k}(h)z\right)\leq\frac{1}{\sqrt{p}}f_{2}(|z|)\leq\frac{1}{\sqrt{p}}f_{1}(e_{p}(h^{*})z).$

Hence it is enough to show there exists at least one $h\in\mathbb{Z}_{p}$ such
that $\left|\arg\left(e_{p}(h)z\right)\right|\leq{\pi}/{p}\leq{\pi}/{3}.$ But
this is a classic consequence of the pigeonhole principle. ∎

######  Theorem 3.

The phase $R(h,p)$ is an open angular wedge of angle $2\pi/p$ of radius 1
centered at $\arg z=2h\pi/p.$

###### Proof.

By Lemma 5, we already know that

$\overline{\mathbb{D}}=\bigcup_{h\in\mathbb{Z}_{p}}\overline{R(h,p)}$

We also have a symmetry that $f_{1}(e_{p}(h)z)=f_{1}(e_{p}(h-1)ze_{p}(1)z)$
implying that if $z\in R(h,p)$ if and only if $e_{p}(1)z\in R(h-1,p).$ Set
wise we have

$R(n,p)=e_{p}(1)R(n-1,p).$

Hence we need only to show $R(1,1)$ contains the wedge $|\arg z|<\pi/p$ and
because phases are disjoint and symmetric we will be done. But by Proposition
4, Equation 6 and the previous lemma we need only determine for $|\arg
z|\leq\pi/p$ that $|\arg z|=\min_{1\leq h\leq p}|\arg\left(e_{p}(h)z\right)|.$

Suppose $|\arg e_{p}(n)|>\frac{3\pi}{2p}$ then by the triangle inequality
$|\arg ze_{p}(n)|>\frac{\pi}{p}$ and by Proposition 4,

$f_{1}(z)\geq f_{1}(e_{2p}(1))>f_{1}(ze_{p}(n)).$

This leaves only two potential candidates which can dominate $f_{1}(z);$
namely, $f_{1}(e_{p}(-1)z)$ and $f_{1}(e_{p}(1)z).$ So long as

$\frac{\pi}{p}-\arg z>\arg z,\quad\arg z+\frac{\pi}{p}>-\arg(z),$

then by Proposition 4, $f_{1}(z)$ dominates the both of them. This concludes
the proof. ∎

Like the other examples all we need now do is apply Theorem 1. Of course, the
boundaries of wedges can be written as the solution to $z^{p}=-1$ and so we
conclude the example.

######  Proposition 6.

Given $p>2$, the zero attractor for partition polynomials corresponding to
partitions into parts congruent to $1\mod p$ is the unit circle and a set of
$p$ spokes.

$S^{1}\cup\\{z\in\mathbb{D}:z^{p}=-1\\}$

###  4.3 Odd Partition Polynomials

When $p=2,$ we obtain a nondegenerate behavior for the odd parts polynomials.
Using Remark 3, we obtain

$\displaystyle\sqrt{2}L_{h,2k-1}(z)=L_{2k-1}(z)$
$\displaystyle=\frac{1}{2k-1}\sqrt{Li_{2}(z^{2k-1})},$
$\displaystyle\sqrt{2}L_{h,2k}(z)=L_{2k}(z)$
$\displaystyle=\frac{1}{k}\sqrt{Li_{2}(-z^{k})}.$

Once again that we note the connection to the root dilogarithm function where
$\Re L_{k}(z)=f_{k}(z)$ for $k$ odd and $\Re L_{2k}(z)=f_{1}(-z^{k})/k$ for
even indices.

######  Theorem 4.

The only nonempty phases are $R(1),$ $R(2),$ and $R(4).$

###### Proof.

Inspection of $L_{k}(z)$ near the points $z=1,-1,i$ proves that $R(1),$
$R(2),$ and $R(4)$ are nonempty. For $k>4,$ we note that $\Re L_{k}(z)\leq
f_{3}(|z|)$ which by Theorem 2 must be bounded by either $\Re
L_{1}(z)=f_{1}(z)$ or $\Re L_{2}(z)=f_{1}(-z)$ depending on whether $\Re z>0$
or $\Re z<0.$ Hence $R(k)$ is empty.

The case of $k=3$ follows by the same Lemma since

$\Re L_{3}(z)=f_{3}(z)\leq f_{3}(|z|)\leq\max\left(\Re L_{1}(z),\Re
L_{2}(z)\right).$

∎

Using Theorem 1 we record the following proposition.

######  Proposition 7.

The zero attractor for the odd parts partition polynomials is given by

$S^{1}\cup\partial R(1)\cup\partial R(2)\cup\partial R(4).$

However, we can go a bit further and identify curves to the boundaries of
$R(m).$

######  Proposition 8.

On $(0,1]$, there is a unique solution $\beta$ to $f_{1}(ir)=f_{2}(r)$;
further, $3/4<\beta<1$.

######  Theorem 5.

There exists exactly three phases $R(1),$ $R(2),$ and $R(4).$

  1. 1.

The boundary of $R(4)$ consists of $\gamma$ $-\gamma,$ $\overline{\gamma},$
and $-\overline{\gamma}$ where $\gamma$ is the level set $\\{z:\Re
L_{1}(z)=\Re L_{4}(z);\Re z,\Im z\geq 0\\}$.

  2. 2.

The boundary of $R(1)$ is $i[0,\beta]\cup
i[-\beta,0]\cup\gamma\cup\overline{\gamma}.$

  3. 3.

The boundary of $R(2)$ is $i[0,\beta]\cup
i[-\beta,0]\cup-\gamma\cup-\overline{\gamma}.$

###### Proof.

Since $\Re L_{1}(-z)=\Re L_{2}(z)$ and $\Re L_{4}(z)=\Re L_{4}(-z)$, we need
only classify the phases on the upper quarter disk $\mathbb{D}^{++}.$ On this
quarter disk, we have by Proposition 4 that $\Re L_{1}(z)>\Re L_{2}(z)$,
$z\neq 0$. It remains to examine $\Re L_{1}(z)$ and $\Re L_{4}(z).$ Consider
the function

$h(z)=\Re L_{1}(z)-\Re L_{4}(z)=f_{1}(z)-\frac{1}{2}f_{1}(-z^{2}).$

If $z\in\mathbb{D}^{++}$ and $h(z)>0$ then $z\in R(1)$ and if $h(z)<0$ then
$z\in R(4).$ But by Proposition 4 $h(re^{it})$ is decreasing.

The theorem now follows from Proposition 8, and the intermediate values
theorem. ∎

The more detailed reduction of the zero attractor now follows directly from
Theorem 1.

######  Theorem 6.

If $A$ is the zero attractor of $F_{n}(z)$ then

$A=S^{1}\cup\gamma\cup-\gamma\cup\overline{\gamma}\cup-\overline{\gamma}\cup
i[-\beta,\beta].$

Figure 3: Closeups of zero attractor inside unit disk: (a) Closeup: parts are
congruent to 0 or 2 modulo 3 (b); parts are congruent to 0 or 2 modulo 3 (c)
parts are congruent to 1 or 4 modulo 5,

Figure 4: Closeups of zero attractor inside unit disk: (a) all parts, zeros of
degree 25,000; (b) parts are odd, zeros of degree $2^{k}$, $k=12,\dots,15$;
(c) parts are congruent to 1 or 2 modulo 3, degree 2500

##  5 Further Examples and Conclusion

This framework for the zeros of partition polynomials is quite deep and
general. For mathematical convenience we restricted ourselves to the cases
that can be fully worked out analytically, but this framework generalizes
beyond just these cases.

Here are a few more examples, where we have done a numerical setup and
computed potential phase functions but have yet to identify $\partial R(m).$
In these cases, we require that the partition parts are solutions to
$x^{2}\equiv 1\mod p$; that is, they lie in the residue classes $1$ or $p-1$
modulo $p$. The calculation of the corresponding phase functions follow from
the same method as those for linear congruences.

Let $(h,k)=1$ with $1\leq h<k$. Then set

$L_{h,k}(z)^{2}=\frac{(k,p)^{2}}{k^{2}p}\left(Li_{2}(e_{(k,p)}(h)z^{k/(k,p)})+Li_{2}(e_{(k,p)}(-h)z^{k/(k,p)}\right).$

When we let $p=3,$

$L_{h,3k}(z)=L_{k}(z)=\frac{1}{k\sqrt{p}}\sqrt{2Li_{2}(z^{k})}$

and for $k$ not divisible by 3,

$L_{h,k}(z)=\frac{1}{k\sqrt{p}}\sqrt{\frac{1}{3}Li_{2}(z^{3k})-Li_{2}(z^{k})}.$

The zero attractor can be seen in Figure (4) where we have identified the
curves which appear to be portions of the level sets $f_{1}=f_{3}$ and
$f_{2}=f_{3}.$

For $p=5,$ we obtain the same kind of phenomenon except things cannot be
reduced as easily. As before, the phase functions are

$L_{h,5k}(z)=L_{k}(z)=\frac{1}{k\sqrt{p}}\sqrt{2Li_{2}(z^{k})}$

and for $k$ not divisible by 5,

$L_{h,k}(z)=\frac{1}{k\sqrt{p}}\sqrt{\frac{1}{3}Li_{2}(e_{5}(1)z^{k})+Li_{2}(e_{5}(-1)z^{k})}.$

Figure (3) shows with a little bit stronger evidence that the unit disk is
made up of three phases $R(1),$ $R(2),$ and $R(3).$

This framework reaches beyond the narrow scope we discussed here. After all,
we first worked out the asymptotics of polynomials for plane partitions whose
coefficients are indexed by the trace. The corresponding phase functions
involved a cube root rather than a square root [13]. Other more interesting
applications would be the polynomial analogue of Wright’s partitions into
powers ($a_{m}=\chi_{m^{k}}(m)$ for $k\in\mathbb{N}$ fixed) and the polynomial
analogue of Cayley double partitions [21] ($a_{m}=p(n)$ where $p(n)$ counts
the number of partitions of $n$).

The topological behavior of all the examples studied is also quite intriguing.
It seems that every zero attractor drawn so far is a connected. Does this
generalize the behavior of the eigenvalues of the truncations of infinite
banded Toeplitz matrices [32]? Is it possible to classify zero attractors by
their fundamental group properties or even perhaps identify each phase as the
interior of a specific set of level curves using the Jordan curve theorem?
These are all interesting questions which beg an answer for them.

##  Appendix A Analysis of the Root Dilogarithm

For convenience, we repeat the definition of the root dilogarithm

$f_{k}(z)=\frac{1}{k}\Re\left[\sqrt{{\mathrm{Li}}_{2}(z^{k})}\right],\quad
k\in{\mathbb{Z}}^{+},$ (7)

where the real part of the square root is chosen as nonnegative. By
construction each $f_{k}(z)$ is harmonic on the sectors in the unit disk
determined by the rays $\arg z=\pi j/k$, $0\leq j<k$, and symmetric about the
real axis. Further $f_{1}(z)=0$ if and only if $z\in[-1,0]$.

From this we learn several facts about $f_{k}(z)$ which derive from
$Li_{2}(z).$ For handling the real component of the square root we note that
for any $z\in{\mathbb{C}}$, $z\neq 0$,

$\sqrt{z}=\pm\frac{1}{\sqrt{2}}\left[\sqrt{\Re(z)+|z|}+i\,\textrm{sign}(\Im
z)\sqrt{-\Re(z)+|z|}\right].$ (8)

There are two special values of the polylogarithm worthy of note
$Li_{2}(1)=\zeta(2)={\pi^{2}}/{6}$ and $Li_{2}(-1)=-\eta(2)=-{\pi^{2}}/{12}.$
The goal of this appendix is to prove the facts in Section 3. We begin with
computing the behavior of $f_{1}(z)$ on circles.

######  Lemma 6.

For a fixed value of $r\in(0,1],$ the function $t\mapsto\arg Li_{2}(re^{it})$
is increasing on $[0,\pi]$.

###### Proof.

Let $g(z)$ be a univalent function on the unit disk normalized by $g(0)=0$
and$g^{\prime}(0)=1$. Then $g(z)$ is star-like if its derivative satisfies

$\frac{\partial\arg g(re^{it})}{\partial t}>0,$

for $0<r<1$ [18, p. 42]. By a result of Lewis [23], any polylogarithm
$Li_{s}(z)$, $s>0$, is univalent and star-like so the result follows ∎

By equation (8), $f_{1}(z)=0$ if and only if $\Im Li_{2}(z)=0$ and $\Re
Li_{2}(z)\leq 0$. Since $Li_{2}(z)$ is negative on $[-1,0]$, we find
$f_{1}(z)=0$ if and only if $z\in[-1,0]$.

######  Lemma 7.

For a fixed value of $r\in(0,1],$ the function $t\mapsto|Li_{2}(re^{it})|$ is
decreasing on $[0,\pi]$.

###### Proof.

Let $h(z)=\sum_{n=1}^{\infty}c_{n}z^{n}$ be a power series convergent in the
unit disk with $c_{n}>0$. For fixed $0<r\leq 1$, the theorem of Fejér
described in [19, p. 513] states that $t\mapsto|h(re^{it})|$ is decreasing
provided $\Delta^{4}(r^{n}c_{n})\geq 0$ for all $n$. For $Li_{2}(z)$, Fejér’s
conditon becomes $\Delta^{4}(r^{n}/n^{2})\geq 0$ for all $n$. These
inequalities hold because $a_{n}=r^{n}/(n+1)^{2}$ is a moment sequence for the
probability measure on $[0,1]$ with density function $-(1/r)\ln(x/r)$; in
fact,

$\frac{r^{n}}{(n+1)^{2}}=-\int_{0}^{1}\frac{x^{n}}{r}\log(x/r)\,dx$

It follows that $(-1)^{j}\Delta^{j}(a_{n})>0$; in particular, the required
fourth-order difference is also nonnegative. ∎

These two lemmas combine to prove our first objective.

######  Proposition 9 (Proposition 4 in main text).

For a fixed value of $r\in(0,1],$ the function $t\mapsto f_{1}(re^{it})$ is
decreasing on $[0,\pi]$.

###### Proof.

We begin by writing $f_{1}(re^{it})$ in polar form

$\displaystyle f_{1}(re^{it})$
$\displaystyle=|Li_{2}(re^{it})|^{1/2}\,\cos(\tfrac{1}{2}\arg
Li_{2}(re^{it})).$

Lemmas 6 and 7 show that both factors in this factorization are decreasing
functions of $t$. In particular, since $\arg Li_{2}(re^{it})$ is an increasing
function of $t$ which is $0$ on the positive real axis and $\pi$ on the
negative axis, the function $\cos(\tfrac{1}{2}\arg Li_{2}(re^{it}))$ is
decreasing on $[0,\pi]$. ∎

An immediate application of this proposition is to obtain the elementary
bounds we use throughout the appendix.

######  Lemma 8.

For $|z|\leq 1$, we have the bounds

  1. 1.

$\displaystyle\frac{\pi^{2}}{12}|z|\leq|Li_{2}(z)|\leq\frac{\pi^{2}}{6}|z|$,

  2. 2.

$\displaystyle 0\leq f_{k}(z)\leq\frac{\pi}{k\sqrt{6}}|z|^{k/2}$.

###### Proof.

(1) Apply the maximum and minimum modulus principle to $Li_{2}(z)/z$. Since
$Li_{2}(z)$ is univalent in the unit disk, it has a unique zero at $z=0$ which
is simple. The maximum occurs at $z=1$ and minimum at $z=-1$ which is observed
using Proposition 7.
(2) This follows immediately from part (1) and the fact that
$Li_{2}(1)=\pi^{2}/6$. ∎

###  A.1 Behavior of the root dilogarithm on the imaginary axis

As a heurstic, studying $f_{k}(z)$ on radial lines and circles is how one
proves facts about phases. Proposition 4 is conclusive on $f_{k}(z)$’s
behavior on the circles $|z|=r\leq 1$ but the behavior on radial lines is much
harder. We can scrape by with knowing how $f_{1}(z)$ behaves on the “main”
radial lines; the real and imaginary axes. On the negative axis $f_{1}(z)=0,$
and on the positive axis, $f_{1}(z)=\sqrt{Li_{2}(z)}$ which is routine. So we
focus on the imaginary axis and prove

######  Proposition 10.

[Proposition 3 in the main text] The function $r\to f_{1}(ir)$ is a positive,
increasing, concave function of $r\in(0,1).$

We break up this theorem into several lemmas; for convenience, we let

$r\mapsto\theta=\arg(Li_{2}(ir)).$

We will begin with an improvement to the bound of Lemma 8 when $z$ lies on the
imaginary axis. Recall that the real and imaginary parts of $Li_{2}(ir)$ have
the explicit forms [22, Chapter 5]:

$\displaystyle\Re Li_{2}(ir)=Li_{2}(-r^{2})/4\leq 0,\quad\Im
Li_{2}(ir)=\int_{0}^{r}\tan^{-1}(y)\,\frac{dy}{y}\geq 0.$ (9)

######  Lemma 9.

For $0<r\leq 1$, $\displaystyle\frac{\pi\sqrt{2}}{8}\sqrt{r}<f_{1}(ir)$.

###### Proof.

To find this bound for $f_{1}(ir)$ on $(0,1]$, we start by explicitly writing
$f_{1}(ir)$ using equation (8)

$\displaystyle f_{1}(ir)$
$\displaystyle=\Re\sqrt{Li_{2}(ir)}=\frac{1}{\sqrt{2}}\sqrt{\Re
Li_{2}(ir)+|Li_{2}(ir)|}$
$\displaystyle>\frac{1}{\sqrt{2}}\sqrt{-\frac{1}{48}\pi^{2}r+\frac{1}{12}\pi^{2}r}=\frac{\sqrt{2}}{8}\pi\,\sqrt{r},$

where

$\Re
Li_{2}(ir)=\frac{1}{4}Li_{2}(-r^{2})>\frac{1}{4}Li_{2}(-1)r=-\frac{1}{48}\pi^{2}r,\quad
0<r<1.$

Here we used that $Li_{2}(-r^{2})$ is concave so its secant line on $[0,1]$
gives a lower bound. ∎

######  Lemma 10.

The function $r\mapsto\theta=\arg(Li_{2}(ir))$ is increasing on $[0,1]$.

###### Proof.

For convenience, write $x(r)=\Re Li_{2}(ir),$ $y(r)=\Im Li_{2}(ir)$, and
$dy/dx=\tan\theta$. We will use the dot notation for derivative. By equation
(9), we note that $\dot{x}$, $\ddot{x}$, and $\ddot{y}$ are negative while
$\dot{y}$ is positive. Consequently, the sign of the desired derivative is
positive:

$\frac{d}{dr}\tan\theta=\frac{\dot{x}\,\ddot{y}-\ddot{x}\,\dot{y}}{(\dot{x})^{2}}>0$

∎

######  Remark 4.

Since $\theta(r)$ is increasing, $\Re Li_{2}(ir)<0$, and $\Im Li_{2}(ir)>0$,
the range of $\theta(r)$ must lie in $[\pi/2,3\pi/2]$. By equation (9),
$Li_{2}(i)$ is given explicitly by $-\pi^{2}/48+iG$, where $G$ is the Catalan
constant $\sum_{n=0}^{\infty}(-1)^{n}/(2n+1)^{2}$. We find that $\arg
Li_{2}(i)=\pi-\arctan(48G/\pi^{2})\simeq 1.79161$ so the range of $\theta(r)$
is exactly $[\pi/2,\arg Li_{2}(i)]$ which lies inside $[\pi/2,\pi/3]$.
Further, using equation (8), we record the explicit form for $f_{1}(i)$

$\displaystyle
f_{1}(i)=\frac{1}{4\sqrt{6}}\sqrt{-\pi^{2}+\sqrt{\pi^{4}+2304G^{2}}}$

so we can verify $f_{1}(i)<f_{2}(1)=\pi\sqrt{6}/12$ by direct calculation.

To complete the proofs in the section, we need the bounds

$\displaystyle\frac{1}{2}\leq\cos\left(\frac{\theta(r)}{2}\right)\leq\frac{\sqrt{2}}{2}\leq\sin\left(\frac{\theta(r)}{2}\right)\leq\frac{\sqrt{3}}{2}.$

If needed, these bounds can be tighten by using

$\displaystyle\cos(\theta(1)/2)$ $\displaystyle=\sqrt{(a-\pi^{2})/(2a)}\simeq
0.62488,$ $\displaystyle\sin\left(\theta(1)/2\right)$
$\displaystyle=\sqrt{(a+\pi^{2})/(2a)}\simeq 0.78071$

with $a=\sqrt{\pi^{4}+2304G^{2}}.$

######  Lemma 11.

$f_{1}(ir)$ is increasing on $[0,1]$.

###### Proof.

We write out the real part of the derivative

$\frac{d}{dr}\sqrt{Li_{2}(ir)}=-\,\frac{\ln(1-ir)}{2r\sqrt{Li_{2}(ir)}}.$

using the polar form of
$\displaystyle\sqrt{Li_{2}(ir)}=|Li_{2}(ir)|^{1/2}e^{i\theta(r)/2}$ to get

$\frac{d}{dr}\Re\sqrt{Li_{2}(ir)}=-\,\Re\left(\frac{\frac{1}{2}\ln(1+r^{2})-i\arctan(r)}{2r|Li_{2}(ir)|^{1/2}e^{i\theta(r)/2}}\right).$

Now its sign is the same as the sign of

$\sin(\theta(r)/2)\arctan(r)-\frac{1}{2}\cos(\theta(r)/2)\ln(1+r^{2})$

which is bounded below by

$\frac{\sqrt{2}}{2}\left(\arctan(r)-\frac{1}{2}\ln(1+r^{2})\right)=\frac{\sqrt{2}}{2}\,\int_{0}^{r}\frac{1-t}{1+t^{2}}\,dt\geq
0$

since $\theta(r)/2$ lies in the interval $[\pi/4,\pi/3]$. ∎

######  Lemma 12.

For $r\in(0,1),$ $f_{1}(ir)$ is a concave function.

###### Proof.

We use polar form again this time for the second derivative of $f_{1}(ir)$.
For the sake of notation, we introduce $g_{1}(r)$ and $g_{2}(r)$ by

$\frac{d^{2}}{dr^{2}}\Re\sqrt{Li_{2}(ir)}=\Re\left(g_{1}(r)+g_{2}(r)\right)$

where

$\displaystyle g_{1}(r)$
$\displaystyle=-\frac{1}{4}\frac{(\frac{1}{2}\ln(1+r^{2})-i\arctan(r))^{2}}{Li_{2}(ir)^{3/2}r^{2}},$
$\displaystyle g_{2}(r)$
$\displaystyle=\frac{i/2}{Li_{2}(ir)^{1/2}(1-ir)r}+\frac{1}{2}\frac{\frac{1}{2}\ln(1+r^{2})-i\arctan(r)}{Li_{2}(ir)^{1/2}r^{2}}.$

Since $\pi/2\leq\theta(r)\leq 2\pi/3$, we have $\sin(3\theta(r)/2)$ is
positive while $\cos(3\theta(r)/2)$ is negative. It is enough show that the
signs of $\Re g_{1}(r)$ and $\Re g_{2}(r)$ are negative. Now the sign of $\Re
g_{1}(r)$ is the same as the sign of

$\displaystyle-\Re$
$\displaystyle\left[\big{(}\tfrac{1}{4}(\ln(1+r^{2}))^{2}-i(\arctan(r))^{2}\big{)}\,\left(\cos(3\theta(r)/2)-i\sin(3\theta(r)/2)\right)\right]$
$\displaystyle\qquad=\cos(3\theta(r)/2)\left\\{(\arctan(r))^{2}-\tfrac{1}{4}(\ln(1+r^{2}))^{2}\right\\}$
$\displaystyle\qquad\qquad-\sin(3\theta(r)/2)\ln(1+r^{2})\arctan(r)$

which is negative since $\sin(3\theta(r)/2)\ln(1+r^{2})\arctan(r)$ is positive
and

$\displaystyle(\arctan(r))^{2}-\tfrac{1}{4}(\ln(1+r^{2}))^{2}$
$\displaystyle=\int_{0}^{r}\int_{0}^{r}\frac{(1-s)(1+t)}{(1+s^{2})(1+t^{2})}\,dsdt>0$

as well. The sign of $\Re g_{2}(r)$ is the same as the sign of

$\displaystyle\frac{1}{1+r^{2}}\left(-r\cos(\theta(r)/2)+\sin(\theta(r)/2)\right)$
$\displaystyle\qquad+\frac{1}{r}\left(\tfrac{1}{2}\ln(1+r^{2})\cos(\theta(r)/2)-\arctan(r)\sin(\theta(r)/2)\right)$
$\displaystyle\,\,=\frac{1}{r}\int_{0}^{r}\frac{r\cos(\theta(r)/2)+\sin(\theta(r)/2)}{1+r^{2}}\,dt+\frac{1}{r}\int_{0}^{r}\frac{t\cos(\theta(r)/2)-\sin(\theta(r)/2)}{1+t^{2}}\,dt$
$\displaystyle\qquad\qquad=\frac{\cos(\theta(r)/2)}{r}\int_{0}^{r}\left(\frac{t}{1+t^{2}}-\frac{r}{1+r^{2}}\right)\,dt$
$\displaystyle\qquad\qquad\qquad+\,\,\frac{\sin(\theta(r)/2)}{r}\int_{0}^{r}\left(\frac{1}{1+r^{2}}-\frac{1}{1+t^{2}}\right)\,dt$

which is negative since both integrals are negative while $\sin(\theta(r)/2)$
and $\cos(\theta(r)/2)$ are positive. ∎

Although the second derivative of $f_{1}(r)$ changes sign, $f_{2}(r)$ is
convex.

######  Lemma 13.

$f_{2}(r)$ is convex.

###### Proof.

The second derivative of $f_{2}(r)$ is

$\frac{-(\ln(1-r^{2}))^{2}}{r^{2}(Li_{2}(r^{2}))^{3/2}}+\frac{\ln(1-r^{2})}{r^{2}\sqrt{Li_{2}(r^{2})}}+\frac{2}{(1-r^{2})\sqrt{Li_{2}(r^{2})}}$

which has the same sign as

$h(r)=2r^{2}Li_{2}(r^{2})-(1-r^{2})(\ln(1-r^{2}))^{2}+(1-r^{2})\ln(1-r^{2}).$

It is elementary that the Taylor expansion of each term above has only even
powers with nonnegative coefficents starting with $r^{6}$. ∎

We record that $f_{k}(r)$ is convex for $k\geq 2$.

######  Proposition 11 (Proposition 8 in the main text).

There is a unique solution $\beta$ to $f_{1}(ir)=f_{2}(r)$ in $(0,1]$ further,
it satisfies $3/4<\beta<1$. For $r\in(0,\beta),$ $f_{1}(ir)>f_{2}(r)$ while
for $r\in(\beta,1),$ $f_{1}(ir)<f_{2}(r).$

###### Proof.

Define $h(r)=f_{2}(r)-f_{1}(ir)$ with $\in(0,1).$ By the last two Lemmas 12
and 13, $h(r)$ must be a convex function and therefore has at most one
critical point $h^{\prime}(r)=0$ in $(0,1)$. This also implies that it can
have at most two roots since if $h(x_{1})=h(x_{2})=h(x_{3})=0$ for some
$x_{1}<x_{2}<x_{3}$ then by Rolle’s theorem
$h^{\prime}(\xi_{1})=h^{\prime}(\xi_{2})=0$ for $\xi_{1}\neq\xi_{2}.$ We then
identify that $h(0)=0$ trivially and from the inequalities in Lemma 8, we find
that

$f_{2}(r)<\frac{\pi r}{2\sqrt{6}}<\frac{\pi\sqrt{2r}}{8}<f_{1}(ir)$

for $0<r<3/4$. Further, $f_{2}(i)>f_{1}(i)$ by Remark 4. Hence, by continuity
there exists $\beta\in(3/4,1)$ such that $f_{2}(i\beta)=f_{1}(i\beta)$.

∎

###  A.2 Generic Dominance of $f_{k}(z)$ on the Unit Disk

The last major goal of this appendix is to show $f_{1}(z)$, $f_{2}(z)$, and
$f_{3}(z)$ dominate all the other $f_{k}(z)$ in the unit disk and determine
where this dominance holds. In particular we need to show

######  Theorem 7.

[Theorem 2 in the main text] For $0<|z|\leq 1$,

  1. 1.

$f_{k}(z)\leq f_{k}(|z|)\leq f_{2}(|z|)<f_{1}(z),$ $k\geq 2,$ $|\arg
z|\leq\pi/3,$

  2. 2.

$f_{k}(z)\leq f_{k}(|z|)<f_{1}(z),$ $k\geq 3,$ $|\arg z|\leq\pi/2$,

  3. 3.

$f_{k}(z)<f_{1}(z)$, $k\geq 2$, $0\leq\arg z\leq\pi/2$,

  4. 4.

$f_{k}(z)<\max[f_{1}(z),f_{2}(z),f_{3}(z)],$ $k\geq 4$, $\pi/2\leq\arg
z\leq\pi$.

As with the previous theorems, we will prove this using a series of lemmas and
propositions. Part (1) of this theorem is an immediate consequence of the next
lemma.

######  Lemma 14.

The following are true:

  1. 1.

Let $0<|z|<1$. If $\Re Li_{2}(z)>0$, then $f_{k}(z)<f_{2}(|z|)<f_{1}(z)$ for
$k\geq 2$.

  2. 2.

For every $r\in(0,1)$ and $\theta\in(0,\pi/3),$ $\Re Li_{2}(re^{i\theta})>0.$

###### Proof.

We begin with the simple bound

$\displaystyle f_{1}(z)$ $\displaystyle=\frac{1}{\sqrt{2}}\sqrt{\Re
Li_{2}(z)+|Li_{2}(z)|}>\frac{1}{\sqrt{2}}\sqrt{|Li_{2}(z)|}\geq\frac{1}{\sqrt{2}}\sqrt{-Li_{2}(-1)|z|}$
$\displaystyle=\frac{1}{\sqrt{2}}\sqrt{\frac{1}{2}Li_{2}(1)|z|}=\frac{1}{2}\sqrt{Li_{2}(1)|z|}\geq\frac{1}{2}\sqrt{Li_{2}(|z|)}\geq
f_{2}(z).$

To check that $\Re Li_{2}(re^{i\pi/3})>0$, we use the identity that $\Re
Li_{2}(re^{i\pi/3})=\frac{1}{6}Li_{2}(-r^{3})-\frac{1}{2}Li_{2}(-r)$ [22, p.
133] and Lemma 6. ∎

By symmetry we need only consider the upper unit disk. For convenience we
divide the upper disk into three sectors: $\arg(z)\in[0,\pi/2]$,
$\arg(z)\in[\pi/2,3\pi/4]$, and $\arg(z)\in[3\pi/4,\pi]$.

######  Proposition 12.

For $0<r\leq 1$, we have the following bounds:
(a) $f_{3}(r)<f_{1}(ir)$, (b) $f_{5}(r)<f_{2}(re^{i3\pi/4})$, (c)
$f_{7}(r)<f_{3}(ir)$.

###### Proof.

Note that $f_{2}(re^{i3\pi/4})=f_{1}(ir^{2})/2$ and $f_{3}(ir)=f_{1}(ir)/3$.
In other words, the desired inequalities reduce to the following easily to
verify bounds which follow from Lemma 9 and Equation (2):

$\displaystyle f_{3}(r)\leq\frac{\pi}{3\sqrt{6}}r^{3/2}$
$\displaystyle<\frac{\pi\sqrt{2}}{8}r^{1/2}\leq f_{1}(ir),$ $\displaystyle
f_{5}(r)\leq\frac{\pi}{5\sqrt{6}}r^{5/2}$
$\displaystyle<\frac{\pi\sqrt{2}}{16}r\leq f_{2}(re^{i3\pi/4}),$
$\displaystyle f_{7}(r)\leq\frac{\pi}{7\sqrt{6}}r^{7/2}$
$\displaystyle<\frac{\pi\sqrt{2}}{24}r^{3/2}\leq f_{3}(ir).$

∎

By the above Proposition together with Proposition 4 and the bounds
$f_{k}(z)\leq f_{k}(|z|)\leq f_{j}(|z|)$, for $j\leq k$, we can extend the
bounds on radial lines to the three above designated sectors inside the unit
disk.

######  Corollary 1.

Let $0<|z|\leq 1$.
(1) For $k\geq 3$, $f_{k}(z)\leq f_{k}(|z|)<f_{1}(z)$,
$0\leq\arg(z)\leq\pi/2$.
(2) For $k\geq 5$, $f_{k}(z)\leq f_{k}(|z|)<f_{2}(z)$,
$3\pi/4\leq\arg(z)\leq\pi$.
(3) For $k\geq 7$, $f_{k}(z)\leq f_{k}(|z|)<f_{3}(z)$, $\pi/2\leq\arg(z)\leq
3\pi/4$.

Part (2) of Theorem 2 is part (1) of this corollary.

To sum up, we know that, for $0<|z|\leq 1$, $f_{k}(z)<f_{1}(z)$ with
$\arg(z)\in[0,\pi/2]$, $k\geq 3$ and $f_{k}(z)<\max(f_{2}(z),f_{3}(z))$ with
$\arg(z)\in[\pi/2,\pi]$, $k\geq 7$. In order to complete the proof of parts
(3) and (4) of Theorem 2, we must establish the five remaining cases which
requires working throught the remaining three special sectors.

###  A.3 Refining the Dominance of $f_{k}(z)$ on the Open Unit Disk

We now show for $k\geq 4,$
$f_{k}(z)<\max\left(f_{1}(z),f_{2}(z),f_{3}(z)\right)$ for $0<|z|\leq 1$. The
maximum modulus principle applies in these exceptional cases since the
functions $f_{k}$ are harmonic on the appropriate sector $S$ in the unit disk.
In particular, we need only to check dominance on the two boundary radial
lines, say $\arg(z)=\alpha$ or $\beta$ and on the boundary arc of the unit
circle $e^{it}$, $\alpha\leq t\leq\beta$. On the unit circle, $Li_{2}(e^{it})$
has an explicit representation:

$Li_{2}(e^{it})=r(t)-iCl_{2}(t)$

where

$r(t)=\frac{\pi^{2}}{6}-\frac{t(2\pi-t)}{4},\quad
Cl_{2}(t)=\int_{0}^{t}\ln[2\sin(s/2)]\,ds$

(see [22, p. 101]). $Cl_{2}(t)$ is called the Clausen integral and has special
values $Cl_{2}(\pi/2)=-G$, the Catalan constant, and $Cl_{2}(\pi)=0$. By
construction, $\Im Li_{2}(e^{it})=-Cl_{2}(t)$ is a nonnegative concave
decreasing function on $[\pi/2,\pi]$ so its secant line

$s(t)=2G(\pi-t)/\pi$

gives a lower bound on $[\pi/2,\pi]$. The tangent line $L(t)$ at $s=\pi/2$
yields an upper bound where $L(t)=s/2+(1/2)\ln(2)-\pi/4$. Given the explicit
form of the real part of the square root (equation (8)), we have a lower bound
for $f_{1}(e^{it})$ on $[\pi/2,\pi]$:

$f_{1}(e^{it})\geq\frac{1}{\sqrt{2}}\sqrt{r(t)+\sqrt{r^{2}(t)+s^{2}(t)}}.$

We obtain the following inequalites by direct calculation using the secant
lower bound and the exact forms of $f_{1}(1)$ and $f_{1}(i)$:

$f_{1}(i)/4<f_{1}(e^{i3\pi/4}),\quad f_{1}(1)/4<f_{1}(e^{i2\pi/3}),\quad
f_{1}(e^{i2\pi/3})/5<f_{1}(e^{i3\pi/4})$ (10)

as well as

$f_{1}(e^{i2\pi/3})/5<f_{1}(e^{i4\pi/5})$ (11)

needed in the proofs of Propositions 15 and 17 below. Further, using the lower
bound $\ln[2\sin(\pi/4)]$ for $\ln[2\sin(s/2)]$ on $[\pi/2,2\pi/3]$, we obtain

$\displaystyle-Cl_{2}(2\pi/3)$
$\displaystyle=-\int_{0}^{\pi/2}\ln[2\sin(s/2)]\,ds-\int_{\pi/2}^{2\pi/3}\ln[2\sin(s/2)]\,ds$
$\displaystyle=G-\int_{\pi/2}^{2\pi/3}\ln[2\sin(s/2)]\,ds<G-\ln[2\sin(\pi/4)](2\pi/3-\pi/2)$
$\displaystyle=G-\frac{\pi\ln 2}{12}.$

Substituting, we find that equation (11) now follows.

Note that that $f_{1}(e^{i2\pi/3})<f_{3}(e^{i2\pi/3})=f_{1}(1)/3$ by the
tangent line upper bound $L(t)$ for $-Cl_{2}(2\pi/3)$ on $[\pi/2,2\pi/3]$.
Since $f_{2}(e^{i2\pi/3})=f_{1}(e^{i2\pi/3})/2$, we find that

$f_{2}(e^{i2\pi/3})<f_{1}(e^{i2\pi/3})<f_{3}(e^{i2\pi/3}),$

so all three functions $f_{1}(z),f_{2}(z)$, and $f_{3}(z)$ occur in describing
phases.

We will also make use of the symmetry relation
$f_{1}(e^{i(\pi+t)})=f_{1}(e^{i(\pi-t)})$ on $[0,\pi]$.

######  Lemma 15.

For $0<r\leq 1$, $f_{3}(re^{i5\pi/6})<f_{2}(r^{i5\pi/6})$.

###### Proof.

We record that $f_{3}(re^{i5\pi/6})=f_{1}(ir^{3})/3$ and
$f_{2}(re^{i3\pi/4})=f_{1}(ir^{2})/2$. Since $f_{2}(re^{it})$ is increasing
for $t\in[\pi/2,\pi]$ and $f_{1}(ir)$ is also increasing, the result follows.
∎

####  A.3.1 Dominance of $f_{k}(z)$ on the Quarter Disk $\arg(z)\in[0,\pi/2]$

On the quarter disk, we already know
$f_{k}(z)\leq\max\left(f_{1}(z),f_{2}(z)\right)$ with $k\geq 3$. We must
reduce this to show $f_{2}(z)<f_{1}(z)$. This also concludes part (3) in
Theorem 2.

######  Proposition 13.

For $0<|z|\leq 1$ and $0\leq\arg(z)\leq\pi/2$,

$f_{2}(z)<f_{1}(z).$

###### Proof.

Since both $f_{1}$ and $f_{2}$ are harmonic on the open quarter unit disk, we
can use the maximum modulus principle. On the radial boundary lines,
$f_{2}(r)<f_{1}(r)$ and $0=f_{2}(ir)<f_{1}(ir)$, $r\in(0,1]$. On the arc
$e^{it}$, $t\in[0,\pi/2]$, both $f_{1}(e^{it})$ and $f_{2}(e^{it})$ are
decreasing functions by Proposition 4. On $t\in[0,\pi/4],$

$f_{2}(e^{it})\leq f_{2}(1)\leq f_{1}(e^{i\pi/4})<f_{1}(e^{it}).$

as $f_{2}(1)<f_{1}(e^{i\pi/4})$ by Lemma 14. For $t\in[\pi/4,\pi/2]$:

$f_{2}(e^{it})\leq f_{2}(e^{i{\pi}/{4}})=\frac{1}{2}f_{1}(i)\leq
f_{1}(i)<f_{1}(e^{it}).$

∎

####  A.3.2 Dominance of $f_{k}(z)$ on the sector $\arg(z)\in[3\pi/4,\pi]$

On this sector we have for $k\geq 5,$

$\max\left(f_{1}(z),f_{2}(z),f_{3}(z),f_{4}(z)\right)>f_{k}(z).$

To refine this result, we show $f_{4}(z)<f_{2}(z).$

######  Proposition 14.

On the sector $\arg z\in[3\pi/4,\pi]$ and $0<|z|\leq 1$, $f_{4}(z)<f_{2}(z)$.

###### Proof.

The sector $S$ has boundary lines with $\alpha=3\pi/4$ and $\beta=\pi$. When
$\arg(z)=3\pi/4$, $f_{4}(z)=0$ while $f_{2}(re^{i3\pi/4})=f_{1}(ir^{2})/2>0$
when $0<r\leq 1$. For $\beta=\pi$, $0<f_{4}(-r)=f_{4}(r)<f_{2}(-r)=f_{2}(r)$,
$0<r\leq 1$. On the arc $e^{it}$, $3\pi/4\leq t\leq\pi$, both $f_{2}(e^{it})$
and $f_{4}(e^{it})$ are increasing by Propositon 4. We check the chain of
inequalities

$f_{4}(e^{i5\pi/6})=f_{1}(e^{i2\pi/3})/4<f_{2}(e^{i3\pi/4})=f_{1}(i)/2$

by Propositon 4 and

$f_{4}(1)=(1/2)(f_{1}(1)/2)<f_{1}(e^{i\pi/3})/2=f_{2}(e^{i5\pi/6})$

by Lemma 14. Thus

$f_{4}(e^{it})<f_{4}(1)<f_{2}(e^{i\frac{5\pi}{6}})<f_{2}(e^{i\frac{3\pi}{4}})<f_{2}(e^{it}).$

∎

####  A.3.3 Dominance of $f_{k}(z)$ on the sector $\arg(z)\in[\pi/2,3\pi/4]$

As in the previous two sections, we have to reduce the number of relavent
$f_{k}(z).$ Things are a bit more fine tuned in this section however. There
are three cases.

######  Proposition 15.

On the sector with $\arg z\in[\pi/2,3\pi/4]$, $f_{4}(z)<f_{1}(z)$, $z\neq 0$.

###### Proof.

The sector $S$ has boundary lines with $\alpha=\pi/2$ and $\beta=3\pi/4$. We
need to recall Theorem 2 part (2) (which is already proven) that $f_{3}(r)\leq
f_{1}(ir).$ When $\arg(z)=\pi/2$, $f_{4}(ir)=f_{4}(r)<f_{3}(r)\leq f_{1}(ir)$,
$0<r\leq 1$. When $\arg(z)=3\pi/4$, $f_{4}(re^{i3\pi/4})=0$ while
$f_{1}(re^{i3\pi/4})>0$, $0<r\leq 1$. Hence $f_{4}(z)<f_{1}(z)$ on the radial
boundary lines.

On the arc $e^{it}$, $t\in[\pi/2,3\pi/4],$ we can use Proposition 4 to show
both $f_{1}(e^{it})$ and $f_{4}(e^{it})$ are decreasing. Using the secant line
bounds for $f_{1}(1)/4<f_{1}(e^{i2\pi/3})$ and
$f_{1}(i)/4<f_{1}(e^{i3\pi/4})$, we find that the required inequalities hold
on $t\in[\pi/2,2\pi/3]$:

$f_{4}(e^{it})\leq
f_{4}(i)=f_{1}(1)/4=\frac{\pi}{4\sqrt{6}}<f_{1}(e^{i2\pi/3})\leq
f_{1}(e^{it})$

and for $t\in[2\pi/3,3\pi/4]$

$f_{4}(e^{it})\leq
f_{4}(e^{i2\pi/3})=f_{1}(e^{i2\pi/3})/4<f_{1}(i)/4<f_{1}(e^{i3\pi/4})\leq
f_{1}(e^{it})$

where $f_{1}(e^{i2\pi/3})/4<f_{1}(i)/4$ holds since $f_{1}(e^{it})$ is
decreasing and $f_{1}(i)/4<f_{1}(e^{i3\pi/4})$ by equation (10).

∎

######  Proposition 16.

On the sector with $\arg z\in[\pi/2,3\pi/4]$, $f_{6}(z)<f_{3}(z)$, $z\neq 0$.

###### Proof.

We need to consider two sectors $S_{1}$ and $S_{2}$ so
$\arg(z)\in[\pi/2,2\pi/3]$ and $[2\pi/3,3\pi/4]$ so $f_{6}(z)$ will be
harmonic in each. The boundary lines of the sector $S_{1}$ are $\alpha=\pi/2$
and $\beta=2\pi/3$. For $\arg(z)=\pi/2$, $f_{6}(ir)=f_{1}(-r^{6})=0$ while
$f_{3}(ir)=f_{1}(ir^{3})>0$ for $r\in(0,1]$. For $\arg(z)=2\pi/3$,
$f_{6}(re^{i2\pi/3})=f_{1}(r)/6$ while $f_{3}(re^{i2\pi/3})=f_{1}(r^{3})/3$.
For $\arg(z)=3\pi/4$,

$f_{6}(re^{i3\pi/4})=\frac{1}{6}f_{1}(ir^{6})<\frac{1}{3}f_{1}(ir^{3})<\frac{1}{3}f_{1}(r^{3}e^{i\pi/4})=f_{3}(re^{i3\pi/4})$

since $f_{1}(ir)$ is increasing in $r$ by Proposition 3 and $f_{1}(z)$ is
decreasing in the argument of $z$ by Proposition 4. Hence $f_{6}(z)\leq
f_{3}(z)$ are the radial lines.

On the arc for $S_{1}$, both $f_{3}(e^{it})$ and $f_{6}(e^{it})$ are
increasing on $[\pi/2,2\pi/3]$ by Proposition 4. It convenient to use two
subintervals $[\pi/2,7\pi/12]$ and $[7\pi/12,2\pi/3]$ in this case. We need to
check that

$f_{6}(e^{i7\pi/12})<f_{3}(e^{i\pi/2}),\quad
f_{6}(e^{i2\pi/3})<f_{3}(e^{i7\pi/12}).$

Now

$\displaystyle f_{6}(e^{i7\pi/12})$
$\displaystyle=f_{1}(e^{i\pi/2})/6=f_{1}(i)/6<f_{1}(i)/3=f_{3}(e^{i\pi/2}),$
$\displaystyle f_{6}(e^{i2\pi/3})$
$\displaystyle=f_{1}(1)/6<f_{3}(e^{i7\pi/12})=f_{1}(e^{i\pi/4})/3$

since $\Re Li_{2}(e^{i\pi/4})>0$ and Lemma 14.

On the arc $[2\pi/3,3\pi/4]$ for $S_{2}$, both $f_{3}(e^{it})$ and
$f_{6}(e^{it})$ are decreasing. The required inequality
$f_{6}(e^{i2\pi/3})<f_{3}(e^{i3\pi/4})$ holds because
$f_{3}(e^{i3\pi/4})=f_{1}(e^{i\pi/4})/3$; so this inequality reduces to the
last inequality above. ∎

######  Proposition 17.

On the sector $\arg z\in[\pi/2,5\pi/8]$, $f_{5}(z)<f_{1}(z)$, $z\neq 0$.
(b) On the sector $\arg z\in[3\pi/5,4\pi/5]$, $f_{5}(z)<f_{2}(z)$, $z\neq 0$.

###### Proof.

(a) Since $f_{4}(z)<f_{1}(z)$ on $[\pi/2,3\pi/4]$, it is enough to show
$f_{5}(z)<f_{4}(z)$. For $t\in[\pi/2,5\pi/8]$, $f_{4}(re^{it})$ is decreasing
while $f_{5}(re^{it})$ is decreasing on $[\pi/2,3\pi/5]$ and increasing on
$[3\pi/5,5\pi/8]$. For fixed $r\in(0,1]$, the minimum value of
$f_{4}(re^{it})$ is $f_{4}(re^{i5\pi/8})=f_{1}(ir^{4})/4$ while the maximum
value of $f_{5}(re^{it})$ is $f_{5}(ir)=f_{1}(ir^{5})/5$. Since $f_{1}(ir)$ is
an increasing function, the desired inequality holds.
(b) We apply the maximum modulus principle on a sector $S$ with boundary
radial lines $\arg(z)=\alpha$ or $\beta$ where $\alpha=3\pi/5$ and
$\beta=3\pi/4$.

For $\arg(z)=3\pi/5$, $f_{5}(z)=0$ while $f_{2}(re^{i3\pi/5})>0$ with
$r\in(0,1]$. For $\arg(z)=4\pi/5$, we see by part (2) of Proposition 12 that

$f_{5}(r)<f_{2}(re^{i3\pi/4})<f_{2}(re^{i4\pi/5}).$

On the arc $e^{it}$, $3\pi/5\leq t\leq 4\pi/5$, both $f_{2}(e^{it})$ and
$f_{5}(e^{it})$ are increasing so it is enough to check the two evaluations:

$f_{5}(e^{i2\pi/3})<f_{2}(e^{i3\pi/5}),\quad
f_{5}(e^{i3\pi/4})<f_{2}(e^{i3\pi/4}).$

We just saw that the last inequality automatically holds. For the remaining
inequality, consider

$f_{5}(e^{i2\pi/3})=f_{1}(e^{i2\pi/3})/5,\quad
f_{2}(e^{i3\pi/5})=f_{1}(e^{i4\pi/5})/2$

then we can use equation (11).

∎

## References

  * [1]
  * [2]

## References

* [3] George E. Andrews.  The theory of partitions.  Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998.  Reprint of the 1976 original.
* [4] George E. Andrews.  Basis partition polynomials, overpartitions and the Rogers-Ramanujan identities.  J. Approx. Theory, 197:62–68, 2015.
* [5] Andrew Bakan, Stephan Ruscheweyh, and Luis Salinas.  Universal convexity and universal starlikeness of polylogarithms.  Proc. Amer. Math. Soc., 143(2):717–729, 2015.
* [6] E. T. Bell.  Partition polynomials.  Ann. of Math. (2), 29(1-4):38–46, 1927/28.
* [7] Norman Biggs, Equimodular curves, Discrete Math. Vol. 259, 37–57 (2002).
* [8] Robert Boyer and William M. Y. Goh.  On the zero attractor of the Euler polynomials.  Adv. in Appl. Math., 38(1):97–132, 2007.
* [9] Robert P. Boyer and William M. Y. Goh.  Partition polynomials: asymptotics and zeros.  In Tapas in experimental mathematics, volume 457 of Contemp. Math., pages 99–111. Amer. Math. Soc., Providence, RI, 2008.
* [10] Robert P. Boyer and William M. Y. Goh.  Polynomials associated with partitions: asymptotics and zeros.  In Special functions and orthogonal polynomials, volume 471 of Contemp. Math., pages 33–45. Amer. Math. Soc., Providence, RI, 2008.
* [11] Robert P. Boyer and William J. Keith.  Stabilization of coefficients for partition polynomials.  Integers, 13:Paper No. A56, 15, 2013.
* [12] Robert P. Boyer and Daniel T. Parry.  On the zeros of plane partition polynomials.  Electron. J. Combin., 18(2):Paper 30, 26, 2011.
* [13] Robert P. Boyer and Daniel T. Parry.  Phase calculations for planar partition polynomials.  Rocky Mountain J. Math., 44(1):1–18, 2014.
* [14] Robert P. Boyer and Daniel T. Parry.  Plane partition polynomial asymptotics.  Ramanujan J., 37(3):573–588, 2015.
* [15] Kathrin Bringmann and Byungchan Kim.  On the asymptotic behavior of unimodal rank generating functions.  J. Math. Anal. Appl., 435(1):627–645, 2016.
* [16] C. Díaz Mendoza and R. Orive.  The Szegő curve and Laguerre polynomials with large negative parameters.  J. Math. Anal. Appl., 379(1):305–315, 2011.
* [17] Karl Dilcher.  Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials.  J. Approx. Theory, 49(4):321–330, 1987.
* [18] Peter L. Duren.  Univalent functions.  Springer-Verlag, 1983.
* [19] Alan Gluchoff and Frederick Hartmann.  Univalent polynomials and non-negative trigonometric sums.  Amer. Math. Monthly, 105(6):508–522, 1998.
* [20] Siegfried Grossmann and Martin Holthaus.  Maxwell’s demon at work: Two types of bose condensate fluctuations in power-law traps.  Opt. Express, 1(10):262–271, Nov 1997.
* [21] Ryuji Kaneiwa, An asymptotic formula for Cayley’s double partition fomula $p(2;n)$, Tokyo J. Math. 2, No. 1, 137-158 (1979).
* [22] Leonard Lewin, Polylogarithms and Associated Functions, Elsevier North Holland, 1981
* [23] John L. Lewis.  Convexity of a certain series.  J. London Math. Soc. (2), 27(3):435–446, 1983.
* [24] R. E. Meyer.  A simple explanation of the Stokes phenomenon.  SIAM Rev., 31(3):435–445, 1989.
* [25] Saiful R. Mondal and A. Swaminathan.  Geometric properties of generalized polylogarithm.  Integral Transforms Spec. Funct., 21(9-10):691–701, 2010.
* [26] Patrick Navez, Dmitri Bitouk, Mariusz Gajda, Zbigniew Idziaszek, and Kazimierz Rza¸żewski.  Fourth statistical ensemble for the bose-einstein condensate.  Phys. Rev. Lett., 79:1789–1792, Sep 1997.
* [27] Cormac O’Sullivan.  On the partial fraction decomposition of the restricted partition generating function.  Forum Math., 27(2):735–766, 2015.
* [28] Daniel Parry.  A polynomial variation of Meinardus’ theorem.  Int. J. Number Theory, 11(1):251–268, 2015.
* [29] Wladimir de Azevedo Pribitkin.  Simple upper bounds for partition functions.  Ramanujan J., 18(1):113–119, 2009.
* [30] Igor E. Pritsker and Richard S. Varga.  The Szegő curve, zero distribution and weighted approximation.  Trans. Amer. Math. Soc., 349(10):4085–4105, 1997.
* [31] Alan D. Sokal.  Chromatic roots are dense in the whole complex plane.  Combin. Probab. Comput., 13(2):221–261, 2004.
* [32] J. L. Ullman, A problem of Schmidt and Spitzer, Bull. Amer. Math. Soc. 73, 883-885, 1967.
* [33] Christoph Weiss and Martin Wilkens.  Particle number counting statistics in ideal bose gases.  Opt. Express, 1(10):272–283, Nov 1997.
* [34] E. Maitland Wright.  Asymptotic Partition Formulae: (II) Weighted Partitions.  Proc. London Math. Soc., S2-36(1):117.
* [35] D. Zagier.  The remarkable dilogarithm.  J. Math. Phys. Sci., 22(1):131–145, 1988.


20XX Vol. X No. XX, 000–000

11institutetext: School of Science, Western Sydney University, Locked Bag
1797, Penrith, NSW 2751, Australia<EMAIL_ADDRESS>
Received 20XX Month Day; accepted 20XX Month Day

# Photometric analysis of two extreme low mass ratio contact binary systems

Surjit S Wadhwa 11 Nick F H Tothill 11 Ain Y DeHorta 11 Miroslav Filipović 11

###### Abstract

Multi band photometry and light curve analysis for two newly recognized
contact binary systems, TYC 6995-813-1 and NSVS 13602901 are presented. Both
were found to be of extreme low mass ratios 0.11 and 0.17, respectively. The
secondary components of both systems show evidence of considerable evolution
with elevated densities as well as both luminosity and radii well above their
main sequence counterparts. Even in the absence of significant spot activity
at least one of the systems, TYC 6995-813-1, shows features of magnetic and
chromospheric activity. TYC 6995-813-1 is also determined to be a potential
merger candidate with its current separation near the theoretical instability
separation.

###### keywords:

stars: binaries: eclipsing — stars: evolution — stars: low-mass — techniques:
photometric

##  1 Introduction

The merger of contact binary star components are thought to result in a
transient event referred to as a red nova. Estimates suggest that up to 1 in
500 stars in the Galaxy maybe contact binaries (Rucinski 2007). The merger
event is relatively rare and until recently there had been no confirmed
observed events. The initial eruption of V1309 Scorpii (V1309 Sco) (Nakano et
al. 2008) was thought to be a classical nova eruption because of the rapid
increase in brightness. In the days following the initial eruption, V1309 Sco
did not behave as expected, rapidly cooling in a matter of days (Mason et al.
2010). The spectrum became redder (cooler), as opposed to hotter as in a
classical nova, suggesting the event was most likely a luminous red nova.
Although not known prior to the event, the progenitor of V1309 Sco had been in
the field of view of the OGLE (Optical Gravitational Lensing Experiment) for a
number of years prior to the outburst. Photometric analysis of the OGLE data
indicated that the progenitor was indeed a contact binary system and that
V1309 Sco was indeed a merger event between the components of a contact binary
system with a resultant rapidly rotating single star (Tylenda et al. 2011).
Recently there has been a rise in the interest of contact binary star mergers
with a number of new projects exploring the theoretical (Wadhwa et al. 2021)
and observational aspects (Gazeas et al. 2021)

There has been heightened interest in orbital stability of contact binary
systems as a way of identifying of potential merger candidates since the
eruption of V-1309. The interplay of orbital and spin angular momentum is
critical in maintaining orbital stability with instability likely if orbital
angular momentum decreases below three times the total spin angular momentum
of the components (Hut 1980). Determination of angular momentum is dependent
on accurate determination of absolute parameters and contact binary systems
provide one of the best avenues for determining the basic absolute parameters
such as masses and radii of stars. Previous work (Arbutina 2007) has shown
that in most instances, orbital instability is likely to occur in systems with
extreme low mass ratios.

In this study we report the multi-band photometry of two recently identified
contact binary systems. We performed simultaneous multi-band analysis of the
light curve using a recent version of the Wilson & Devinney (1971) code. We
show that both systems are suitable for light curve analysis with complete
eclipses, both are of extreme low mass ratio and both have significantly
brighter and larger secondary components relative to their main sequence
counterparts. In addition, where possible, we explore the evolution of the
systems and other astrophysical indicators of potential instability.

##  2 Photometry and Light Curves

Images were acquired in B,V and R bands using the Western Sydney University
(WSU) 0.6m telescope equipped with a SBIG 8300T CCD camera and standard
filters. Images were plate solved using the American Association of Variable
Star Observers (AAVSO) VPHOT engine and differential photometry was also
performed using the VPHOT engine. The photometry report from the VPHOT engine
gives an estimate of the error and all observations where the reported error
exceeded 0.01 magnitude were excluded. We used the method of Kwee & van
Woerden (1956) to derive new times of minima and new period was derived using
the PSeach utility available through the AAVSO. The photometric data was
folded using the derived period and time of minima. The deeper of the two
eclipses was designated as the primary eclipse at phase zero. The maximum
brightness for each pass-band at phase 0.25 and 0.75 were determined using
parabolic fit for observations from phase 0.24-0.26 and 0.74-0.76
respectively. In order to determine the absolute magnitude of the primary (see
below) the V band magnitude at mid secondary eclipse was taken as the average
magnitude between phases 0.49-0.51 as all systems showed complete eclipse
during this part of the phase cycle. Parabolic fit between phase 0.95 and 0.05
combined with the maximum magnitude was used to determine the amplitude of
each system. As both systems being imaged are relatively new discoveries no
orbital period change analysis was possible given only scant (or nil)
historical observations.

###  2.1 TYC 6995-813-1

TYC 6995-813-1 ($\alpha_{2000.0}=00\ 06\ 49.98$, $\delta_{2000.0}=-35\ 37\
29.1$) was identified as a contact binary both by the Catalina survey (Drake
et al. 2017) and the All Sky Automated Survey - Super Nova survey (ASAS-SN)
(Jayasinghe et al. 2018) with a preliminary period of 0.38318 days.
Differential photometry was performed with TYC 6995-815-1 as the comparison
star and 2MASS 00071379-3534495 as the check star. The amplitude in V band was
approximately 0.34 magnitudes (12.08 - 12.42) while the difference between the
primary and secondary eclipses, as expected, was small at 0.04 magnitudes. The
red band amplitude was 0.32 magnitudes (11.97 - 12.29) while the blue band
amplitude was 0.37 magnitudes (12.63 - 13.00). The light curve demonstrates a
complete eclipse lasting approximately 28 minutes. A single new minimum time
for the primary eclipse was determined as HJD 2458797.9902 (0.0002) and a more
refined period of 0.383130 (0.00002) days calculated.

###  2.2 NSVS 13602901

NSVS 13602901 ($\alpha_{2000.0}=16\ 46\ 09.83$, $\delta_{2000.0}=-03\ 52\
14.3$) was recognised as a contact binary systems both by the All Sky
Automated Survey (Pojmanski 1997) and Northern Sky Variable Survey (Wozniak et
al. 2003) with a preliminary period of 0.52379 days. As per TYC 6995-813-1
images were acquired using the UWS telescope and photometry performed using
the VPHOT engine with TYC 5054-1601-1 as the comparison star and TYC
5058-642-1 as the check star. The V band amplitude was 0.4 magnitudes (11.83 -
12.23) while difference in the eclipse depths was again 0.04 magnitudes. The
red band amplitude was 0.39 magnitudes (11.47 - 11.86) while the blue band
amplitude was 0.41 magnitudes (12.45 - 12.86). The eclipses are complete
lasting approximately 69 minutes. A single new time for the primary minimum
was determined as HJD 2458737.89608 (0.00003) while the period was refined to
0.5238905 (0.00002) days.

##  3 Light Curve Analysis

Reliable light curve solutions of contact binary systems without spectroscopic
mass ratios is only possible in cases where complete eclipses are present to
constrain the the highly correlated geometric parameters such as the mass
ratio ($q$), inclination ($i$) and fill-out ($f$) (Terrell & Wilson 2005).
Both our systems show complete eclipses and are thus suitable for photometric
analysis without needing spectroscopic observations. All light curves were
solved using a recent (2010) version of the Wilson-Devinney code (WD) (Wilson
& Devinney 1971). As there was no appreciable variation in the two brightest
magnitudes (O’Connell Effect) only unspotted solutions were sought for both
systems. As geometric elements account almost entirely for the shape of
contact binary light curves (Rucinski 1993) certain parameters were fixed for
both systems. As the components of contact binary systems are surrounded by a
common we applied simple reflection treatment (Rucinski 1969) and fixed the
bolometric albedos $A_{1}=A_{2}=0.5$ and the gravity darkening coefficients
$g_{1}=g_{2}=0.32$ (Lucy 1967). Limb darkening coefficients for each pass-band
were interpolated from van Hamme (1993) and logarithmic law applied as per
Nelson & Robb (2015). The other fixed parameter in all cases was the effective
temperature of the primary ($T_{1}$). This was estimated firstly by estimating
the absolute magnitude of the primary ($Mv_{1}$) by using the mid-eclipse
magnitude ($m$) from the light curve (as this represents light from the
primary only), distance ($D$) (Gaia Collaboration 2018), and interstellar
reddening [E(B-V)] (Schlegel et al. 1998) as follows:

$M_{v}=m-(5*log_{10}(D/10))+3.1*E(B-V)$ (1)

Temperature of the primary was then determined from calibration for main
sequence stars from Cox (2000). There are various approaches to estimate the
effective temperature of the primary with wide variations possible. In the
case of NSVS 1360291 the Gaia Collaboration (2018) reports the effective
temperature as 5725K while the LAMOST catalog (Luo et al. 2019) suggests
likely temperature as approximately 7300K. As geometric parameters almost
exclusively determine the shape of contact binary light curves variations in
the estimate of $T_{1}$ will have no significant effect on the light curve
solution with respect to the photometric mass ratio and other major parameters
such as inclination and fill-out (Rucinski 1993).

Geometric parameters such as the mass ratio, inclination and degree of contact
are the dominant variables determining the shape of contact binary light
curves with gravity/limb darkening and reflection playing a minimal role
(Rucinski 1993). As noted by Terrell & Wilson (2005) significant correlation
exists between the mass ratio and the other two major geometric parameters and
reliable photometric solutions cannot be obtained, except circumstances of a
total eclipses, when the mass ratio and the other two combinations are treated
as free. The systemic mass ratio search grid method described by Russo &
Sollazzo (1982) involves obtaining the best solution for a range of fixed
values of the mass ratio $q$, with other parameters such as the temperature of
the secondary ($T_{2}$), potential (fill-out), and inclination acting as free
parameters. The solution with the smallest total error between the observed
and modelled light curve is then selected as representing the true mass ratio.
Terrell & Wilson (2005) have previously shown that such a technique when used
in cases of complete eclipses will yield the true mass ratio. For both systems
simultaneous multi-band solutions were obtained for various fixed incremental
(0.01) values of $q$ to initially find a approximate solution. Finer grid
search at increments of 0.001 for the value of $q$ was conducted near the
approximate value to obtain the true mass ratio. In the final iteration the
mass ratio was also made a free parameter and the software reported standard
deviations were recorded as the errors in the light curve solution. Fitted and
observed light curves and mass ratio search grids for TYC 6995-813-1 and NSVS
13602901 are shown in figures 1 and 2 respectively while 3D representations
are illustrated in figures 3. The light curve solution along with absolute and
other astrophysical parameters (see below) are summarised in table 1.

##  4 Absolute and Astrophysical Parameters

Yildiz & Doğan (2013) showed that the primary components of contact binaries
in terms of mass, radius and luminosity are much closer to that of the normal
zero age main sequence (ZAMS) stars than the secondary components. Despite
their low masses, the secondary component are significantly brighter and
larger than to their ZAMS or terminal age main sequence (TAMS) counterparts.
Using equation 1 it is possible to determine the absolute magnitude of the
primary and using the mass luminosity relation of (Demircan & Kahraman 1991)
and the light curve solution it is possible to determine the masses of both
components. The error in he mass of the primary was estimated from the error
in the distance (Gaia Collaboration 2018) and subsequent errors estimated
through error propagation. As the total luminosity of the system is known and
the luminosity of the primary determined as described the luminosity of the
secondary can be easily derived. The hallmark of a contact binary system is
that both components overflow their Roche lobes and as such their relative
radii are dependent on the Roche geometry and the mass ratio. As part of the
light curve solution the WD code provides the fractional radii of the
component stars in three orientations. In this study we use the geometric mean
of the three directional radii as an estimate of the fractional radii of the
primary ($r_{1}$) and secondary ($r_{2}$). From Kepler’s third law, mass ratio
and period it is possible to determine the separation ($A$) and from the
separation the absolute radii of the components as per (Awadalla & Hanna 2005)
$R_{1,2}$ = $A*r_{1,2}$ can be calculated.

Yildiz & Doğan (2013) compiled these basic parameters for 100 well studies
contact binary systems and showed good agreement between the luminosities of
the primary component and main sequence stars of similar mass. With respect to
the radii they showed that the primaries have somewhat larger radii than ZAMS
stars. Unlike the primaries the luminosities and radii of the secondary do not
correlate with main sequence stars of similar mass. They found that all
secondary components were considerably brighter and larger than TAMS stars.
The relative radii and luminosities and their relationship to ZAMS and TAMS
are summarised in table 1 and confirm the general reported findings.

Yildiz & Doğan (2013) argue that given the distorted nature of the secondaries
they must have different structure and evolutionary path relative to single
stars. They argue that the interior of the secondaries are quite different
from main sequence stars because their initial mass was significantly higher
than the present mass as significant mass loss (or mass transfer) has occurred
from the secondary to the primary. This situation means the present secondary
component had a different evolutionary path compared to main sequence stars.
Through detailed modelling they showed that the present secondary was in fact
larger and the initial primary which lost mass to the initial smaller
secondary through a semi-detached phase. The initial secondary grew larger
while the initial primary grew smaller. Eventually, the initial secondary
became larger than the mass losing primary and assumed the role of the present
primary. Due to loss of mass from the outer layers of the current secondary,
the internal structure of the current secondary is thought to be considerably
denser than that of the primary. Modelling by Kähler (2004) showed that the
difference in the densities $\Delta\rho=\rho_{1}-\rho_{2}$ ($\rho_{1,2}$ =
densities of the primary and secondary respectively) is always negative for
all contact binary systems. Knowing the period, radii, and mass ratio, the
mean density of each component can be easily computed. The density (in
$gcm^{-3}$) of the components can be expressed as a function of the mass
ratio, period and radii of the components (Mochnacki 1981) and the difference
between the densities of the components can be expressed as follows:

$\Delta\rho=\frac{0.0189q}{R_{2}^{3}(1+q)P^{2}}-\frac{0.0189}{R_{1}^{3}(1+q)P^{2}}$
(2)

where $q$ is the current mass ratio, $R_{1,2}$ radii of the current primary
and secondary and $P$ is the period in days. $\Delta\rho$ was calculated for
each of the systems and reported in table 1 and as predicted it is strongly
negative.

Contact binary systems are known to be magnetically active and hence
potentially chromospherically active. Activity signals are common in contact
binary systems. One of these is the asymmetry in the light curve maxima
(O’Connell effect). Apart from star spot modelling, light curve analysis
provides little indication of choromospheric activity although much clearer
indicators such as certain emission lines are more specific for chromospheric
activity. In the optical waveband the chromospheric emissions are obscured by
the intense photospheric activity. The same however is not true for far
ultraviolet wavelengths, especially in the case of late type dwarfs which make
up most of the contact binary systems (Smith & Redenbaugh 2010). The GALEX
(Galaxy Evolution Explorer) satellite imaged the sky in two photometric band-
passes: a near-ultraviolet band (NUV) with an effective wavelength of 2316Å
and a far band (FUV) centered on 1539Å (Morrissey et al. 2007). Only the FUV
band is useful for the detection of chromospheric activity as the NUV band
remains contaminated by photospheric activity (Smith & Redenbaugh 2010). A
large volume of data on the strength of active chromospheric emission lines
for a large number of late type dwarf stars exists as a result of the Mount
Wilson HK project (Vaughan et al. 1978; Duncan et al. 1991). Smith &
Redenbaugh (2010) used the GALEX FUV magnitudes ($m_{FUV}$) combined with the
HK survey data to investigate the sensitivity of FUV brightness as an
indicator of chromospheric activity.

The most accepted measure of chromospheric emission strength is the fraction
of the star’s bolometric luminosity emitted in the active H and K lines (Noyes
et al. 1984). The value is normally expressed as $logR_{HK}^{{}^{\prime}}$ and
stars with $logR_{HK}^{{}^{\prime}}\geq-4.75$ are considered more active
(Henry et al. 1996). Using the large database of $logR_{HK}^{{}^{\prime}}$
values of dwarf stars Smith & Redenbaugh (2010) matched the observations with
GALEX magnitudes to derive two key relationships:

$(m_{FUV}-B)_{base}=6.73(B-V)+7.43$ (3)

which defines the $m_{FUV}-B$ for stars with the weakest emissions and low
activity. This is the used as a correction term when defining the colour
excess:

$\Delta(m_{FUV}-B)=(m_{FUV}-B)-(m_{FUV}-B)_{base}$ (4)

They showed that for active stars i.e those with
$logR_{HK}^{{}^{\prime}}\geq-4.75$ the colour excess was generally less than
-0.5 and more frequently less than -1.0. Less active stars had colour excess
values significantly higher than -0.5. Only one of our systems, TYC
6995-813-1, was observed by GALEX with the reported $m_{FUV}$ of 20.85. Using
this and the relationships above (we use the published B-V value from ASAS-SN
database) we derive the UV colour excess of -2.89 indicating underlying
significant choromospheric activity even in the absence of significant spot
activity.

Parameter | TYC 6995-813-1 | NSVS 13602901
---|---|---
Distance (pc) | $498.0\pm 14$ | $6114.4\pm 40$
$M_{1}/M_{\sun}$ | $1.23\pm 0.01$ | $1.19\pm 0.02$
$M_{2}/M_{\sun}$ | $0.135\pm 0.01$ | $0.203\pm 0.01$
$T_{1}$(K)(Fixed) | 6300 | 6250
$T_{2}$(K) | $6235\pm 27$ | $6222\pm 18$
Inclination(∘) | $84.03\pm 1.51$ | $83.59\pm 1.15$
Fill-out($f$)% | $72\pm 2$ | $44\pm 2$
Potential($\Omega$) | $1.941\pm 0.002$ | $2.11\pm 0.007$
$r_{1}$(mean) | 0.633 | 0.554
$r_{2}$(mean) | 0.288 | 0.260
Mass Ratio ($M_{2}/M_{1}$) | $0.111\pm 0.002$ | $0.171\pm 0.002$
$A$($R_{\sun}$) | $2.46\pm 0.01$ | $3.05\pm 0.01$
$R_{1}/R_{\sun}$ | $1.46\pm 0.01$ | $1.69\pm 0.01$
$R_{2}/R_{\sun}$ | $0.60\pm 0.01$ | $0.79\pm 0.01$
$L_{1}/L_{\sun}$ | $2.29\pm 0.04$ | $2.05\pm 0.04$
$L_{2}/L_{\sun}$ | $0.73\pm 0.04$ | $0.58\pm 0.02$
$Mv_{1}$ | 3.93 | 4.05
$Mv_{2}$ | 5.17 | 5.42
$R_{1}/R_{ZAMS}$ | $1.37\pm 0.01$ | $1.69\pm 0.01$
$R_{2}/R_{TAMS}$ | $1.34\pm 0.01$ | $1.31\pm 0.01$
$L_{2}/L_{TANS}$ | $314.7$ | $61.9$
Table 1: Light curve solution and absolute parameters for TYC 6995-813-1 and
NSVS 13602901  Figure 1: Observed and fitted light curves (left) and mass
ratio search grid (right) for TYC 6995-813-1. For ease of display the blue
curves are shifted up 0.1 units and the red curves shifted down 0.1 units. The
check star flux is shown at the bottom of the light curve is shifted up by 0.1
units. The mass ratio search grid has been normalised to the minimum sum of
the residuals ($2.216^{-2}$) as reported by the WD code  Figure 2: Observed
and fitted light curves (left) and mass ratio search grid (left) for NSVS
13602901. For ease of display the blue curves are shifted up 0.1 units and the
red curves shifted down 0.1 units. The check star flux is shown at the bottom
of the light curve is shifted down by 0.05 units. The mass ratio search grid
has been normalised to the minimum sum of the residuals ($3.51^{-1}$) as
reported by the WD code Figure 3: 3D representations of TYC 6995-813-1 (left)
and NSVS 13602901 (right)

##  5 Merger Potential

As noted above, contact binary systems are thought to merge when the orbital
angular momentum falls below a critical (minimum) value. Recently, Wadhwa et
al. (2021) showed that orbital instability is not independent of the masses of
the components but critically dependent particularly on the mass of the
primary. They concluded that instability is unlikely unless the separation
between the components was less than a critical value ($A_{inst}$) and showed
that this could be expressed a function of the mass ratio, radius of the
primary and the gyration radii of the components and expressed as:

$\frac{A_{\mathrm{\scriptscriptstyle
inst}}}{R_{1}}=\frac{q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q}+\sqrt{(q\frac{k_{2}^{2}}{k_{1}^{2}}{P}{Q})^{2}+3(1+q\frac{k_{2}^{2}}{k_{1}^{2}}{Q}^{2})(q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}+\frac{q}{(1+q)k_{1}^{2}})}}{q\frac{k_{2}^{2}}{k_{1}^{2}}{P}^{2}+\frac{q}{(1+q)k_{1}^{2}}}.$
(5)

Where $k_{1,2}$ is the gyration radius for the primary and secondary
components and

$P=\frac{0.49q^{2/3}-3.26667q^{-2/3}(0.27q-0.12q^{4/3})}{0.6q^{2/3}+\ln(1+q^{1/3})}.$
(6)

$Q=\frac{(0.27q-0.12q^{4/3})({0.6q^{-2/3}+\ln(1+q^{-1/3})})}{0.15(0.6q^{2/3}+\ln(1+q^{1/3}))},$
(7)

Review of the modelled values of the gyration radii ($k$) of low mass
($0.4M_{\sun}<M<1.4M_{\sun}$) rotating and tidally distorted ZAMS stars by
Landin et al. (2009) yields a simple linear relationship between the two as:

$k=-0.2455M+0.5368$ (8)

We use this relationship to derive the values of $k_{1}$ (gyration radius of
the primary) for the two systems. The secondary is a very low mass star and
for a low mass secondary, the convective n=1.5 polytrope is an excellent
approximation (Arbutina 2007) for which $k_{2}=0.453$. This fixed value of
$k_{2}$ and the variable value of $k_{1}$ from equation 8 and absolute
parameters derived above were adopted to calculate the instability separation,
using equation 5, for both systems reported. The instability separation of TYC
6995-813-1 is estimated as 2.35$R_{\sun}$ compared to the current separation
only slightly higher at 2.46$R_{\sun}$ suggesting the the system is
approaching instability. In contrast, the instability separation for NSVS
13602901 at 2.76$R_{\sun}$ is significantly smaller that the current
separation of 3.05$R_{\sun}$ suggesting the system is likely to be stable.

##  6 Summary and Conclusions

First multi-band photometric analysis of two bright southern contact binary
systems are presented. We find both systems show extreme low mass ratios with
evidence of significant age given the relative brightness, size and density of
the ”current” secondary relative to main sequence counterparts of similar
mass. As expected the primaries of both systems have absolute parameters not
far removed from their ZAMS counterparts. Although neither system demonstrates
a significant O’Connell effect at least one of the systems that was observed
by the GALEX mission does have features suggestive of significant
chromospheric activity. Post V1309 Sco, orbital instability of contact binary
systems has received considerable attention and at least one of our systems,
TYC 6995-813-1 would fall into the category of being a potential merger
candidate. Among the most critical changes observed retrospectively for V1309
Sco was the exponential decline in its period in the years prior to the merger
event (Tylenda et al. 2011). Given its relative brightness, TYC 6995-813-1,
represents an easy target for long term follow up by smaller instruments such
as campus telescopes or even advanced amateurs.

###### Acknowledgements.

This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France.

## References

  * Arbutina (2007) Arbutina, B. 2007, MNRAS, 377, 1635
  * Awadalla & Hanna (2005) Awadalla, N. S., & Hanna, M. A. 2005, Journal of Korean Astronomical Society, 38, 43
  * Cox (2000) Cox, A. N. 2000, Allen’s astrophysical quantities (AIP)
  * Demircan & Kahraman (1991) Demircan, O., & Kahraman, G. 1991, Ap&SS, 181, 313
  * Drake et al. (2017) Drake, A. J., Djorgovski, S. G., Catelan, M., et al. 2017, MNRAS, 469, 3688
  * Duncan et al. (1991) Duncan, D. K., Vaughan, A. H., Wilson, O. C., et al. 1991, ApJS, 76, 383
  * Gaia Collaboration (2018) Gaia Collaboration. 2018, VizieR Online Data Catalog, I/345
  * Gazeas et al. (2021) Gazeas, K. D., Loukaidou, G. A., Niarchos, P. G., et al. 2021, MNRAS, 502, 2879
  * Henry et al. (1996) Henry, T. J., Soderblom, D. R., Donahue, R. A., & Baliunas, S. L. 1996, AJ, 111, 439
  * Hut (1980) Hut, P. 1980, A&A, 92, 167
  * Jayasinghe et al. (2018) Jayasinghe, T., Kochanek, C. S., Stanek, K. Z., et al. 2018, MNRAS, 477, 3145
  * Kähler (2004) Kähler, H. 2004, A&A, 414, 317
  * Kwee & van Woerden (1956) Kwee, K. K., & van Woerden, H. 1956, Bull. Astron. Inst. Netherlands, 12, 327
  * Landin et al. (2009) Landin, N. R., Mendes, L. T. S., & Vaz, L. P. R. 2009, A&A, 494, 209
  * Lucy (1967) Lucy, L. B. 1967, ZAp, 65, 89
  * Luo et al. (2019) Luo, A. L., Zhao, Y. H., Zhao, G., & et al. 2019, VizieR Online Data Catalog, V/164
  * Mason et al. (2010) Mason, E., Diaz, M., Williams, R. E., Preston, G., & Bensby, T. 2010, A&A, 516, A108
  * Mochnacki (1981) Mochnacki, S. W. 1981, ApJ, 245, 650
  * Morrissey et al. (2007) Morrissey, P., Conrow, T., Barlow, T. A., et al. 2007, ApJS, 173, 682
  * Nakano et al. (2008) Nakano, S., Nishiyama, K., Kabashima, F., et al. 2008, IAU Circ., 8972, 1
  * Nelson & Robb (2015) Nelson, R. H., & Robb, R. M. 2015, Information Bulletin on Variable Stars, 6134, 1
  * Noyes et al. (1984) Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, A. H. 1984, ApJ, 279, 763
  * Pojmanski (1997) Pojmanski, G. 1997, Acta Astron., 47, 467
  * Rucinski (1969) Rucinski, S. 1969, Postepy Astronomii Krakow, 17, 163
  * Rucinski (1993) Rucinski, S. M. 1993, PASP, 105, 1433
  * Rucinski (2007) Rucinski, S. M. 2007, MNRAS, 382, 393
  * Russo & Sollazzo (1982) Russo, G., & Sollazzo, C. 1982, A&A, 107, 197
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525
  * Smith & Redenbaugh (2010) Smith, G. H., & Redenbaugh, A. K. 2010, PASP, 122, 1303
  * Terrell & Wilson (2005) Terrell, D., & Wilson, R. E. 2005, Ap&SS, 296, 221
  * Tylenda et al. (2011) Tylenda, R., Hajduk, M., Kamiński, T., et al. 2011, A&A, 528, A114
  * van Hamme (1993) van Hamme, W. 1993, AJ, 106, 2096
  * Vaughan et al. (1978) Vaughan, A. H., Preston, G. W., & Wilson, O. C. 1978, PASP, 90, 267
  * Wadhwa et al. (2021) Wadhwa, S. S., De Horta, A., Filipović, M. D., et al. 2021, MNRAS, 501, 229
  * Wilson & Devinney (1971) Wilson, R. E., & Devinney, E. J. 1971, ApJ, 166, 605
  * Wozniak et al. (2003) Wozniak, P. R., Vestrand, W. T., McGowan, K. E., Kinemuchi, K., & ROTSE Collaboration. 2003, in American Astronomical Society Meeting Abstracts, Vol. 203, American Astronomical Society Meeting Abstracts, 57.03
  * Yildiz & Doğan (2013) Yildiz, M., & Doğan, T. 2013, MNRAS, 430, 2029