Inequalities for the $k$-Regular Overpartitions

Yi Peng1, Helen W.J. Zhang1,2 and Ying Zhong1

1School of Mathematics

Hunan University, Changsha 410082, P.R. China

2Hunan Provincial Key Laboratory of

Intelligent Information Processing and Applied Mathematics,

Changsha 410082, P.R. China

Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract. Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the
first to discover the log-subadditivity, log-concavity, and the third-order
Turán inequality of partition function, respectively. Many other important
partition statistics are proved to enjoy similar properties. This paper
focuses on the partition function ${\overline{p}}_{k}(n)$, which counts the
number of overpartitions of $n$ with no parts divisible by $k$. We provide a
combinatorial proof to establish that for any $k\geq 2$, the partition
function ${\overline{p}}_{k}(n)$ exhibits strict log-subadditivity.
Specifically, we show that
${\overline{p}}_{k}(a){\overline{p}}_{k}(b)>{\overline{p}}_{k}(a+b)$ for
integers $a\geq b\geq 1$ and $a+b\geq k$. Furthermore, we investigate the log-
concavity and the satisfaction of the third-order Turán inequality for
${\overline{p}}_{k}(n)$, where $2\leq k\leq 9$.

Keywords: $k$-regular overpartition, strict log-subadditivity, log-concavity,
third order Turán inequalities

AMS Classifications: 11P82, 05A19, 30A10

##  1 Introduction

The aim of this paper is to present certain inequality relations satisfied by
$k$-regular overpartition function, specifically focusing on strict log-
subadditivity, log-concavity, and the third-order Turán inequality. Recall
that a partition of a positive integer $n$ is a non-increasing sequence of
positive integers whose sum is $n$. Let $p(n)$ denote the number of partitions
of $n$. Bessenrodt and Ono [3] showed that $p(n)$ satisfies the strict log-
subadditivity result, that is, for $a,b>1$ and $a+b>9$

$\displaystyle p(a)p(b)>p(a+b).$ (1.1)

Alanazi, Gagola and Munagi [1] resolved the conjecture proposed by Bessenrodt
and Ono [3], providing a combinatorial proof for equation (1.1). Since then,
various partition functions have been proven to enjoy the same strict log-
subadditivity as $p(n)$, for instance, the overpartition function [14, 15],
the $k$-regular partition function [2] and the spt-function [5, 8].

Recall that an overpartition of a nonnegative integer $n$ is a partition of
$n$ where the first occurrence of each distinct part may be overlined. In
[16], Lovejoy initiated the exploration of an overpartition function,
characterized by its count of overpartitions of $n$ with parts that are
indivisible by $k$. Subsequently, Shen [18] reinterpreted this function,
giving it the appellation of $k$-regular overpartition
${\overline{p}}_{k}(n)$, and elucidated its generating function as follows.

$\displaystyle\sum_{n=0}^{\infty}{\overline{p}}_{k}(n)q^{n}=\frac{(q^{k};q^{k})_{\infty}(-q;q)_{\infty}}{(q;q)_{\infty}(-q^{k};q^{k})_{\infty}},$
(1.2)

where $(a;q)_{\infty}$ stands for the $q$-shifted factorial

$(a;q)_{\infty}=\prod_{n=1}^{\infty}(1-aq^{n-1}),\quad|q|<1.$

The first result of this paper is the demonstration, using combinatorial
methods, that ${\overline{p}}_{k}(n)$ satisfies strict log-subadditivity.

######  Theorem 1.1.

For integers $k$, $a$ and $b$, if $k\geq 2$, $a\geq b\geq 1$, $a+b\geq k$,
then

$\displaystyle{\overline{p}}_{k}(a){\overline{p}}_{k}(b)>{\overline{p}}_{k}(a+b).$

Having set the stage with our first result, demonstrating the strict log-
subadditivity of ${\overline{p}}_{k}(n)$ through combinatorial methods, we now
transition towards a broader context. We delve into the realm of log-
concavity, also known as Turán inequalities, along with their higher-order
counterparts, known as cubic inequalities, originate from the examination of
the Maclaurin coefficients of real entire functions within the Laguerre-Pólya
class. A sequence $\\{\alpha_{n}\\}_{n\geq 0}$ is considered to be log-concave
if it fulfills the Turán inequalities, which are given by

$\alpha_{n}^{2}\geq\alpha_{n-1}\alpha_{n+1},~{}~{}\text{for}~{}n\geq 1.$

Moreover, a sequence $\\{\alpha_{n}\\}_{n\geq 0}$ is defined as satisfying the
third order Turán inequalities if for $n\geq 1$,

$\displaystyle
4(\alpha_{n}^{2}-\alpha_{n-1}\alpha_{n+1})(\alpha_{n+1}^{2}-\alpha_{n}\alpha_{n+2})\geq(\alpha_{n}\alpha_{n+1}-\alpha_{n-1}\alpha_{n+2})^{2}.$

The initial exploration of the Turán inequalities, along with their higher-
order variants for the partition function, can be credited to the research
undertaken by Chen, Jia and Wang [6], DeSalvo and Pak [9]. In the wake of
these seminal contributions, a substantial body of related research has
burgeoned. This includes noteworthy contributions from scholars such as
Bringmann, Kane, Rolen, and Tripp [4], Dong and Ji [10], Engel [11], Griffin,
Ono, Rolen, and Zagier [12], Jia [13], Liu and Zhang [15], and Ono, Pujahari,
and Rolen [17]. In furthering this line of research, our work reveals new
insights about the properties of $k$-regular overpartitions. We establish that
when $2\leq k\leq 9$, these overpartitions fulfill the conditions of log-
concavity and adhere to the third-order Turán inequalities.

In order to establish these inequality relations, our initial undertaking
involves developing the asymptotic formulae for ${\overline{p}}_{k}(n)$,
specifically when $2\leq k\leq 9$. This critical step in our methodology
ultimately allows us to unveil the stated inequality relations. For
convenience, we adopt the notation below, set

${\mu_{2}(n)}=\frac{\pi}{2}\sqrt{2n},\quad\mu_{3}(n)=\frac{\pi}{3}\sqrt{6n},\quad\mu_{4}(n)=\frac{\pi}{2}\sqrt{3n},\quad\mu_{5}(n)=\frac{2\pi}{5}\sqrt{5n},$
$\quad\mu_{6}(n)=\frac{\pi}{6}\sqrt{30n},\quad\mu_{7}(n)=\frac{\pi}{7}\sqrt{42n},\quad\mu_{8}(n)=\frac{\pi}{4}\sqrt{14n},\quad\mu_{9}(n)=\frac{2\pi}{3}\sqrt{2n}.$

For the sake of convenience, we use the following notations throughout this
paper:

${\mu_{k}^{-}}=\mu_{k}(n-1),\quad{\mu_{k}}=\mu_{k}(n),\quad{\mu_{k}^{+}}=\mu_{k}(n+1).$

######  Theorem 1.2.

Let $I_{1}(s)$ denote the first modified Bessel function of the first kind.
For each fixed $2\leq k\leq 9$, and $\mu_{k}\geq n_{k}$, we have

$\displaystyle{\overline{p}}_{k}(n)=C_{k}(n)I_{1}({\mu_{k}})+R_{k}(n),$

where

$|R_{k}(n)|\leq R_{k}^{\prime}(n),$

and all these values are defined in Table 1.1.

$k$ | $n_{k}$ | $C_{k}(n)$ | $R_{k}^{\prime}(n)$ | $k$ | $n_{k}$ | $C_{k}(n)$ | $R_{k}^{\prime}(n)$
---|---|---|---|---|---|---|---
$2$ | $22$ | $\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}$ |  $\frac{\sqrt{3}\pi^{\frac{3}{2}}}{2\sqrt{\mu_{2}}}$ $\exp\left(\frac{{\mu_{2}}}{3}\right)$ | $6$ | $130$ | $\frac{5\sqrt{6}\pi^{2}}{18\mu_{6}}$ |  $\frac{10\sqrt{15}\pi^{\frac{3}{2}}}{27\sqrt{\mu_{6}}}$ $\exp\left(\frac{\mu_{6}}{5}\right)$
$3$ | $49$ | $\frac{2\sqrt{3}\pi^{2}}{9\mu_{3}}$ |  $\frac{4\sqrt{30}\pi^{\frac{3}{2}}}{27\sqrt{\mu_{3}}}$ $\exp\left(\frac{\mu_{3}}{5}\right)$ | $7$ | $102$ | $\frac{18\sqrt{7}\pi^{2}}{49\mu_{7}}$ |  $\frac{108\sqrt{42}\pi^{\frac{3}{2}}}{343\sqrt{\mu_{7}}}$ $\exp\left(\frac{\mu_{7}}{3}\right)$
$4$ | $41$ | $\frac{3\pi^{2}}{4\mu_{4}}$ |  $\frac{3\sqrt{6}\pi^{\frac{3}{2}}}{4\sqrt{\mu_{4}}}$ $\exp\left(\frac{\mu_{4}}{3}\right)$ | $8$ | $129$ | $\frac{7\sqrt{2}\pi^{2}}{8\mu_{8}}$ |  $\frac{7\sqrt{3}\pi^{\frac{3}{2}}}{4\sqrt{\mu_{8}}}$ $\exp\left(\frac{\mu_{8}}{3}\right)$
$5$ | $58$ | $\frac{8\sqrt{5}\pi^{2}}{25\mu_{5}}$ |  $\frac{32\sqrt{30}\pi^{\frac{3}{2}}}{125\sqrt{\mu_{5}}}$ $\exp\left(\frac{\mu_{5}}{3}\right)$ | $9$ | $268$ | $\frac{8\pi^{2}}{9\mu_{9}}$ |  $\frac{16\sqrt{10}\pi^{\frac{3}{2}}}{27\sqrt{\mu_{9}}}$ $\exp\left(\frac{\mu_{9}}{5}\right)$
Table 1.1: Values of $n_{k}$, $C_{k}(n)$ and $R_{k}^{\prime}(n)$ for $2\leq
k\leq 9$.

######  Theorem 1.3.

For $2\leq k\leq 9$, ${\overline{p}}_{k}(n)$ satisfies log-concavity for
$n\geq\bar{n}_{k}$ and the third-order Turán inequality for
$n\geq\hat{n}_{k}$. The specific ranges are provided in the following table.

$k$ |  $2$ |  $3$ |  $4$ |  $5$ |  $6$ |  $7$ |  $8$ |  $\quad 9$
---|---|---|---|---|---|---|---|---
$\bar{n}_{k}$ |  $21$ |  $4$ |  $5$ |  $6$ |  $1$ |  $1$ |  $1$ |  $\quad 1$
$\hat{n}_{k}$ |  $65$ |  $23$ |  $28$ |  $26$ |  $11$ |  $22$ |  $23$ |  $~{}~{}10$

The paper is organized as follows. In Section 2, we rigorously prove the
strict log-subadditivity of ${\overline{p}}_{k}(n)$ using combinatorial
methods. Section 3 utilizes the results established by Chern, Ji and Dong to
derive the asymptotic formulas for ${\overline{p}}_{k}(n)$ when $2\leq k\leq
9$. Lastly, in Section 4, the derived asymptotic formula is applied to
demonstrate the log-concavity and third-order Turán inequalities of
${\overline{p}}_{k}(n)$ for $2\leq k\leq 9$.

##  2 Proof of Theorem 1.1

In this section, we establish a series of lemmas to support the proof.
Throughout this section, we employ combinatorial methods to demonstrate the
log-subadditivity property of ${\overline{p}}_{k}(n)$, considering that
exponents refer to multiplicities.

######  Lemma 2.1.

For an integer $k$ with $k\geq 2$, if $a$, $b$ are integers with $a,b\geq 1$,
then

${\overline{p}}_{k}(a|\ {\rm no\ 1^{\prime}s}){\overline{p}}_{k}(b|\ {\rm no\
2^{\prime}s})\geq{\overline{p}}_{k}(a+b|\ {\rm no\ 1^{\prime}s\ and\ no\
2^{\prime}s}).$

###### Proof.

For $\lambda=(\lambda_{1},\dots,\lambda_{t})\in\overline{P}_{k}(a+b)$, we
define

$i=i(\lambda)=\mathrm{m}ax\\{j\in\mathbb{N}|1\leq j\leq
t,\lambda_{j}+\dots+\lambda_{t}\geq b\\}.$

Let $\lambda_{i}=x+y$ such that

$y+\lambda_{1}+\dots+\lambda_{i-1}=a~{}~{}\text{and}~{}~{}x+\lambda_{i+1}+\dots+\lambda_{t}=b,$

where $x\geq 1$. With these definitions, we introduce the map

$f_{1}:\ \overline{P}_{k}(a+b|\text{ no 1's and no
2's})\rightarrow\overline{P}_{k}(a|\text{ no
1's})\oplus\overline{P}_{k}(b|\text{ no 2's}).$

Next, we aim to establish the injectivity of $f_{1}$ by examining the parity
of $k$. For $\lambda=(\lambda_{1},\dots,\lambda_{t})\in\overline{P}_{k}(a+b|\
{\rm no\ 1^{\prime}s\ and\ no\ 2^{\prime}s})$. When $k$ is odd, we consider
the case of $k=2m+1$ ($m\geq 2$), specifically focusing on the situation when
$k\geq 5$. The proof method for $k=3$ follows a similar approach. We will
divide our analysis into six cases for further discussion.

  1. 1.

If $y=0$, we set

$\displaystyle
f_{1}(\lambda)=(\lambda_{1},\dots,\lambda_{i-1};\lambda_{i},\dots,\lambda_{t}).$

  2. 2.

If $y=1$, we consider two separate cases by distinguishing whether
$\lambda_{i}$ is overlined.

(1). When $\lambda_{i}$ is overlined, we have
$f_{1}(\lambda)=(\lambda_{1},\dots,\lambda_{i-1},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}).$

(2). When $\lambda_{i}$ is non-overlined and $\lambda_{i-1}$ is overlined, it
leads to the following expression.

$\displaystyle
f_{1}(\lambda)=\begin{dcases}(\lambda_{1},\dots,\overline{\lambda_{i-1}-k},2^{m+1},\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i-1}\geq{k+2},\\\\[3.0pt]
(\lambda_{1},\dots,\lambda_{i-2},\overline{2},2^{c};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&3\leq\lambda_{i-1}=2c+1\leq{k+1},\\\\[3.0pt]
(\lambda_{1},\dots,\lambda_{i-2},\overline{2},2^{c-1},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&3\leq\lambda_{i-1}=2c\leq{k+1}.\end{dcases}$

(3). When both $\lambda_{i}$ and $\lambda_{i-1}$ are non-overlined, we observe
the following mapping.

$\displaystyle
f_{1}(\lambda)=\begin{dcases}(\lambda_{1},\dots,\lambda_{i-1}-k,2^{m+1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i-1}\geq{k+2},\\\
(\lambda_{1},\dots,\lambda_{i-2},2^{c+1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&3\leq\lambda_{i-1}=2c+1\leq{k+1},\\\
(\lambda_{1},\dots,\lambda_{i-2},2^{c},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&3\leq\lambda_{i-1}=2c\leq{k+1}.\end{dcases}$

  3. 3.

If $y\equiv i\pmod{k},\ (i=2,\dots,k-1)$, we have $y\geq i$, which leads to
the following mapping.

$\displaystyle
f_{1}(\lambda)=\begin{cases}(\lambda_{1},\dots,\lambda_{i-1},\overline{y};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i}\
\mathrm{is\ overlined},\\\\[5.0pt]
(\lambda_{1},\dots,\lambda_{i-1},y;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i}\
\mathrm{is}\ \mathrm{non\mbox{-}overlined}.\end{cases}$

  4. 4.

If $y\equiv 1\pmod{k}$ and $y\geq{k+1}$, we obtain the following mapping.

$\displaystyle
f_{1}(\lambda)=\begin{cases}(\lambda_{1},\dots,\lambda_{i-1},\overline{y};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i}\
\mathrm{is\ overlined},\\\\[5.0pt]
(\lambda_{1},\dots,\lambda_{i-1},y;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i}\
\mathrm{is}\ \mathrm{non\mbox{-}overlined}.\end{cases}$

  5. 5.

If $y=k$, we represent the mapping in the following form.

$\displaystyle
f_{1}(\lambda)=\begin{cases}(\lambda_{1},\dots,\lambda_{i-1},\overline{m+1},m;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i}\geq{k+2}\
\mathrm{and\ overlined},\\\
(\lambda_{1},\dots,\lambda_{i-1},{m+1},m;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&\lambda_{i}\geq{k+2}\
\mathrm{and\ non\mbox{-}overlined},\\\
(\lambda_{1},\dots,\lambda_{i-1},\overline{2},2^{m-1},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1),&\lambda_{i}=k+1\
\mathrm{and\ overlined},\\\
(\lambda_{1},\dots,\lambda_{i-1},2^{m},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1),&\lambda_{i}=k+1\
\mathrm{and\ non\mbox{-}overlined}.\end{cases}$

  6. 6.

If $y\equiv 0\pmod{k}$ and $y\geq{2k}$, we divide our analysis into two
distinct cases based on whether $\lambda_{i}$ is overlined or not.

(1). When $\lambda_{i}$ is overlined and $\lambda_{i}\equiv j\pmod{k}$, we
observe that

$\displaystyle
f_{1}(\lambda)=\begin{dcases}(\lambda_{1},\dots,\lambda_{i-1},\overline{y-(k-j)},k-j;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&1\leq
j\leq k-2,\\\\[5.0pt]
(\lambda_{1},\dots,\lambda_{i-1},\overline{y-1},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&j=k-1.\end{dcases}$

(2). When $\lambda_{i}$ is non-overlined and $\lambda_{i}\equiv j\pmod{k}$,
the following observation becomes apparent.

$\displaystyle
f_{1}(\lambda)=\begin{dcases}(\lambda_{1},\dots,\lambda_{i-1},{y-(k-j)},k-j;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&1\leq
j\leq k-2,\\\\[5.0pt]
(\lambda_{1},\dots,\lambda_{i-1},{y-1},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&j=k-1.\end{dcases}$

Notice that in cases 3 and 4, $y\geq 2$ and $2$ (or $\overline{2}$) as a part
appear at most once, which has no intersection with case 2. In case 5, if
$\lambda_{i}={k+1}$, then $x=1$. However, in case 2 when $\lambda_{i}$ is non-
overlined, we have $\lambda_{i}\geq 3$ and $x\geq 2$, which implies there is
no intersection between the two cases. Similarly, in case 2 where
$\lambda_{i}$ is overlined, if $x=1$, then $\lambda_{i}=y+x=\overline{2}$.
According to the principles of notations,
$\lambda_{i-1}>\lambda_{i}=\overline{2}$, which means the appearance of two is
zero. Thus, case 5 has no intersection with case 2 either. By analogy, case 5
shares no intersection with cases 3 and 4. In case 6, where $j\neq{k-1}$, the
multiplicity of two is at most one. When $j=k-1$, we have
$x\equiv{k-1}\pmod{k}$. However, in case 2, when $\lambda_{i}$ is overlined
and $x\equiv{k-1}\pmod{k}$, we have $\lambda_{i}=y+x=1+x\equiv 0\pmod{k}$,
which implies that $k$ divides $\lambda_{i}$, a contradiction. Hence, case 6
is disjoint from case 2. The same reasoning applies when comparing case 6 with
cases 3 and 4. Regarding the comparison between case 5 and case 6, we note
that in case 6, the difference between the last two parts of the partition,
namely $y-(k-j)$ and $k-j$, is greater than $1$. Therefore, when $k$ is odd,
$f_{1}$ is one-to-one.

When $k$ is even, let $k=2m,$ here we present the cases below for $m\geq 3$,
that is $k\geq 6$. The proof approaches to the cases for $k=2$ and $k=4$ are
similar to $k\geq 6$.

$f_{1}^{\prime}(\lambda)=\begin{dcases}(\lambda_{1},\dots,\overline{\lambda_{i-1}-k},2^{m},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&y=1,\lambda_{i}\
\mathrm{is\ non\mbox{-}overlined},\lambda_{i-1}\geq{k+2},\\\ &\lambda_{i-1}\
\mathrm{is\ overlined};\\\
(\lambda_{1},\dots,\lambda_{i-1}-k,2^{m},\overline{1};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&y=1,\lambda_{i}\
\mathrm{is\ non\mbox{-}overlined},\lambda_{i-1}\geq{k+2},\\\ &\lambda_{i-1}\
\mathrm{is\ non\mbox{-}overlined};\\\
(\lambda_{1},\dots,\lambda_{i-1},\overline{m},m;\lambda_{i+1},\dots,\lambda_{t},1^{x}),&y=k,\lambda_{i}\
\mathrm{is\ overlined};\\\
(\lambda_{1},\dots,\lambda_{i-1},m^{2};\lambda_{i+1},\dots,\lambda_{t},1^{x}),&y=k,\lambda_{i}\
\mathrm{is\ non\mbox{-}overlined};\\\
f_{1}(\lambda),&\mathrm{else}.\end{dcases}$

With minor adjustments, it is obvious to know $f_{1}^{\prime}$ is also one-to-
one.

Therefore, through all cases above, we can always construct a map that is one-
to-one for $k\geq 2$ to meet the inequality. Hence, the proof for the
inequalities is completed.

######  Lemma 2.2.

If $k$, $a$ are integers with $k\geq 2$, $a\geq 1$, then

${\overline{p}}_{k}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}_{k}(1)>{\overline{p}}_{k}(a+1|\ \mathrm{no\
2^{\prime}s}).$

###### Proof.

Let
$\lambda=(\lambda_{1},\dots,\lambda_{t},{\overline{1}}^{r},1^{s})\in\overline{P}_{k}(a+1|{\rm
no\ 2^{\prime}s})$ with $r\in\\{0,1\\}$. We introduce the mapping

$f_{2}:\overline{P}_{k}(a+1|\ \mathrm{no\
2^{\prime}s})\rightarrow\overline{P}_{k}(a|\ \mathrm{no\
2^{\prime}s})\oplus\overline{P}_{k}(1)$

as

$\displaystyle
f_{2}(\lambda)=\begin{cases}(\lambda_{1},\dots,\lambda_{t},\overline{1}^{r},1^{s-1};1),&{\rm
if}\ s\geq 1,\\\
(\lambda_{1},\dots,\lambda_{t-1},1^{\lambda_{t}-1};\overline{1}),&{\rm if}\
s=0,r=0,\lambda_{t}\ {\rm is\ non\mbox{-}overlined},\\\
(\lambda_{1},\dots,\lambda_{t-1},\overline{1},1^{\lambda_{t}-2};\overline{1}),&{\rm
if}\ s=0,r=0,\lambda_{t}\ {\rm is\ overlined},\\\
(\lambda_{1},\dots,\lambda_{t};\overline{1}),&{\rm if}\ s=0,r=1.\end{cases}$

The mapping $f_{2}$ is well-defined and injective. However, it is evident that
$(\lambda_{1},\dots,\lambda_{t},1;\overline{1})$, with $\lambda_{t}>1$, is not
in the image of $f_{2}$. Therefore, the inequality holds, which completes the
proof.

######  Lemma 2.3.

If $k$, $a$ are integers with $k\geq 2$, $a\geq 1$, then

${\overline{p}}_{k}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}_{k}(2)>{\overline{p}}_{k}(a+2|\ \mathrm{no\
2^{\prime}s}).$

###### Proof.

Define a map

$f_{3}:\overline{P}_{k}(a+2|\ \mathrm{no\
2^{\prime}s})\rightarrow\overline{P}_{k}(a|\ \mathrm{no\
2^{\prime}s})\oplus\overline{P}_{k}(2)$

by

$\displaystyle
f_{3}(\lambda)=\begin{cases}(\lambda_{1},\dots,\lambda_{t},\overline{1}^{r},1^{s-2};2),&{\rm
if}\ s\geq 2,\\\ (\lambda_{1},\dots,\lambda_{t};\overline{2}),&{\rm if}\
s=1,~{}r=1,\\\
(\lambda_{1},\dots,\lambda_{t-1},\overline{1},1^{\lambda_{t}-2};1^{2}),&{\rm
if}\ s=1,~{}r=0,~{}\lambda_{t}\ {\rm is\ overlined},\\\
(\lambda_{1},\dots,\lambda_{t-1},1^{\lambda_{t}-1};1^{2}),&{\rm if}\
s=1,~{}r=0,~{}\lambda_{t}\ {\rm is\ non\mbox{-}overlined},\\\
(\lambda_{1},\dots,\lambda_{t-1};1^{2}),&{\rm if}\
s=0,~{}r=0,~{}\lambda_{t}=\overline{2},\\\
(\lambda_{1},\dots,\lambda_{t-1},\overline{1},1^{\lambda_{t}-3};\overline{1},1),&{\rm
if}\ s=0,~{}r=0,~{}\lambda_{t}\geq 3,\ {\rm is\ overlined},\\\
(\lambda_{1},\dots,\lambda_{t-1},1^{\lambda_{t}-2};\overline{1},1),&{\rm if}\
s=1,~{}r=0,~{}\lambda_{t}\geq 3,\ {\rm is\ non\mbox{-}overlined},\\\
(\lambda_{1},\dots,\lambda_{t-1},\overline{1},1^{\lambda_{t}-2};\overline{2}),&{\rm
if}\ s=0,~{}r=1,~{}\lambda_{t}\ {\rm is\ overlined},\\\
(\lambda_{1},\dots,\lambda_{t-1},1^{\lambda_{t}-1};\overline{2}),&{\rm if}\
s=0,~{}r=1,~{}\lambda_{t}\ {\rm is\ non\mbox{-}overlined}.\end{cases}$

The mapping $f_{3}$ is well-defined and injective. It is clear that
$(\lambda_{1},\dots,\lambda_{t};\overline{1},1)$, with $\lambda_{t}>1$, is not
in the image of $f_{3}$. This completes the proof.

######  Lemma 2.4.

If $k$, $a$, $b$, are integers with $k\geq 2$, $a\geq 1$, $b\geq 3$ and
$a+b\geq{k+1}$, then

${\overline{p}}_{k}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}_{k}(b)>{\overline{p}}_{k}(a+b|\ \mathrm{no\
2^{\prime}s}).$

###### Proof.

Let $n=a+b$. We will apply induction on $n$ to prove the inequality.

Base cases: The inequality holds for $n=k+1$. In this case, we have $a+b=k+1$,
where $b\geq 3$ and $a\geq 1$. It follows that $a\leq k-2$ and $b\leq k$. When
$a,b<k$, we have

${\overline{p}}_{k}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}_{k}(b)={\overline{p}}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}(b)\geq{\overline{p}}(a+b|\ \mathrm{no\
2^{\prime}s})>{\overline{p}}_{k}(a+b|\ \mathrm{no\ 2^{\prime}s}).$

If $b=k$ and $a=1$, we can construct a mapping to show that the result holds.

Inductive step: Suppose $n\geq{k+2}$ and the inequality holds for $n-1$. Now,
if $b\geq 4$, then ${\overline{p}}_{k}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}_{k}(b-1)>{\overline{p}}_{k}(a+b-1|\ \mathrm{no\
2^{\prime}s})$ by the inductive hypothesis. Similarly, if $b=3$, we can apply
Lemma 2.3 to conclude that

${\overline{p}}_{k}(a|\ \mathrm{no\
2^{\prime}s}){\overline{p}}_{k}(b-1)>{\overline{p}}_{k}(a+b-1|\ \mathrm{no\
2^{\prime}s}).$

Using Lemma 2.1, we have:

$\displaystyle{\overline{p}}_{k}(a+b|\ {\rm no\ 2^{\prime}s})$
$\displaystyle={\overline{p}}_{k}(a+b|\ {\rm no\ 1^{\prime}s\ and\ no\
2^{\prime}s})+{\overline{p}}_{k}(a+b-1|\ {\rm no\ 2^{\prime}s})$
$\displaystyle\leq{\overline{p}}_{k}(a|\ {\rm no\
2^{\prime}s}){\overline{p}}_{k}(b|\ {\rm no\
1^{\prime}s})+{\overline{p}}_{k}(a+b-1|\ {\rm no\ 2^{\prime}s})$
$\displaystyle<{\overline{p}}_{k}(a|\ {\rm no\
2^{\prime}s})({\overline{p}}_{k}(b|\ {\rm no\
1^{\prime}s})+{\overline{p}}_{k}(b-1))$ $\displaystyle={\overline{p}}_{k}(a|\
{\rm no\ 2^{\prime}s}){\overline{p}}_{k}(b).$

Therefore, by the principle of mathematical induction, the inequality holds.

Based on the preceding lemmas and employing mathematical induction, we are
ready to commence the proof of Theorem 1.1.

Proof of Theorem 1.1. Let $n=a+b$. We apply induction on $n$.

Base cases: When $n=k,\ k+1$, it holds. If $n=k$, then $a+b=k$ and $a,b<k$, we
have

${\overline{p}}_{k}(a){\overline{p}}_{k}(b)={\overline{p}}(a){\overline{p}}(b)>{\overline{p}}(a+b)>{\overline{p}}_{k}(a+b).$

If $a+b=k+1$, when $a,b<k$, it can be proved in a similar way. By symmetry,
set $a\geq b$, when $a=k$, $b=1$, the case belongs to Lemma 2.2 that $a=k$, so
the inequality is established.

Inductive step: Suppose $n\geq{k+2}$ and the inequality holds for $n-2$.
Therefore, by the inductive hypothesis,
${\overline{p}}_{k}(a+b-2)<{\overline{p}}_{k}(a-2){\overline{p}}_{k}(b)$. By
Lemmas 2.2-2.4,

${\overline{p}}_{k}(a+b|\ {\rm no\ 2^{\prime}s})<{\overline{p}}_{k}(a|\ {\rm
no\ 2^{\prime}s}){\overline{p}}_{k}(b).$

Hence,

$\displaystyle{\overline{p}}_{k}(a+b)$ $\displaystyle={\overline{p}}_{k}(a+b|\
{\rm no\ 2^{\prime}s})+{\overline{p}}_{k}(a+b-2)$
$\displaystyle<{\overline{p}}_{k}(a|\ {\rm no\
2^{\prime}s}){\overline{p}}_{k}(b)+{\overline{p}}_{k}(a-2){\overline{p}}_{k}(b)$
$\displaystyle=({\overline{p}}_{k}(a|\ {\rm no\
2^{\prime}s})+{\overline{p}}_{k}(a-2)){\overline{p}}_{k}(b)$
$\displaystyle={\overline{p}}_{k}(a){\overline{p}}_{k}(b).$

Thus, by the principle of mathematical induction, the inequality holds.

##  3 Proof of Theorem 1.2

In this section, our aim is to establish the asymptotic formula for
${\overline{p}}_{k}(n)$ when $2\leq k\leq 9$. To accomplish this, we first
utilize the following lemma given by Ji and Dong [10, Lemma 2.2] to obtain
inequalities involving
$I_{1}({\mu_{k}(n-1)})I_{1}({\mu_{k}(n+1)})/I_{1}({\mu_{k}(n)})^{2}$, which
are necessary for the subsequent proofs.

######  Lemma 3.1.

Let

$\displaystyle
E_{I}(s):=1-\frac{3}{8s}-\frac{15}{128s^{2}}-\frac{105}{1024s^{3}}-\frac{4725}{32768s^{4}}-\frac{72765}{262144s^{5}}.$

Then for $s\geq 26$,

$\displaystyle\frac{e^{s}}{\sqrt{2\pi
s}}\left(E_{I}(s)-\frac{31}{s^{6}}\right)\leq
I_{1}(s)\leq\frac{e^{s}}{\sqrt{2\pi
s}}\left(E_{I}(s)+\frac{31}{s^{6}}\right).$ (3.1)

######  Lemma 3.2.

For $2\leq k\leq 9$ and ${\mu_{k}}\geq n^{\prime}_{k}$,

$\displaystyle\frac{{\mu_{k}}}{\sqrt{{\mu_{k}^{-}}{\mu_{k}^{+}}}}U_{k}(n)\leq\frac{I_{1}({\mu_{k}^{-}})I_{1}({\mu_{k}^{+}})}{I_{1}({\mu_{k}})^{2}}\leq\frac{{\mu_{k}}}{\sqrt{{\mu_{k}^{-}}{\mu_{k}^{+}}}}V_{k}(n),$

where all these values are defined in following table.

$k$ |  $n_{k}^{\prime}$ |  $U_{k}(n)$ |  $V_{k}(n)$
---|---|---|---
$2$ |  $167$ |  $\left(1-\frac{\pi^{4}}{16{\mu_{2}}^{3}}-\frac{5\pi^{8}}{512{\mu_{2}}^{7}}\right)\left(1-\frac{7}{{\mu_{2}}^{5}}-\frac{115}{{\mu_{2}}^{6}}\right)$ |  $\left(1-\frac{\pi^{4}}{16{\mu_{2}}^{3}}+\frac{\pi^{8}}{256{\mu_{2}}^{6}}\right)\left(1-\frac{6}{{\mu_{2}}^{5}}+\frac{115}{{\mu_{2}}^{6}}\right)$
$3$ |  $47$ |  $\left(1-\frac{\pi^{4}}{9{\mu_{3}}^{3}}-\frac{5\pi^{8}}{162{\mu_{3}}^{7}}\right)\left(1-\frac{13}{{\mu_{3}}^{5}}-\frac{141}{{\mu_{3}}^{6}}\right)$ |  $\left(1-\frac{\pi^{4}}{9{\mu_{3}}^{3}}+\frac{\pi^{8}}{81{\mu_{3}}^{6}}\right)\left(1-\frac{6}{{\mu_{3}}^{5}}+\frac{141}{{\mu_{3}}^{6}}\right)$
$4$ |  $58$ |  $\left(1-\frac{9\pi^{4}}{64{\mu_{4}}^{3}}-\frac{405\pi^{8}}{8192{\mu_{4}}^{7}}\right)\left(1-\frac{16}{{\mu_{4}}^{5}}-\frac{145}{{\mu_{4}}^{6}}\right)$ |  $\left(1-\frac{9\pi^{4}}{64{\mu_{4}}^{3}}+\frac{81\pi^{8}}{4096{\mu_{4}}^{6}}\right)\left(1-\frac{15}{{\mu_{4}}^{5}}+\frac{145}{{\mu_{4}}^{6}}\right)$
$5$ |  $66$ |  $\left(1-\frac{4\pi^{4}}{25{\mu_{5}}^{3}}-\frac{8\pi^{8}}{125{\mu_{5}}^{7}}\right)\left(1-\frac{18}{{\mu_{5}}^{5}}-\frac{148}{{\mu_{5}}^{6}}\right)$ |  $\left(1-\frac{4\pi^{4}}{25{\mu_{5}}^{3}}+\frac{16\pi^{8}}{625{\mu_{5}}^{6}}\right)\left(1-\frac{17}{{\mu_{5}}^{5}}+\frac{148}{{\mu_{5}}^{6}}\right)$
$6$ |  $47$ |  $\left(1-\frac{25\pi^{4}}{144{\mu_{6}}^{3}}-\frac{3125\pi^{8}}{41472{\mu_{6}}^{7}}\right)\left(1-\frac{20}{{\mu_{6}}^{5}}-\frac{150}{{\mu_{6}}^{6}}\right)$ |  $\left(1-\frac{25\pi^{4}}{144{\mu_{6}}^{3}}+\frac{625\pi^{8}}{20736{\mu_{6}}^{6}}\right)\left(1-\frac{19}{{\mu_{6}}^{5}}+\frac{150}{{\mu_{6}}^{6}}\right)$
$7$ |  $51$ |  $\left(1-\frac{9\pi^{4}}{49{\mu_{7}}^{3}}-\frac{25920\pi^{8}}{307328{\mu_{7}}^{7}}\right)\left(1-\frac{21}{{\mu_{7}}^{5}}-\frac{151}{{\mu_{7}}^{6}}\right)$ |  $\left(1-\frac{9\pi^{4}}{49{\mu_{7}}^{3}}+\frac{81\pi^{8}}{2401{\mu_{7}}^{6}}\right)\left(1-\frac{20}{{\mu_{7}}^{5}}+\frac{151}{{\mu_{7}}^{6}}\right)$
$8$ |  $300$ |  $\left(1-\frac{98\pi^{4}}{512{\mu_{8}}^{3}}-\frac{48020\pi^{8}}{524288{\mu_{8}}^{7}}\right)\left(1-\frac{21}{{\mu_{8}}^{5}}-\frac{152}{{\mu_{8}}^{6}}\right)$ |  $\left(1-\frac{98\pi^{4}}{512{\mu_{8}}^{3}}+\frac{9604\pi^{8}}{262144{\mu_{8}}^{6}}\right)\left(1-\frac{20}{{\mu_{8}}^{5}}+\frac{152}{{\mu_{8}}^{6}}\right)$
$9$ |  $81$ |  $\left(1-\frac{16\pi^{4}}{81{\mu_{9}}^{3}}-\frac{640\pi^{8}}{6561{\mu_{9}}^{7}}\right)\left(1-\frac{22}{{\mu_{9}}^{5}}-\frac{153}{{\mu_{9}}^{6}}\right)$ |  $\left(1-\frac{16\pi^{4}}{81{\mu_{9}}^{3}}+\frac{256\pi^{8}}{6561{\mu_{9}}^{6}}\right)\left(1-\frac{21}{{\mu_{9}}^{5}}+\frac{153}{{\mu_{9}}^{6}}\right)$
Table 3.2: Values of $n^{\prime}_{k}$, $U_{k}(n)$ and $V_{k}(n)$ for $2\leq
k\leq 9$.

###### Proof.

By (3.1), we find that for $2\leq k\leq 9$ and ${\mu_{k}}\geq 26$,

$\displaystyle\frac{{\mu_{k}}}{\sqrt{{\mu_{k}^{-}}{\mu_{k}^{+}}}}e^{{\mu_{k}^{-}}+{\mu_{k}^{+}}-2{\mu_{k}}}U^{\prime}_{k}(n)\leq\frac{I_{1}({\mu_{k}^{-}})I_{1}({\mu_{k}^{+}})}{I_{1}({\mu_{k}})^{2}}\leq\frac{{\mu_{k}}}{\sqrt{{\mu_{k}^{-}}{\mu_{k}^{+}}}}e^{{\mu_{k}^{-}}+{\mu_{k}^{+}}-2{\mu_{k}}}V^{\prime}_{k}(n),$
(3.2)

where

$\displaystyle
U_{k}^{\prime}(n)=\frac{\left(E_{I}({\mu_{k}^{-}})-\frac{31}{{\mu_{k}^{-}}^{6}}\right)\left(E_{I}({\mu_{k}^{+}})-\frac{31}{{\mu_{k}^{+}}^{6}}\right)}{\left(E_{I}({\mu_{k}})+\frac{31}{{\mu_{k}}^{6}}\right)^{2}},$
(3.3) $\displaystyle
V_{k}^{\prime}(n)=\frac{\left(E_{I}({\mu_{k}^{-}})+\frac{31}{{\mu_{k}^{-}}^{6}}\right)\left(E_{I}({\mu_{k}^{+}})+\frac{31}{{\mu_{k}^{+}}^{6}}\right)}{\left(E_{I}({\mu_{k}})-\frac{31}{{\mu_{k}}^{6}}\right)^{2}}.$
(3.4)

We consider the case of $k=2$. First, we will estimate
$\exp({\mu_{2}^{-}}+{\mu_{2}^{+}}-2{\mu_{2}})$, $U^{\prime}_{2}(n)$ and
$V^{\prime}_{2}(n)$ in terms of ${\mu_{2}}$. For ${\mu_{2}}\geq 3$, we observe
the following relationships:

${\mu_{2}^{-}}=\sqrt{{\mu_{2}}^{2}-\frac{\pi^{2}}{2}},\qquad{\mu_{2}^{+}}=\sqrt{{\mu_{2}}^{2}+\frac{\pi^{2}}{2}}.$
(3.5)

Thus we can observe

$\displaystyle{\mu_{2}^{-}}$
$\displaystyle={\mu_{2}}-\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}-\frac{\pi^{6}}{128{\mu_{2}}^{5}}-\frac{5\pi^{8}}{128\times
16{\mu_{2}}^{7}}+O\left(\frac{1}{{\mu_{2}}^{8}}\right),$
$\displaystyle{\mu_{2}^{+}}$
$\displaystyle={\mu_{2}}+\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}+\frac{\pi^{6}}{128{\mu_{2}}^{5}}-\frac{5\pi^{8}}{128\times
16{\mu_{2}}^{7}}+O\left(\frac{1}{{\mu_{2}}^{8}}\right).$

It is easy to check that for ${\mu_{2}}\geq 3$,

$\displaystyle
d_{\mu_{2}}<{\mu_{2}^{-}}<w_{\mu_{2}},\quad\overline{d}_{\mu_{2}}<{\mu_{2}^{+}}<\overline{w}_{\mu_{2}},$
(3.6)

where

$\displaystyle\left\\{\begin{lgathered}d_{\mu_{2}}={\mu_{2}}-\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}-\frac{\pi^{6}}{128{\mu_{2}}^{5}}-\frac{5\pi^{8}}{128\times
8{\mu_{2}}^{7}},\\\\[3.0pt]
w_{\mu_{2}}={\mu_{2}}-\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}-\frac{\pi^{6}}{128{\mu_{2}}^{5}},\\\\[3.0pt]
\overline{d}_{\mu_{2}}={\mu_{2}}+\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}+\frac{\pi^{6}}{128{\mu_{2}}^{5}}-\frac{5\pi^{8}}{128\times
8{\mu_{2}}^{7}},\\\\[3.0pt]
\overline{w}_{\mu_{2}}={\mu_{2}}+\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}+\frac{\pi^{6}}{128{\mu_{2}}^{5}}.\end{lgathered}d_{\mu_{2}}={\mu_{2}}-\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}-\frac{\pi^{6}}{128{\mu_{2}}^{5}}-\frac{5\pi^{8}}{128\times
8{\mu_{2}}^{7}},\\\\[3.0pt]
w_{\mu_{2}}={\mu_{2}}-\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}-\frac{\pi^{6}}{128{\mu_{2}}^{5}},\\\\[3.0pt]
\overline{d}_{\mu_{2}}={\mu_{2}}+\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}+\frac{\pi^{6}}{128{\mu_{2}}^{5}}-\frac{5\pi^{8}}{128\times
8{\mu_{2}}^{7}},\\\\[3.0pt]
\overline{w}_{\mu_{2}}={\mu_{2}}+\frac{\pi^{2}}{4{\mu_{2}}}-\frac{\pi^{4}}{32{\mu_{2}}^{3}}+\frac{\pi^{6}}{128{\mu_{2}}^{5}}.\right.$
(3.11)

Applying (3.11) into (3.6), we find when ${\mu_{2}}\geq 3$,

$-\frac{\pi^{4}}{16{\mu_{2}}^{3}}-\frac{5\pi^{8}}{128\times
4{\mu_{2}}^{7}}<{\mu_{2}^{-}}+{\mu_{2}^{+}}-2{\mu_{2}}<-\frac{\pi^{4}}{16{\mu_{2}}^{3}}.$

It follows that,

$\displaystyle\exp{\left(-\frac{\pi^{4}}{16{\mu_{2}}^{3}}-\frac{5\pi^{8}}{128\times
4{\mu_{2}}^{7}}\right)}<\exp{({\mu_{2}^{-}}+{\mu_{2}^{+}}-2{\mu_{2}})}$
$\displaystyle<\exp{\left(-\frac{\pi^{4}}{16{\mu_{2}}^{3}}\right)}.$

Note that for $s<0$,

$1+s<e^{s}<1+s+s^{2},$

here we come that

$\displaystyle 1-\frac{\pi^{4}}{16{\mu_{2}}^{3}}-\frac{5\pi^{8}}{128\times
4{\mu_{2}}^{7}}<\exp{({\mu_{2}^{-}}+{\mu_{2}^{+}}-2{\mu_{2}})}$
$\displaystyle<1-\frac{\pi^{4}}{16{\mu_{2}}^{3}}+\frac{\pi^{8}}{256{\mu_{2}}^{6}}.$
(3.12)

Next we estimate $U^{\prime}_{2}(n)$ and $V^{\prime}_{2}(n)$. Let

$\displaystyle P_{l}(n)$
$\displaystyle=\frac{1}{{\mu_{2}^{-}}^{6}{\mu_{2}^{+}}^{6}}\left({\mu_{2}^{-}}^{6}-\frac{3}{8}{\mu_{2}^{-}}^{4}w_{\mu_{2}}-\frac{15}{128}{\mu_{2}^{-}}^{4}-\frac{105}{1024}{\mu_{2}^{-}}^{2}w_{\mu_{2}}\right.$
$\displaystyle\qquad\qquad\qquad\left.-\frac{4725}{32768}{\mu_{2}^{-}}^{2}-\frac{72765}{262144}w_{\mu_{2}}-31\right)$
$\displaystyle\qquad\qquad\times\left({\mu_{2}^{+}}^{6}-\frac{3}{8}{\mu_{2}^{+}}^{4}\overline{w}_{\mu_{2}}-\frac{15}{128}{\mu_{2}^{+}}^{4}-\frac{105}{1024}{\mu_{2}^{+}}^{2}\overline{w}_{\mu_{2}}\right.$
$\displaystyle\left.\qquad\qquad\qquad-\frac{4725}{32768}{\mu_{2}^{+}}^{2}-\frac{72765}{262144}\overline{w}_{\mu_{2}}-31\right)$
(3.13)

and

$\displaystyle P_{r}(n)$
$\displaystyle=\frac{1}{{\mu_{2}^{-}}^{6}{\mu_{2}^{+}}^{6}}\left({\mu_{2}^{-}}^{6}-\frac{3}{8}{\mu_{2}^{-}}^{4}d_{\mu_{2}}-\frac{15}{128}{\mu_{2}^{-}}^{4}-\frac{105}{1024}{\mu_{2}^{-}}^{2}d_{\mu_{2}}\right.$
$\displaystyle\left.\qquad\qquad\qquad-\frac{4725}{32768}{\mu_{2}^{-}}^{2}-\frac{72765}{262144}d_{\mu_{2}}+31\right)$
$\displaystyle\qquad\qquad\times\left({\mu_{2}^{+}}^{6}-\frac{3}{8}{\mu_{2}^{+}}^{4}\overline{d}_{\mu_{2}}-\frac{15}{128}{\mu_{2}^{+}}^{4}-\frac{105}{1024}{\mu_{2}^{+}}^{2}\overline{d}_{\mu_{2}}\right.$
$\displaystyle\left.\qquad\qquad\qquad-\frac{4725}{32768}{\mu_{2}^{+}}^{2}-\frac{72765}{262144}\overline{d}_{\mu_{2}}+31\right).$
(3.14)

Applying (3.6) into (3.3) and (3.4), we have for ${\mu_{2}}\geq 3$,

$\displaystyle
U^{\prime}_{2}(n)\geq\frac{P_{l}(n)}{\left(E_{I}({\mu_{2}})+\frac{31}{{\mu_{2}}^{6}}\right)^{2}},\quad
V^{\prime}_{2}(n)\leq\frac{P_{r}(n)}{\left(E_{I}({\mu_{2}})-\frac{31}{{\mu_{2}}^{6}}\right)^{2}}.$
(3.15)

To bound $U^{\prime}_{2}(n)$ and $V^{\prime}_{2}(n)$ in terms of ${\mu_{2}}$,
we shall compute that for ${\mu_{2}}\geq 167$,

$\displaystyle\frac{P_{l}(n)}{\left(E_{I}({\mu_{2}})+\frac{31}{{\mu_{2}}^{6}}\right)^{2}}-\left(1-\frac{7}{{\mu_{2}}^{5}}-\frac{115}{{\mu_{2}}^{6}}\right)\geq
0,$ (3.16)

and for ${\mu_{2}}\geq 31$

$\displaystyle\frac{P_{r}(n)}{\left(E_{I}({\mu_{2}})-\frac{31}{{\mu_{2}}^{6}}\right)^{2}}-\left(1-\frac{6}{{\mu_{2}}^{5}}+\frac{115}{{\mu_{2}}^{6}}\right)\leq
0.$ (3.17)

Rewrite (3.16) and (3.17) as

${\mu_{2}}^{12}P_{l}(n)-\left(1-\frac{7}{{\mu_{2}}^{5}}-\frac{115}{{\mu_{2}}^{6}}\right)\left({\mu_{2}}^{6}E_{I}({\mu_{2}})+31\right)^{2}\geq
0$

and

${\mu_{2}}^{12}P_{r}(n)-\left(1-\frac{6}{{\mu_{2}}^{5}}+\frac{115}{{\mu_{2}}^{6}}\right)\left({\mu_{2}}^{6}E_{I}({\mu_{2}})-31\right)^{2}\leq
0.$

Substituting (3.11) into (3) and (3), respectively, we can therefore compute
that

$\displaystyle{\mu_{2}}^{12}P_{l}(n)-\left(1-\frac{7}{{\mu_{2}}^{5}}-\frac{115}{{\mu_{2}}^{6}}\right)\left({\mu_{2}}^{6}E_{I}({\mu_{2}})+31\right)^{2}=\frac{\sum_{i=0}^{25}a_{i}{\mu_{2}}^{i}}{17592186044416{\mu_{2}}^{6}\left(4{\mu_{2}}^{4}-\pi^{4}\right)^{3}},$
$\displaystyle{\mu_{2}}^{12}P_{r}(n)-\left(1-\frac{6}{{\mu_{2}}^{5}}+\frac{115}{{\mu_{2}}^{6}}\right)\left({\mu_{2}}^{6}E_{I}({\mu_{2}})-31\right)^{2}=-\frac{\sum_{i=0}^{25}b_{i}{\mu_{2}}^{i}}{1125899906842624{\mu_{2}}^{6}\left(4{\mu_{2}}^{4}-\pi^{4}\right)^{3}},$

where $a_{i}$ and $b_{i}$ are real numbers, here we just list
$a_{23}$-$a_{25}$ and $b_{23}$-$b_{25}$:

$\displaystyle a_{23}$
$\displaystyle=-45493393110859776-86586540687360\pi^{4},$ $\displaystyle
a_{24}$ $\displaystyle=-16044073672507392-46179488366592\pi^{4},$
$\displaystyle a_{25}$ $\displaystyle=7881299347898368-79164837199872\pi^{4},$
$\displaystyle b_{23}$
$\displaystyle=-2823756966361300992+5541538603991040\pi^{4},$ $\displaystyle
b_{24}$ $\displaystyle=-324259173170675712+2955487255461888\pi^{4},$
$\displaystyle b_{25}$
$\displaystyle=-432345564227567616+5066549580791808\pi^{4}.$

It can be checked that for any $0\leq i\leq 22$ and ${\mu_{2}}\geq 24$,

$-|a_{i}|{\mu_{2}}^{i}\geq-|a_{23}|{\mu_{2}}^{23},\quad-|b_{i}|{\mu_{2}}^{i}\geq-|b_{23}|{\mu_{2}}^{23}.$

So we could obtain that for ${\mu_{2}}\geq 24$,

$\displaystyle\sum_{i=0}^{25}a_{i}{\mu_{2}}^{i}$
$\displaystyle\geq-\sum_{i=0}^{23}|a_{i}|{\mu_{2}}^{i}+a_{24}{\mu_{2}}^{24}+a_{25}{\mu_{2}}^{25}\geq-24|a_{23}|{\mu_{2}}^{23}+a_{24}{\mu_{2}}^{24}+a_{25}{\mu_{2}}^{25}$

and

$\displaystyle\sum_{i=0}^{25}b_{i}{\mu_{2}}^{i}$
$\displaystyle\geq-\sum_{i=0}^{23}|b_{i}|{\mu_{2}}^{i}+b_{24}{\mu_{2}}^{24}+b_{25}{\mu_{2}}^{25}\geq-24|b_{23}|{\mu_{2}}^{23}+b_{24}{\mu_{2}}^{24}+b_{25}{\mu_{2}}^{25}.$

Meanwhile, we can calculate that for ${\mu_{2}}\geq 167$,

$(-24|a_{23}|+a_{24}{\mu_{2}}+a_{25}{\mu_{2}}^{2}){\mu_{2}}^{23}\geq 0,$

and for ${\mu_{2}}\geq 31,$

$(-24|b_{23}|+b_{24}{\mu_{2}}+b_{25}{\mu_{2}}^{2}){\mu_{2}}^{23}\geq 0.$

So (3.16) and (3.17) are established. Combining them with (3.15), we attain
for ${\mu_{2}}\geq 167$,

$\displaystyle U^{\prime}_{2}(n)\geq
1-\frac{7}{{\mu_{2}}^{5}}-\frac{115}{{\mu_{2}}^{6}},\quad
V^{\prime}_{2}(n)\leq 1-\frac{6}{{\mu_{2}}^{5}}+\frac{115}{{\mu_{2}}^{6}}.$
(3.18)

Applying (3.12) and (3.18) into (3.2), we establish the validity of the Lemma
3.2 when $k=2$. The remaining cases follow a similar pattern to the case of
$k=2$, and their proofs are omitted here. Thus, the proof is completed.

In order to provide the asymptotic formula for ${\overline{p}}_{k}(n)$, we
begin by reviewing some definitions and notations introduced by Chern [7]. Let
us define

$G(q)=G(e^{2\pi\mathnormal{i}\tau}):=\prod_{r=1}^{R}{(q^{m_{r}};q^{m_{r}})}_{\infty}^{\delta_{r}},$

where $\mathbf{m}=(m_{1},\dots,m_{R})$ and
$\mathbf{\delta}=(\delta_{1},\dots,\delta_{R})$. Assuming that $k$ and $h$ are
positive integers with $\text{gcd}(h,k)=1$, we define

$\displaystyle\Delta_{1}=-\frac{1}{2}\sum_{r=1}^{R}\delta_{r},\qquad\Delta_{2}=\sum_{r=1}^{R}m_{r}\delta_{r},$
$\displaystyle\Delta_{3}(k)=-\sum_{r=1}^{R}\frac{\delta_{r}\mathrm{gcd}^{2}(m_{r},k)}{m_{r}},\qquad\
\Delta_{4}(k)=\prod_{r=1}^{R}\left(\frac{m_{r}}{\mathrm{gcd}(m_{r},k)}\right)^{-\frac{\delta_{r}}{2}},$
$\displaystyle\hat{A}_{k}(n)=\mathop{\sum}_{0\leq h<k\atop\
\mathrm{gcd}(h,k)=1}\exp{\left(-\frac{2\pi nhi}{k}-\pi
i\sum_{r=1}^{R}\delta_{r}s\left(\frac{m_{r}h}{\mathrm{gcd}(m_{r},k)},\frac{k}{\mathrm{gcd}(m_{r},k)}\right)\right)},$

where $s(h,j)$ is the Dedekind sum defined by

$s(h,j)=\sum_{r=1}^{j-1}\left(\frac{r}{j}-\left[\frac{r}{j}\right]-\frac{1}{2}\right)\left(\frac{hr}{j}-\left[\frac{hr}{j}\right]-\frac{1}{2}\right).$

Let $L=lcm(m_{1},\dots,m_{R})$, we divide the set $\\{1,2,\dots,L\\}$ into two
disjoint subsets:

$\mathcal{L}_{>0}:=\\{1\leq l\leq L:\Delta_{3}(l)>0\\},\qquad\mathcal{L}_{\leq
0}:=\\{1\leq l\leq L:\Delta_{3}(l)\leq 0\\}.$

If we write

$G(q)=\sum_{n=0}^{\infty}g(n)q^{n}.$

Chern [7, Theorem 1.1] obtained the following asymptotic formula for $g(n)$
with $\Delta_{1}\leq 0$.

######  Lemma 3.3.

If $\Delta_{1}\leq 0$ and the inequality

$\min_{1\leq r\leq
R}\left(\frac{\mathrm{gcd}^{2}(m_{r},l)}{m_{r}}\right)\geq\frac{\Delta_{3}(l)}{24}$
(3.19)

holds for all $1\leq l\leq L$, then for positive integers
$n>-\frac{\Delta_{2}}{24}$, we have

$\displaystyle g(n)=E(n)+$
$\displaystyle\sum_{l\in\mathcal{L}_{>0}}2\pi\Delta_{4}(l)\left(\frac{24n+\Delta_{2}}{\Delta_{3}(l)}\right)^{-\frac{\Delta_{1}+1}{2}}\sum_{1\leq
k<N\atop\ k\equiv
L^{l}}\frac{I_{-\Delta_{1}-1}(\frac{\pi}{6k}\sqrt{\Delta_{3}(l)(24n+\Delta_{2})})}{k}\hat{A}_{k}(n),$

where $\zeta(\cdot)$ is the Riemann zeta-function, $I_{v}(s)$ is the $v$-th
modified Bessel function of the first kind,

$\displaystyle\Xi_{\Delta_{1}}(t):=\begin{cases}1,&\Delta_{1}=0,\\\
2\sqrt{t},&\Delta_{1}=-\frac{1}{2},\\\ t\log{(t+1)},&\Delta_{1}=-1,\\\
\zeta(-\Delta_{1})t^{-2\Delta_{1}-1},&\mathrm{otherwise},\\\ \end{cases}$

and

$\displaystyle|E(n)|\leq$
$\displaystyle\frac{2^{-\Delta_{1}}\pi^{-1}N^{-\Delta_{1}+2}}{n+\frac{\Delta_{2}}{24}}\exp\left(2\pi\left(n+\frac{\Delta_{2}}{24}\right)N^{-2}\right)\sum_{l\in\mathcal{L}_{>0}}\Delta_{4}(l)\exp\left(\frac{\Delta_{3}(l)\pi}{3}\right)$
$\displaystyle+2\exp\left(2\pi\left(n+\frac{\Delta_{2}}{24}\right)N^{-2}\right)\Xi_{\Delta_{1}}(N)$
$\displaystyle\times\left(\sum_{1\leq l\leq
L}\Delta_{4}(l)\exp\left(\frac{\pi\Delta_{3}(l)}{24}+\sum_{r=1}^{R}\frac{|\delta_{r}|\exp(-\pi\mathrm{gcd}^{2}(m_{r},l)/m_{r})}{(1-\exp(-\pi\mathrm{gcd}^{2}(m_{r},l)/m_{r}))^{2}}\right)\right.$
$\displaystyle\left.\qquad-\sum_{l\in\mathcal{L}_{>0}}\Delta_{4}(l)\exp\left(\frac{\pi\Delta_{3}(l)}{24}\right)\right).$

We will now use Lemma 3.3 to prove Theorem 1.2.

Proof of Theorem 1.2. For $2\leq k\leq 9$, the proof process is similar.
Therefore, here we only demonstrate the case of $k=2$. Let $k=2$ in (1.2), and
based on the notation defined by Chern, we have $\mathbf{m}=(1,2,2,4)$ and
$\delta=(-2,1,2,-1)$. The calculations show that $\Delta_{1}=0$ and
$\Delta_{2}=0$. Furthermore, we find that $L=4$. The values of $\Delta_{3}(l)$
and $\Delta_{4}(l)$ for $1\leq l\leq L$ are presented in following table.

$l$ |  $1$ |  $2$ |  $3$ |  $4$
---|---|---|---|---
$\Delta_{3}(l)$ |  $\frac{3}{4}$ |  $-3$ |  $\frac{3}{4}$ |  $0$
$\Delta_{4}(l)$ |  $\frac{\sqrt{2}}{2}$ |  $\sqrt{2}$ |  $\frac{\sqrt{2}}{2}$ |  $1$
Table 3.3: The values of $\Delta_{3}(l)$ and $\Delta_{4}(l)$ for $1\leq l\leq
4$.

As a result, we have $\mathcal{L}_{>0}=\\{1,3\\}$.

Since in this case (3.19) is always satisfied, we can apply Lemma 3.3, which
yields the following result:

$\overline{p}_{2}(n)=E_{2}(n)+\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}\mathop{\sum}_{1\leq
k\leq N\atop\ 2\nmid
k}I_{1}\left(\frac{{\mu_{2}}}{k}\right)\frac{\hat{A}_{k}(n)}{k},$

where

$\displaystyle|E_{2}(n)|$
$\displaystyle\leq\frac{\pi^{-1}N^{2}}{n}\exp{\left(2\pi nN^{-2}\right)}\cdot
2\frac{\sqrt{2}}{2}\exp\left(\frac{\pi}{4}\right)+2\exp\left(2\pi
nN^{-2}\right)\Bigg{\\{}\sum_{1\leq l\leq 4}\Delta_{4}(l)$
$\displaystyle\quad\times\exp\left(\frac{\pi\Delta_{3}(l)}{24}+\sum_{r=1}^{4}\frac{|\delta_{r}|\exp(-\pi\mathrm{gcd}^{2}(m_{r},l)/m_{r})}{(1-\exp(-\pi\mathrm{gcd}^{2}(m_{r},l)/m_{r}))^{2}}\right)-\sqrt{2}\exp\left(\frac{\pi}{32}\right)\Bigg{\\}}.$

Let’s assume $N=\lfloor{\mu_{2}}\rfloor$. Then, we have the following:

$\displaystyle|E_{2}(n)|$
$\displaystyle\leq\frac{\sqrt{2}\pi}{2}\frac{{\lfloor{\mu_{2}}\rfloor}^{2}}{{\mu_{2}}^{2}}\exp\left(\frac{4{\mu_{2}}^{2}}{\pi{\lfloor{\mu_{2}}\rfloor}^{2}}+\frac{\pi}{4}\right)+2\exp\left(\frac{4{\mu_{2}}^{2}}{\pi{\lfloor{\mu_{2}}\rfloor}^{2}}\right)$
$\displaystyle\quad\times\left\\{\sum_{1\leq l\leq
4}\Delta_{4}(l)\exp\left(\frac{\pi\Delta_{3}(l)}{24}+\sum_{r=1}^{4}\frac{|\delta_{r}|\exp(-\pi\mathrm{gcd}^{2}(m_{r},l)/m_{r})}{(1-\exp(-\pi\mathrm{gcd}^{2}(m_{r},l)/m_{r}))^{2}}\right)\right.$
$\displaystyle\quad\qquad-\sqrt{2}\exp\left(\frac{\pi}{32}\right)\Bigg{\\}},$
(3.20)

and using the following inequalities for ${\mu_{2}}\geq 4$,

$\frac{\lfloor{\mu_{2}}\rfloor^{2}}{{\mu_{2}}^{2}}\leq
1,\quad\frac{{\mu_{2}}^{2}}{\lfloor{\mu_{2}}\rfloor^{2}}<\frac{{\mu_{2}}^{2}}{({\mu_{2}}-1)^{2}}<2,$

connecting (3) with values in Table 3.3, it follows $|E_{2}(n)|\leq 631.$
Observing $\hat{A}_{1}(n)=1$, we can further conclude that

$\overline{p}_{2}(n)=\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}I_{1}({\mu_{2}})+R_{2}(n),$

where

$R_{2}(n)=E_{2}(n)+\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}\mathop{\sum}_{3\leq k\leq
N\atop\ 2\nmid
k}I_{1}\left(\frac{{\mu_{2}}}{k}\right)\frac{\hat{A}_{k}(n)}{k}.$

Next, we come to show that for ${\mu_{2}}\geq 22,$

$|R_{2}(n)|\leq\frac{\sqrt{3}\pi^{\frac{3}{2}}}{2{\mu_{2}}^{\frac{1}{2}}}\exp\left(\frac{{\mu_{2}}}{3}\right).$
(3.21)

Since for any $n\geq 0$, $k\geq 1$, $|\hat{A}_{k}(n)|\leq k$ and $|e^{2\pi
si}|=1$ for any $s\in\mathbb{R}$, we obtain that

$\displaystyle\left|\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}\mathop{\sum}_{3\leq
k\leq N\atop\ 2\nmid
k}I_{1}\left(\frac{{\mu_{2}}}{k}\right)\frac{\hat{A}_{k}(n)}{k}\right|\leq\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}\mathop{\sum}_{3\leq
k\leq N\atop\ 2\nmid
k}I_{1}\left(\frac{{\mu_{2}}}{k}\right)\leq\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}\frac{\lfloor{\mu_{2}}\rfloor}{2}I_{1}\left(\frac{{\mu_{2}}}{3}\right)\leq\frac{\pi^{2}}{2\sqrt{8}}I_{1}\left(\frac{{\mu_{2}}}{3}\right).$

Recall that Bringmann, Kane, Rolen and Trippin [4] stated that for $s\geq 1$,

$\displaystyle I_{1}(s)\leq\sqrt{\frac{2}{\pi s}}e^{s},$

which yields that

$\displaystyle\left|\frac{\pi^{2}}{\sqrt{8}{\mu_{2}}}\mathop{\sum}_{3\leq
k\leq N\atop\ 2\nmid
k}I_{1}\left(\frac{{\mu_{2}}}{k}\right)\frac{\hat{A}_{k}(n)}{k}\right|\leq\frac{\pi^{2}}{2\sqrt{8}}\sqrt{\frac{2}{\pi\frac{{\mu_{2}}}{3}}}\exp{\left(\frac{{\mu_{2}}}{3}\right)}\leq\frac{\sqrt{3}\pi^{\frac{3}{2}}}{4{\mu_{2}}^{\frac{1}{2}}}\exp{\left(\frac{{\mu_{2}}}{3}\right)}.$

Thus, we get for ${\mu_{2}}\geq 4$,

$\displaystyle|R_{2}(n)|\leq
631+\frac{\sqrt{3}\pi^{\frac{3}{2}}}{4{\mu_{2}}^{\frac{1}{2}}}\exp{\left(\frac{{\mu_{2}}}{3}\right)}.$

Moreover, for ${\mu_{2}}\geq 22$, it can be easily checked that,

$631<\frac{\sqrt{3}\pi^{\frac{3}{2}}}{4{\mu_{2}}^{\frac{1}{2}}}\exp{\left(\frac{{\mu_{2}}}{3}\right)}.$

So (3.21) is valid, which completes the proof.

In accordance with Theorem 1.2, we are able to derive the upper and lower
bounds for ${\overline{p}}_{k}(n)$ when $2\leq k\leq 9$.

######  Corollary 3.4.

For $2\leq k\leq 9$ and $\mu_{k}\geq\dot{n}_{k}$, we have

$\displaystyle{M_{k}(n)}\left(1-\frac{1}{{\mu_{k}}^{6}}\right)\leq{\overline{p}}_{k}(n)\leq{M_{k}(n)}\left(1+\frac{1}{{\mu_{k}}^{6}}\right),$
(3.22)

where

$\displaystyle\dot{n}_{2}=43,\quad\dot{n}_{3}=49,\quad\dot{n}_{4}=43,\quad\dot{n}_{5}=58,$
$\displaystyle\dot{n}_{6}=130,\quad\dot{n}_{7}=102,\quad\dot{n}_{8}=129,\quad\dot{n}_{9}=268.$

Here, $M_{k}(n)=C_{k}(n)I_{1}(\mu_{k})$, and $C_{k}(n)$ is defined in Theorem
1.2.

###### Proof.

Let us define

$G_{k}(n):=\frac{R_{k}^{\prime}(n)}{C_{k}(n)I_{1}({\mu_{k}})},$

where $R_{k}^{\prime}(n)$ is defined in Theorem 1.2.

By utilizing Theorem 1.2, we can establish the following inequalities for
$2\leq k\leq 9$ when $\mu_{k}\geq n_{k}$:

${M_{k}(n)}(1-G_{k}(n))\leq\overline{p}_{k}(n)\leq{M_{k}(n)}(1+G_{k}(n)).$

First, we will prove the case of $k=2$. The expression for $G_{2}$ can be
simplified as follows.

$G_{2}(n):=\sqrt{\frac{6{\mu_{2}}}{\pi}}\frac{\exp{\left(\frac{{\mu_{2}}}{3}\right)}}{I_{1}({\mu_{2}})}.$

According to the proof of Theorem 1.3 in [10], we can demonstrate that for
$x\geq 43$,

$G(x):=\sqrt{\frac{6{x}}{\pi}}\frac{\exp{\left(\frac{{x}}{3}\right)}}{I_{1}({x})}\leq\frac{1}{{x}^{6}},$

which in turn implies that for $\mu_{2}\geq 43$,

$G_{2}(n)\leq\frac{1}{{\mu_{2}}^{6}}.$

Thus, we have verified the validity of (3.22) when $k=2$. The proofs for other
cases are similar, and we omit the details here. This completes the proof.

##  4 Proof of Theorem 1.3

In this section, our goal is to demonstrate the log-concavity and third-order
Turán inequalities of ${\overline{p}}_{k}(n)$ for $2\leq k\leq 9$. To begin,
we will establish both upper and lower bounds for $Q_{k}(n)$, where

$\displaystyle
Q_{k}(n)=\frac{{\overline{p}}_{k}(n-1){\overline{p}}_{k}(n+1)}{{\overline{p}}_{k}(n)^{2}}.$

######  Theorem 4.1.

For $\mu_{k}\geq\ddot{n}_{k}$, that is $n\geq\tilde{n}_{k}$, we establish the
following inequalities:

$\displaystyle\tilde{L}_{k}(n)<Q_{k}(n)<\tilde{R}_{k}(n),$ (4.1)

where all these values are defined in following table.

$k$ | $\ddot{n}_{k}$ | $\tilde{n}_{k}$ | $\tilde{L}_{k}(n)$ | $\tilde{R}_{k}(n)$
---|---|---|---|---
$2$ | $167$ | $5652$ |  $1$ $-\frac{\pi^{4}}{16{\mu_{2}}^{3}}+\frac{3\pi^{4}}{16{\mu_{2}}^{4}}-\frac{7}{{\mu_{2}}^{5}}-\frac{130}{{\mu_{2}}^{6}}$ |  $1$$-\frac{\pi^{4}}{16{\mu_{2}}^{3}}+\frac{3\pi^{4}}{16{\mu_{2}}^{4}}-\frac{6}{{\mu_{2}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{\mu_{2}}^{6}}$
$3$ | $49$ | $365$ |  $1$ $-\frac{\pi^{4}}{9\mu_{3}^{3}}+\frac{\pi^{4}}{3\mu_{3}^{4}}-\frac{13}{\mu_{3}^{5}}-\frac{200}{\mu_{3}^{6}}$ |  $1$ $-\frac{\pi^{4}}{9\mu_{3}^{3}}+\frac{\pi^{4}}{3\mu_{3}^{4}}-\frac{6}{\mu_{3}^{5}}+\frac{146+\frac{\pi^{8}}{81}}{\mu_{3}^{6}}$
$4$ | $58$ | $455$ |  $1$ $-\frac{9\pi^{4}}{64\mu_{4}^{3}}+\frac{27\pi^{4}}{64\mu_{4}^{4}}-\frac{16}{\mu_{4}^{5}}-\frac{300}{\mu_{4}^{6}}$ |  $1$ $-\frac{9\pi^{4}}{64\mu_{4}^{3}}+\frac{27\pi^{4}}{64\mu_{4}^{4}}-\frac{15}{\mu_{4}^{5}}+\frac{150+\frac{81\pi^{8}}{4096}}{\mu_{4}^{6}}$
$5$ | $94$ | $1120$ |  $1$ $-\frac{4\pi^{4}}{25\mu_{5}^{3}}+\frac{12\pi^{4}}{25\mu_{5}^{4}}-\frac{18}{\mu_{5}^{5}}-\frac{400}{\mu_{5}^{6}}$ |  $1$ $-\frac{4\pi^{4}}{25\mu_{5}^{3}}+\frac{12\pi^{4}}{25\mu_{5}^{4}}-\frac{17}{\mu_{5}^{5}}+\frac{400}{\mu_{5}^{6}}$
$6$ | $130$ | $2055$ |  $1$ $-\frac{25\pi^{4}}{144\mu_{6}^{3}}+\frac{25\pi^{4}}{48\mu_{6}^{4}}-\frac{20}{\mu_{6}^{5}}-\frac{441}{\mu_{6}^{6}}$ |  $1$ $-\frac{25\pi^{4}}{144\mu_{6}^{3}}+\frac{25\pi^{4}}{48\mu_{6}^{4}}-\frac{19}{\mu_{6}^{5}}+\frac{441}{\mu_{6}^{6}}$
$7$ | $102$ | $1230$ |  $1$ $-\frac{9\pi^{4}}{49\mu_{7}^{3}}+\frac{27\pi^{4}}{49\mu_{7}^{4}}-\frac{21}{\mu_{7}^{5}}-\frac{500}{\mu_{7}^{6}}$ |  $1$ $-\frac{9\pi^{4}}{49\mu_{7}^{3}}+\frac{27\pi^{4}}{49\mu_{7}^{4}}-\frac{20}{\mu_{7}^{5}}+\frac{500}{\mu_{7}^{6}}$
$8$ | $300$ | $10422$ |  $1$ $-\frac{49\pi^{4}}{256\mu_{8}^{3}}+\frac{147\pi^{4}}{256\mu_{8}^{4}}-\frac{21}{\mu_{8}^{5}}-\frac{505}{\mu_{8}^{6}}$ |  $1$ $-\frac{49\pi^{4}}{256\mu_{8}^{3}}+\frac{147\pi^{4}}{256\mu_{8}^{4}}-\frac{20}{\mu_{8}^{5}}+\frac{505}{\mu_{8}^{6}}$
$9$ | $268$ | $8187$ |  $1$ $-\frac{16\pi^{4}}{81\mu_{9}^{3}}+\frac{16\pi^{4}}{27\mu_{9}^{4}}-\frac{22}{\mu_{9}^{5}}-\frac{524}{\mu_{9}^{6}}$ |  $1$ $-\frac{16\pi^{4}}{81\mu_{9}^{3}}+\frac{16\pi^{4}}{27\mu_{9}^{4}}-\frac{21}{\mu_{9}^{5}}+\frac{529}{\mu_{9}^{6}}$
Table 4.4: Values of $\tilde{n}_{k}$, $\tilde{L}_{k}(n)$ and
$\tilde{R}_{k}(n)$ for $2\leq k\leq 9$.

###### Proof.

Define

$\displaystyle A_{k}(n)$
$\displaystyle=\frac{M_{k}(n-1)M_{k}(n+1)}{{M_{k}(n)}^{2}},$

where $M_{k}(n)$ is defined in Corollary 3.4.

From Corollary 3.4, we see that for $\mu_{k}\geq\tilde{n}_{k}$,

$A_{k}(n)L_{Q_{k}(n)}\leq{Q_{k}(n)}\leq A_{k}(n)R_{Q_{k}(n)},$ (4.2)

where

$L_{Q_{k}(n)}=\frac{\left(1-\frac{1}{{{\mu_{k}^{-}}}^{6}}\right)\left(1-\frac{1}{{{\mu_{k}^{+}}}^{6}}\right)}{\left(1+\frac{1}{{{\mu_{k}}}^{6}}\right)^{2}},\quad
R_{Q_{k}(n)}=\frac{\left(1+\frac{1}{{{\mu_{k}^{-}}}^{6}}\right)\left(1+\frac{1}{{{\mu_{k}^{+}}}^{6}}\right)}{\left(1-\frac{1}{{{\mu_{k}}}^{6}}\right)^{2}}.$
(4.3)

According to Theorem 1.2, $A_{k}(n)$ can be simplified as

$\displaystyle A_{k}(n)$
$\displaystyle=\frac{{\mu_{k}}^{2}I_{1}({{\mu_{k}^{-}}})I_{1}({{\mu_{k}^{+}}})}{{{\mu_{k}^{-}}}{{\mu_{k}^{+}}}I_{1}({\mu_{k}})^{2}}.$
(4.4)

By Lemma 3.2, the upper and lower bounds of (4.4) are given by

$\displaystyle\frac{{{\mu_{k}}}^{3}}{\sqrt{{{\mu_{k}^{-}}}^{3}{{\mu_{k}^{+}}}^{3}}}U_{k}(n)\leq
A_{k}(n)\leq\frac{{{\mu_{k}}}^{3}}{\sqrt{{{\mu_{k}^{-}}}^{3}{{\mu_{k}^{+}}}^{3}}}V_{k}(n).$
(4.5)

Next, we aim to provide upper and lower bounds for

$\frac{{{\mu_{k}}}^{3}}{\sqrt{{{\mu_{k}^{-}}}^{3}{{\mu_{k}^{+}}}^{3}}}.$

We begin by considering the case of $k=2$. It can be established that for
${{\mu_{2}}}\geq 3$,

$1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\leq\frac{{{\mu_{2}}}^{3}}{\sqrt{{{\mu_{2}^{-}}}^{3}{{\mu_{2}^{+}}}^{3}}}\leq
1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}},$ (4.6)

which is equivalent to

$\displaystyle{{\mu_{2}}}^{12}-{{\mu_{2}^{-}}}^{6}{{\mu_{2}^{+}}}^{6}{\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\right)}^{4}$
$\displaystyle\geq 0,$ (4.7)
$\displaystyle{{\mu_{2}}}^{12}-{{\mu_{2}^{-}}}^{6}{{\mu_{2}^{+}}}^{6}{\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}}\right)}^{4}$
$\displaystyle\leq 0.$ (4.8)

By utilizing (3.5), we can express (4.7) and (4.8) in the following form:

$\displaystyle{{\mu_{2}}}^{12}-{{\mu_{2}^{-}}}^{6}{{\mu_{2}^{+}}}^{6}\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\right)^{4}$
$\displaystyle=\frac{1}{6294077440000{{\mu_{2}}}^{32}}\left(313298944000\pi^{8}{{\mu_{2}}}^{36}+177899008000\pi^{12}{{\mu_{2}}}^{32}\right.$
$\displaystyle\left.\qquad+3408753600\pi^{16}{{\mu_{2}}}^{28}-5446075600\pi^{20}{{\mu_{2}}}^{24}-1992890060\pi^{24}{{\mu_{2}}}^{20}\right.$
$\displaystyle\qquad-167858335\pi^{28}{{\mu_{2}}}^{16}+19789376\pi^{32}{{\mu_{2}}}^{12}+8611328\pi^{36}{{\mu_{2}}}^{8}$
$\displaystyle\left.\qquad+933888\pi^{40}{{\mu_{2}}}^{4}+65536\pi^{44}\right)$

and

$\displaystyle{{\mu_{2}}}^{12}-{\mu_{2}^{-}}^{6}{\mu_{2}^{+}}^{6}{\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}}\right)}^{4}$
$\displaystyle=-\frac{1}{339738624{{\mu_{2}}}^{32}}\left(57507840\pi^{8}{{\mu_{2}}}^{36}-23556096\pi^{12}{{\mu_{2}}}^{32}+2714688\pi^{16}{{\mu_{2}}}^{28}\right.$
$\displaystyle\left.\qquad-3100464\pi^{20}{\mu_{2}}^{24}+580044\pi^{24}{{\mu_{2}}}^{20}-94689\pi^{28}{{\mu_{2}}}^{16}+51952\pi^{32}{{\mu_{2}}}^{12}\right.$
$\displaystyle\left.\qquad-4704\pi^{36}{{\mu_{2}}}^{8}+768\pi^{40}{{\mu_{2}}}^{4}-256\pi^{44}\right).$

We can further compute that when ${{\mu_{2}}}\geq 3$,

$\displaystyle
313298944000\pi^{8}{{\mu_{2}}}^{36}+177899008000\pi^{12}{{\mu_{2}}}^{32}+3408753600\pi^{16}{{\mu_{2}}}^{28}$
$\displaystyle\quad-5446075600\pi^{20}{{\mu_{2}}}^{24}-1992890060\pi^{24}{{\mu_{2}}}^{20}-167858335\pi^{28}{{\mu_{2}}}^{16}$
$\displaystyle\quad+19789376\pi^{32}{{\mu_{2}}}^{12}+8611328\pi^{36}{{\mu_{2}}}^{8}+933888\pi^{40}{{\mu_{2}}}^{4}+65536\pi^{44}\geq
0$

and when $\mu_{2}\geq 4$,

$\displaystyle
57507840\pi^{8}{{\mu_{2}}}^{36}-23556096\pi^{12}{{\mu_{2}}}^{32}+2714688\pi^{16}{{\mu_{2}}}^{28}$
$\displaystyle\quad-3100464\pi^{20}{\mu_{2}}^{24}+580044\pi^{24}{{\mu_{2}}}^{20}-94689\pi^{28}{{\mu_{2}}}^{16}+51952\pi^{32}{{\mu_{2}}}^{12}$
$\displaystyle\quad-4704\pi^{36}{{\mu_{2}}}^{8}+768\pi^{40}{{\mu_{2}}}^{4}-256\pi^{44}\geq
0.$

Therefore, we have verified (4.7) and (4.8), and thus obtained (4.6).
Substituting (4.6) into (4.5), we get that for ${\mu_{2}}\geq 167$, i.e.
$n\geq 5652$,

$\displaystyle A_{2}(n)$
$\displaystyle\geq\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}-\frac{5\pi^{8}}{512{{\mu_{2}}}^{7}}\right)\left(1-\frac{7}{{{\mu_{2}}}^{5}}-\frac{115}{{{\mu_{2}}}^{6}}\right)$
(4.9)

and

$\displaystyle A_{2}(n)$
$\displaystyle\leq\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{\pi^{8}}{256{{\mu_{2}}}^{6}}\right)\left(1-\frac{6}{{{\mu_{2}}}^{5}}+\frac{115}{{{\mu_{2}}}^{6}}\right).$
(4.10)

Then, we will compute the upper bound for $R_{Q_{2}(n)}$ and the lower bound
for $L_{Q_{2}(n)}$ in (4.3). Applying (3.5) into (4.3), we find that

$L_{Q_{2}(n)}=\frac{{{\mu_{2}}}^{12}\left(\left({{\mu_{2}}}^{2}+\frac{\pi^{2}}{2}\right)^{3}-1\right)\left(\left({{\mu_{2}}}^{2}-\frac{\pi^{2}}{2}\right)^{3}-1\right)}{({{\mu_{2}}}^{6}+1)^{2}\left({{\mu_{2}}}^{4}-\frac{\pi^{4}}{4}\right)^{3}}$

and

$R_{Q_{2}(n)}=\frac{{{\mu_{2}}}^{12}\left(\left({{\mu_{2}}}^{2}+\frac{\pi^{2}}{2}\right)^{3}+1\right)\left(\left({{\mu_{2}}}^{2}-\frac{\pi^{2}}{2}\right)^{3}+1\right)}{({{\mu_{2}}}^{6}-1)^{2}\left({{\mu_{2}}}^{4}-\frac{\pi^{4}}{4}\right)^{3}}.$

Since $L_{Q_{2}(n)},R_{Q_{2}(n)}$ are both just polynomial about ${\mu_{2}}$,
it can be readily checked that when ${\mu_{2}}\geq 57$,

$L_{Q_{2}(n)}\geq 1-\frac{5}{{{\mu_{2}}}^{6}}\quad\mathrm{and}\quad
R_{Q_{2}(n)}\leq 1+\frac{5}{{{\mu_{2}}}^{6}}.$ (4.11)

By substituting (4.9), (4.10), and (4.11) into (4.2), we obtain that for
${{\mu_{2}}}\geq 167$, i.e. $n\geq 5652$,

$\displaystyle{Q_{2}(n)}$
$\displaystyle\geq\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}-\frac{5\pi^{8}}{512{{\mu_{2}}}^{7}}\right)$
$\displaystyle\quad\times\left(1-\frac{7}{{{\mu_{2}}}^{5}}-\frac{115}{{{\mu_{2}}}^{6}}\right)\left(1-\frac{5}{{{\mu_{2}}}^{6}}\right)$
(4.12)

and

$\displaystyle{Q_{2}(n)}$
$\displaystyle\leq\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{\pi^{8}}{256{{\mu_{2}}}^{6}}\right)$
$\displaystyle\quad\times\left(1-\frac{6}{{{\mu_{2}}}^{5}}+\frac{115}{{{\mu_{2}}}^{6}}\right)\left(1+\frac{5}{{{\mu_{2}}}^{6}}\right).$
(4.13)

Now we shall compute that when ${\mu_{2}}\geq 38$,

$\displaystyle\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}-\frac{5\pi^{8}}{512{{\mu_{2}}}^{7}}\right)\left(1-\frac{7}{{{\mu_{2}}}^{5}}-\frac{115}{{{\mu_{2}}}^{6}}\right)\left(1-\frac{5}{{{\mu_{2}}}^{6}}\right)$
$\displaystyle\quad>1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{7}{{{\mu_{2}}}^{5}}-\frac{130}{{{\mu_{2}}}^{6}},$
(4.14)

and when ${{\mu_{2}}}\geq 16$,

$\displaystyle\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{\pi^{8}}{256{{\mu_{2}}}^{6}}\right)\left(1-\frac{6}{{{\mu_{2}}}^{5}}+\frac{115}{{{\mu_{2}}}^{6}}\right)\left(1+\frac{5}{{{\mu_{2}}}^{6}}\right)$
$\displaystyle\quad<1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{6}{{{\mu_{2}}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}}}^{6}}.$
(4.15)

Notice that

$\displaystyle\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{35{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}-\frac{5\pi^{8}}{512{{\mu_{2}}}^{7}}\right)\left(1-\frac{7}{{{\mu_{2}}}^{5}}-\frac{115}{{{\mu_{2}}}^{6}}\right)\left(1-\frac{5}{{{\mu_{2}}}^{6}}\right)$
$\displaystyle\quad-\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{7}{{{\mu_{2}}}^{5}}-\frac{130}{{{\mu_{2}}}^{6}}\right)=\frac{1}{286720{{\mu_{2}}}^{27}}\sum_{i=0}^{21}c_{i}{{\mu_{2}}}^{i},$

and

$\displaystyle\left(1+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{\pi^{8}}{12{{\mu_{2}}}^{8}}\right)\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{\pi^{8}}{256{{\mu_{2}}}^{6}}\right)\left(1-\frac{6}{{{\mu_{2}}}^{5}}+\frac{115}{{{\mu_{2}}}^{6}}\right)\left(1+\frac{5}{{{\mu_{2}}}^{6}}\right)$
$\displaystyle\quad-\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{6}{{{\mu_{2}}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}}}^{6}}\right)=-\frac{1}{12288{{\mu_{2}}}^{26}}\sum_{j=0}^{20}d_{j}{{\mu_{2}}}^{j},$

where $c_{i}$ and $d_{j}$ are real numbers. We list the values of
$c_{19}-c_{21}$ and $d_{18}-d_{20}$:

$\displaystyle c_{19}=125440\pi^{4}+8192\pi^{8},\quad\quad
c_{20}=-6160\pi^{8},\qquad\qquad c_{21}=2867200,$ $\displaystyle
d_{18}=-4608\pi^{4}-1024\pi^{8},\quad\quad d_{19}=144\pi^{8},\qquad\qquad\quad
d_{20}=96\pi^{8}.$

It can be measured that when ${\mu_{2}}\geq 3$, for $0\leq i\leq 18$,

$-|c_{i}|{{\mu_{2}}}^{i}\geq-|c_{19}|{{\mu_{2}}}^{19},$

and when ${{\mu_{2}}}\geq 3$, for $0\leq j\leq 17$,

$-|d_{j}|{{\mu_{2}}}^{j}\geq-|d_{18}|{{\mu_{2}}}^{18}.$

Further, we can get

$\displaystyle\sum_{i=0}^{21}c_{i}{{\mu_{2}}}^{i}$
$\displaystyle\geq-\sum_{i=0}^{19}|c_{i}|{{\mu_{2}}}^{19}+c_{20}{{\mu_{2}}}^{20}+c_{21}{{\mu_{2}}}^{21}\geq-20|c_{19}|{{\mu_{2}}}^{19}+c_{20}{{\mu_{2}}}^{20}+c_{21}{{\mu_{2}}}^{21}$

and similarly,

$\displaystyle\sum_{j=0}^{20}d_{j}{{\mu_{2}}}^{j}$
$\displaystyle\geq-\sum_{j=0}^{18}|d_{j}|{{\mu_{2}}}^{18}+d_{19}{{\mu_{2}}}^{19}+d_{20}{{\mu_{2}}}^{20}\geq-19|d_{18}|{{\mu_{2}}}^{18}+d_{19}{{\mu_{2}}}^{19}+d_{20}{{\mu_{2}}}^{20}.$

Since for ${{\mu_{2}}}\geq 38$,

$-20|c_{19}|+c_{20}{{\mu_{2}}}+c_{21}{{\mu_{2}}}^{2}>0$

and for ${{\mu_{2}}}\geq 14$,

$-19|d_{18}|+d_{19}{{\mu_{2}}}+d_{20}{{\mu_{2}}}^{2}>0,$

we see that (4) and (4) are true. By plugging (4) and (4) into (4) and (4),
respectively, we can establish that (4.1) holds when $k=2$. The proofs for
other cases are similar, and therefore, we omit them here. This completes the
proof.

Subsequently, we proceed with the proof of Theorem 1.3.

Proof of Theorem 1.3. To establish log-concavity, we need to show that
${Q_{k}(n)}\leq 1$ for $2\leq k\leq 9$. From Theorem 4.1, we see that for
$2\leq k\leq 9$ and ${n}\geq\tilde{n}_{k}$,

${Q_{k}(n)}<\tilde{R}_{k}<1,$

where $\tilde{n}_{k}$ and $\tilde{R}_{k}$ are defined in Table 4.4. Besides,
${Q_{k}(n)}<1$ holds as long as one can verify it holds for $\bar{n}_{k}\leq
n\leq\tilde{n}_{k}$, where $\bar{n}_{k}$ is defined in Theorem 1.3. So
${Q_{k}(n)}<1$ for $n\geq\bar{n}_{k}$ and ${\overline{p}}_{k}(n)$ is log-
concave for $n\geq\bar{n}_{k}$ when $2\leq k\leq 9$.

To demonstrate that ${\overline{p}}_{k}(n)$ for $2\leq k\leq 9$ satisfies the
third order Turán inequalities for $n\geq\hat{n}_{k}$, we rely on Chen, Jia,
and Wang’s work [6]. It is sufficient to prove that when $n\geq\hat{n}_{k}$,

$\displaystyle 4(1-{Q_{k}(n)})(1-Q_{k}(n+1))-(1-{Q_{k}(n)}Q_{k}(n+1))^{2}>0.$
(4.16)

Recall that Jia [13, Lemma 5.1] obtained the following result. If $u$ and $v$
are positive real numbers satisfying $\frac{15}{16}\leq u<v<1$ and
$u+\sqrt{(1-u)^{3}}>v$ holds, then

$\displaystyle 4(1-u)(1-v)-(1-uv)^{2}>0.$ (4.17)

Using (4.17), we come to prove that for $\mu_{k}\geq\tilde{n}_{k}$ and $2\leq
k\leq 9$,

$\frac{15}{16}\leq{Q_{k}(n)}<Q_{k}(n+1),$ (4.18)

and

$Q_{k}(n+1)<{Q_{k}(n)}+\sqrt{(1-{Q_{k}(n)})^{3}}.$ (4.19)

Next, we will prove the case when $k=2$. By Theorem 4.1, we have for
${\mu_{2}}\geq 167$,

${Q_{2}(n)}>1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{7}{{{\mu_{2}}}^{5}}-\frac{130}{{{\mu_{2}}}^{6}}.$

It can be checked that when ${{\mu_{2}}}\geq 4$,

$1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{7}{{{\mu_{2}}}^{5}}-\frac{130}{{{\mu_{2}}}^{6}}\geq\frac{15}{16}$

Hence, for ${\mu_{2}}\geq 167$,

${Q_{2}(n)}>\frac{15}{16}.$

Moreover, by Theorem 4.1, it can be computed directly that for
${{\mu_{2}}}\geq 11$,

$\displaystyle Q_{2}(n+1)-{Q_{2}(n)}>$
$\displaystyle\left(1-\frac{\pi^{4}}{16{{\mu_{2}^{+}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}^{+}}}^{4}}-\frac{7}{{{\mu_{2}^{+}}}^{5}}-\frac{130}{{{\mu_{2}^{+}}}^{6}}\right)$
$\displaystyle-\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{6}{{{\mu_{2}}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}}}^{6}}\right)$
$\displaystyle=$
$\displaystyle\frac{\pi^{4}}{16}\left(\frac{1}{{{\mu_{2}}}^{3}}-\frac{1}{{{\mu_{2}^{+}}}^{3}}+\frac{3}{{{\mu_{2}^{+}}}^{4}}-\frac{3}{{{\mu_{2}}}^{4}}\right)$
$\displaystyle+\frac{6}{{{\mu_{2}}}^{5}}-\frac{7}{{{\mu_{2}^{+}}}^{5}}-\frac{250+\frac{\pi^{8}}{256}}{{{\mu_{2}}}^{6}}$
$\displaystyle>$ $\displaystyle 0.$

Here we realize (4.18) is valid. And we find that for ${\mu_{2}}\geq 167$,

$\displaystyle Q_{2}(n+1)-{Q_{2}(n)}$
$\displaystyle<\left(1-\frac{\pi^{4}}{16{{\mu_{2}^{+}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}^{+}}}^{4}}-\frac{6}{{{\mu_{2}^{+}}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}^{+}}}^{6}}\right)$
$\displaystyle\quad-\left(1-\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}+\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{7}{{{\mu_{2}}}^{5}}-\frac{130}{{{\mu_{2}}}^{6}}\right)$
$\displaystyle=\frac{\pi^{4}}{16}\left(\frac{1}{{{\mu_{2}}}^{3}}-\frac{1}{{{\mu_{2}^{+}}}^{3}}\right)-\frac{3\pi^{4}}{16}\left(\frac{1}{{{\mu_{2}}}^{4}}-\frac{1}{{{\mu_{2}^{+}}}^{4}}\right)$
$\displaystyle\quad-\frac{6}{{{\mu_{2}^{+}}}^{5}}+\frac{7}{{{\mu_{2}}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}^{+}}}^{6}}+\frac{130}{{{\mu_{2}}}^{6}}.$
(4.20)

Notice that when ${{\mu_{2}}}>0$,

$\frac{1}{{{\mu_{2}}}^{3}}-\frac{1}{{{\mu_{2}^{+}}}^{3}}<\frac{3\pi^{2}}{4{{\mu_{2}}}^{5}},$

and when ${{\mu_{2}}}\geq 60$,

$\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}^{+}}}^{6}}+\frac{130}{{{\mu_{2}}}^{6}}<\frac{288}{{{\mu_{2}}}^{6}}<\frac{\pi^{2}}{2{{\mu_{2}}}^{5}}.$

Hence, we can rewrite (4) as

$\displaystyle Q_{2}(n+1)-{Q_{2}(n)}$
$\displaystyle<\frac{\pi^{4}}{16}\cdot\frac{3\pi^{2}}{4{{\mu_{2}}}^{5}}+\frac{7}{{{\mu_{2}}}^{5}}+\frac{\pi^{2}}{2{{\mu_{2}}}^{5}}$
$\displaystyle=\frac{\frac{3}{64}\pi^{6}+7+\frac{1}{2}\pi^{2}}{{{\mu_{2}}}^{5}}.$

By Theorem 4.1,

$\displaystyle 1-{Q_{2}(n)}$
$\displaystyle>\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}-\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}+\frac{6}{{{\mu_{2}}}^{5}}-\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}}}^{6}}.$

Note for ${{\mu_{2}}}\geq 73$,

$\frac{1}{4{{\mu_{2}}}^{3}}>\frac{3\pi^{4}}{16{{\mu_{2}}}^{4}}-\frac{6}{{{\mu_{2}}}^{5}}+\frac{120+\frac{\pi^{8}}{256}}{{{\mu_{2}}}^{6}},$

so we have

$\displaystyle
1-{Q_{2}(n)}>\frac{\pi^{4}}{16{{\mu_{2}}}^{3}}-\frac{1}{4{{\mu_{2}}}^{3}}=\frac{\pi^{4}-4}{16{{\mu_{2}}}^{3}}.$

It can be verified that for ${{\mu_{2}}}\geq 17$,

$\sqrt{\left(\frac{\pi^{4}-4}{16{{\mu_{2}}}^{3}}\right)^{3}}>\frac{\frac{3}{64}\pi^{6}+7+\frac{1}{2}\pi^{2}}{{{\mu_{2}}}^{5}}.$

So (4.19) holds for ${{\mu_{2}}}\geq 167$. This prove the cases that for
${{\mu_{2}}}\geq 167$. When $65\leq n\leq 5652$, we can compute that (4.16) is
also met. Thus for $n\geq 65$, ${\overline{p}}_{2}(n)$ satisfies the third
order Turán inequalities. The proof for other cases is similar, and we omit it
here. This completes the proof.

Acknowledgements

The second author would like to acknowledge that the research was supported by
the National Natural Science Foundation of China (Grant Nos. 12001182 and
12171487), the Fundamental Research Funds for the Central Universities (Grant
No. 531118010411) and Hunan Provincial Natural Science Foundation of China
(Grant No. 2021JJ40037).

## References

  * [1] A.A. Alanazi, S.M.III Gagola and A.O. Munagi, Combinatorial proof of a partition inequality of Bessenrodt-Ono, Ann. Comb. 21 (3) (2017) 331–337.
  * [2] O. Beckwith and C. Bessenrodt, Multiplicative properties of the number of $k$-regular partitions, Ann. Comb. 20 (2) (2016) 231–250.
  * [3] C. Bessenrodt and K. Ono, Maximal multiplicative properties of partitions, Ann. Comb. 20 (1) (2016) 59–64.
  * [4] K. Bringmann, B. Kane, L. Rolen and Z. Tripp, Fractional partitions and conjectures of Chern-Fu-Tang and Heim-Neuhauser, Trans. Amer. Math. Soc. Ser. B 8 (2021) 615–634.
  * [5] W.Y.C. Chen, The spt-function of Andrews, Surveys in Combinatorics 2017, 141–203, London Math. Soc. Lecture Note Ser., 440, Cambridge Univ. Press, Cambridge, 2017.
  * [6] W.Y.C. Chen, D.X.Q. Jia and L.X.W. Wang, Higher order Turán inequalities for the partition function, Trans. Amer. Math. Soc. 372 (3) (2019) 2143–2165.
  * [7] S. Chern, Asymptotics for the Fourier coefficients of eta-quotients, J. Number Theory 199 (2019) 168–191.
  * [8] M.L. Dawsey and R. Masri, Effective bounds for the Andrews spt-function, Forum Math. 31 (3) (2019) 743–767.
  * [9] S. DeSalvo and I. Pak, Log-concavity of the partition function, Ramanujan J. 38 (1) (2015) 61–73.
  * [10] J.J.W. Dong and K.Q. Ji, Higher order Turán inequalities for the distinct partition function, arXiv:2303.05243.
  * [11] B. Engel, Log-concavity of the overpartition function, Ramanujan J. 43 (2) (2017) 229–241.
  * [12] M. Griffin, K. Ono, L. Rolen and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA 116 (23) (2019) 11103–11110.
  * [13] D.X.Q. Jia, Inequalities for the broken $k$-diamond partition functions, J. Number Theory 249 (2023) 314–347.
  * [14] X. Li, Polynomization of the Liu-Zhang inequality for the overpartition function, Ramanujan J. DOI:10.1007/s11139-023-00711-7.
  * [15] E.Y.S. Liu and H.W.J. Zhang, Inequalities for the overpartition function, Ramanujan J. 54 (3) (2021) 485–509.
  * [16] J. Lovejoy, Gordon’s theorem for overpartitions, J. Combin. Theory Ser. A 103 (2) (2003) 393–401.
  * [17] K. Ono, S. Ken and L. Rolen, Turán inequalities for the plane partition function, Adv. Math. 409 (2022) part B, Paper No. 108692, 31 pp.
  * [18] E.Y.Y. Shen, Arithmetic properties of $l$-regular overpartitions, Int. J. Number Theory 12 (3) (2016) 841–852.


Hsin-Hao Chen1,2, Yung-Lun Chien1,2, Ming-Chi Yen2, Shu-Wei Tsai3,
Yu Tsao2, Tai-shih Chi1, and Hsin-Min Wang2

# Mandarin Electrolaryngeal Speech Voice Conversion
using Cross-domain Features

###### Abstract

Patients who have had their entire larynx removed, including the vocal folds,
owing to throat cancer may experience difficulties in speaking. In such cases,
electrolarynx devices are often prescribed to produce speech, which is
commonly referred to as electrolaryngeal speech (EL speech). However, the
quality and intelligibility of EL speech are poor. To address this problem, EL
voice conversion (ELVC) is a method used to improve the intelligibility and
quality of EL speech. In this paper, we propose a novel ELVC system that
incorporates cross-domain features, specifically spectral features and self-
supervised learning (SSL) embeddings. The experimental results show that
applying cross-domain features can notably improve the conversion performance
for the ELVC task compared with utilizing only traditional spectral features.

Index Terms: electrolaryngeal speech, voice conversion, self-supervised
learning

##  1 Introduction

Some patients with laryngeal diseases, such as laryngeal cancer, may need to
undergo laryngectomy, which is a surgical procedure that involves removing the
larynx, including the vocal folds. Without the vibration of the vocal folds,
these patients lose their ability to generate excitation signals and cannot
produce speech normally. Electrolarynx (EL) is a medical device used to
restore the ability to speak in these patients. Specifically, the
electrolarynx generates surrogate excitation signals that enable patients to
produce speech. However, the sound quality and intelligibility of the speech
produced by the electrolarynx are often poor and accompanied by constant
mechanical noise, and consequently, the speech does not resemble a natural
human voice.

Voice conversion (VC)[1, 2, 3] techniques have been widely used to overcome
this problem to convert electrolaryngeal (EL) speech to natural (NL) speech
without altering the underlying content. This task is commonly referred to as
EL speech voice conversion (ELVC) [4, 5, 6, 7, 8, 9, 10]. Existing ELVC
approaches can be classified into two categories: sequence-to-sequence and
frame-based approaches.

The implementation of a frame-based ELVC system typically involves three
steps: first, extracting features from both EL and NL speech; second,
converting the features of the EL speech to those of the target NL speech
using a conversion model; and finally, generating speech waveforms from the
converted features, which is often performed by a vocoder. Traditional frame-
based ELVC techniques, such as those used in [4, 5, 6], employ Gaussian
mixture models to build the conversion model. Recently, deep neural networks
have been widely used to build the conversion model [7, 8, 9].

In addition to frame-based ELVC systems, previous studies have proposed
implementing ELVC systems using sequence-to-sequence (seq2seq) models. A
seq2seq ELVC system can effectively avoid frame-alignment difficulties that
can occur in frame-based ELVC systems [10]. However, a relatively large amount
of computation is required during training and conversion, which makes such
systems challenging to implement in real-world scenarios.

Figure 1: The overall architecture of the proposed ELVC system. F1 and F2
represent features 1 and 2, respectively, and y represents the speaker
identity. During NLVC pre-training, the encoder and decoder (pink parts) are
tuned based on NL speech. During ELVC training, the gray part is fixed and the
pink part is tuned based on aligned NL and EL speech. $Z_{F1}$ and $Z_{F2}$
denote the latent representations generated from the F1 and F2 encoders,
respectively, and the G and D components of the PWG vocoder refer to the
generator and discriminator of the GAN-based network.

Self-supervised learning (SSL) has become popular in recent years. It is a
promising alternative to traditional supervised learning, which requires large
amounts of labeled data, which can be both time-consuming and labor-intensive.
By contrast, SSL can train high-performance models without the need for
labeled data, thereby overcoming the aforementioned limitations of supervised
learning. The representations obtained from SSL (referred to as SSL
representations) have proven to be highly effective in various speech-related
tasks, as demonstrated in several studies [11, 12, 13]. In the context of the
VC task, SSL representations have also been incorporated and shown promising
results in one-to-one, any-to-one, and any-to-any modes [14, 15].

In [16], the use of SSL representations improved speech recognition for
patients with dysarthria, which is a type of impaired speech. However,
dysarthric speech and EL speech have distinct characteristics. The dysarthric
speech dataset used in [16] includes speech signals from patients with
cerebral palsy and Parkinson’s disease, which cause motor control dysfunction,
primarily affecting pronunciation. In contrast, EL speech is generated from
surrogate excitation signals, which are significantly different from the real
excitation signals generated from the lungs. In [17], the authors investigated
an SSL pre-trained model for pathological speech, focusing on fine-tuning the
pre-trained model for speech recognition.

In this paper, we propose a novel frame-based Mandarin ELVC system that
utilizes both traditional spectral features and SSL representations (or called
SSL features) extracted using WavLM [13]. The ELVC system comprises three
steps, and we employ SSL and cross-domain features in all these steps. First,
we used the SSL features to implement a dynamic time warping (DTW) algorithm
to align the EL and NL speech signals. Subsequently, we employed the cross-
domain variational autoencoder (CDVAE) [18] as a conversion model to convert
EL speech to NL speech. Finally, we trained a vocoder using cross-domain
features to generate speech audio from the converted features. In summary, the
contributions of this study are as follows: (1) confirming the effectiveness
of the SSL features for temporal alignment, (2) confirming the effectiveness
of the cross-domain features, including the SSL features, for the ELVC task,
and (3) training vocoders based on the SSL and cross-domain features. To the
best of our knowledge, this is the first study to investigate this topic with
promising results. We also demonstrate the effectiveness of the proposed
method through subjective and objective evaluations.

##  2 Methods

In this section, we introduce our overall architecture (see Fig. 1). The
frame-based ELVC system is implemented in three steps: feature extraction,
feature conversion, and audio generation. Before generating the audio file, we
used the converted features to fine-tune the vocoder to obtain better results.

###  2.1 Feature extraction

We used cross-domain features in two parts: (1) traditional features: mel
spectrum (Mel), mel cepstral coefficients (MCC), and STRAIGHT spectra (SP);
(2) SSL features, which refers to the embeddings extracted by the pre-trained
WavLM model [19] from the input waveform.

###  2.2 Feature conversion

Owing to the scarcity of EL data, we propose a two-stage feature conversion
approach. In the first stage, we trained an NL speech conversion (NLVC) model
as a pre-trained model. In the second stage, we fine-tuned the pre-trained
NLVC model using EL speech data to obtain an ELVC model. This approach
leverages the abundance of NL speech data to improve the conversion
performance of the limited EL speech data.

####  2.2.1 Stage1. NLVC model training

We trained the NLVC model using utterances from 18 speakers. The model adopts
a CDVAE architecture [18]. This model benefits from the simultaneous use of
multiple spectral features. In this study, we tested two sets of cross-domain
features: Mel+SP and Mel+SSL. In the CDVAE model, the encoder uses cross-
domain features as input and maps them to a latent space. The speaker identity
was then concatenated with the latent representation before it was passed to
the decoder to generate the predicted features.

####  2.2.2 Stage2. Speech alignment and ELVC model training

Before training the ELVC model, we aligned the EL and NL speech. We first
hand-labeled word boundaries to split the entire sentence into word-by-word
segments and then used the DTW algorithm to align the NL and EL segments. The
DTW algorithm minimizes the distance between the NL and EL segments based on
the mean square error (MSE). We implemented the DTW algorithm with different
features, such as the Mel, MCC, and SSL features, as input, and the
corresponding experimental results are presented in Section 4. With the
aligned EL and NL data, we fed the NL features to the fixed NL encoder and the
EL features to the EL encoder to train the VC model to minimize the difference
between the two latent representations, where the L1 loss was used as the
objective function to measure the difference. We then appended the speaker
code to the latent representations and fed them to a decoder that was well-
trained in Stage 1 to obtain the converted features. We derived another loss
(reconstruction loss) to measure the difference between the converted and NL
speech features.

###  2.3 Audio generation

This section describes the process of generating audio files from converted
features, which involves vocoder training and fine-tuning using cross-domain
features. In this study, we used the Parallel WaveGAN (PWG) [20] vocoder
because of its ability to produce high-quality audio and its efficient
training and synthesis processes.

We trained several vocoders using the Mel, SSL, and cross-domain features and
compared the quality of the audio produced by them. Our experimental results
for the NLVC task indicate that the PWG trained with cross-domain features is
capable of producing high-quality audio. In contrast, for ELVC, the PWG
trained with SSL features produces audio with better intelligibility. The
above results demonstrate the effectiveness of incorporating SSL features into
the PWG vocoder for both NLVC and ELVC tasks. Furthermore, we attempted to
fine-tune the PWG model based on the converted features to enhance its
performance specifically for the ELVC task.

##  3 Experiments

###  3.1 Experimental setup

We recruited a doctor (medical) to prepare the EL speech data by imitating a
patient reading the Taiwanese Mandarin Noise Test (TMHINT) phonetically
balanced script [21] using an electrolarynx device. The EL data contained 320
utterances, of which 240 utterances were used for training. For the NL data,
18 speakers were recruited to read prompts in the TMHINT script. All the
speech signals were recorded at a sampling rate of 16 kHz. For feature
extraction, we used the World vocoder [22] to extract 513-dimensional SP and
24-dimensional MCCs. The frame size and number of hops were 1024 and 256, and
the time shift was 20 ms. We also used this setting to extract the
80-dimensional mel spectrogram (Mel). For the SSL feature, we used a pre-
trained WavLM model with a frame/hop size of 400/320, resulting in a time
shift of 25 ms, and each SSL feature had 768 dimensions. When using SSL
features, we adopted the same frame size and hop size (400/320) to extract Mel
features for consistency.

####  3.1.1 CDVAE Model structure and parameters

The CDVAE model consisted of two encoders and two decoders, each consisting of
five convolutional neural network (CNN) layers. Assuming that the cross-domain
features have N dimensions, the input dimension of the CNN layer in the
encoder was (N, 1024, 512, 256, 128), and the output dimension of the decoder
was (128, 256, 512, 1024, N). All the CNN layers in the model shared the same
stride and kernel size of 1 and 5, respectively. The optimizer used in this
model was RAdam (Rectified Adam), The batch size was set to 16, and the
learning rate was set to 0.0001.

####  3.1.2 Evaluation metrics

Three objective metrics were employed to evaluate the proposed system: (1)
mel-cepstral distortion (MCD) in dB, which measures spectral distortions; (2)
fundamental frequency root mean square error (F0 RMSE) in Hz, which measures
the accuracy of F0 information; and (3) fundamental frequency correlation
coefficient (F0 CORR) in Hz, which measures the correlation of F0 features.

In addition, we conducted a subjective listening test in which participants
rated the intelligibility and quality (cleanliness) of the converted audio
files on a five-point scale, where 1 denotes the worst score and 5 denotes the
best score. The listening test involved 12 untrained, normal hearing
participants, of whom 8 were male and 4 were female. The average age of the
participants was 24 years old. For each test sample, participants were unaware
which ELVC system was used to generate it. We selected 20 speech utterances
from each ELVC system and asked subjects to rate them in terms of
intelligibility and quality.

###  3.2 Experimental results

####  3.2.1 DTW using different features

In our initial experiments, we tested the performance of DTW using different
features, including Mel, MCC, and WavLM. The CDVAE model for NLVC was trained
using Mel as the NLF1 features and SP as the NLF2 features (see Fig. 1). The
ELVC model was trained using Mel as the NLF1 and ELF1 features. The PWG
vocoder was trained to generate speech audio from Mel. The results are shown
in Table 1. The SSL features yielded the best results (lowest F0 RMSE and
highest F0 CORR), indicating that they are a better choice than the other
features. We also tested DTW performance using cross-domain features, but no
further improvements were observed. Therefore, we used the SSL features as
input to the DTW algorithm in the following experiments.

####  3.2.2 Vocoder using cross-domain features

This section provides a performance comparison of different features for
training the vocoder using NL speech from 18 speakers. Table 2 shows that the
Mel features outperformed the SSL features, but the cross-domain features
(SSL+Mel) yielded the best performance among the three systems. These results
demonstrate the significant advantages of using cross-domain features to build
an NL vocoder.

Table 1: ELVC results when using different features in DTW. DTW | MCD | F0 RMSE | F0 CORR
---|---|---|---
Mel | 8.43 | 35.48 | 0.078
MCC | 8.46 | 35.41 | 0.100
SSL | 8.46 | 34.22 | 0.131
Table 2: NL speech self-reconstruction results using vocoders with different features. Vocoder | MCD | F0 RMSE | F0 CORR
---|---|---|---
Mel | 3.67 | 39.63 | 0.223
SSL | 5.40 | 42.50 | 0.124
SSL+Mel | 3.97 | 30.98 | 0.384
Table 3: Overall results of ELVC, where the features used in CDVAE are denoted as F1+F2, and FT denotes the model after a further fine-tuning process. CDVAE | Vocoder | MCD | F0 RMSE | F0 CORR
---|---|---|---|---
Mel+SP | Mel | 8.46 | 34.22 | 0.131
Mel+SSL | Mel | 7.68 | 42.01 | 0.041
SSL+Mel | SSL | 7.03 | 51.21 | 0.041
SSL+Mel | SSL(FT) | 6.90 | 38.62 | 0.157
SSL+Mel | SSL+Mel | 7.60 | 39.68 | 0.118
SSL+Mel | SSL+Mel(FT) | 7.44 | 39.46 | 0.083

####  3.2.3 Overall ELVC results

In this section, we compare the performance of CDVAE models and vocoders
trained using different feature combinations. We first pre-trained the CDVAE
model using Mel as the NLF1 features and either SP or SSL as the NLF2 features
(see Fig. 1). We then used Mel as the NLF1 and ELF1 features to train the ELVC
model, which takes Mel as input and generates Mel as output. The Mel features
were then fed to the PWG vocoder to generate speech audio. For the first
combination, we used Mel+SP for the CDVAE model and Mel for the vocoder. For
the second combination, we used Mel+SSL for the CDVAE model and Mel for the
vocoder. The results are presented in the first and second rows of Table 3.

We also investigated ELVC systems whose vocoders used the SSL features. In the
third row of Table 3, we used SSL+Mel for the CDVAE model and SSL for the
vocoder. To further improve the performance, we fine-tuned the vocoder using
the converted SSL features obtained by the CDVAE model. The results of this
approach are shown in the fourth row of Table 3.

Finally, we evaluated ELVC systems where both the CDVAE model and the vocoder
used SSL+Mel features. The results without and with vocoder fine-tuning are
presented in the fifth and sixth rows of Table 3, respectively. Note that we
had to implement the CDVAE model twice with Mel+SSL and SSL+Mel feature
combinations to obtain the converted Mel and SSL features for the vocoder.

Analyzing the results in the first three rows of Table 3, we observe that
incorporating the SSL features improves the performance of the ELVC task,
especially when using the SSL vocoder to generate the converted audio.
Furthermore, we achieved the best overall performance by fine-tuning the
vocoder with the converted features, as evidenced by the results in the fourth
row of Table 3.

As shown in the fifth and sixth rows of Table 3, the use of cross-domain
features (SSL+Mel) in the vocoder did not improve the performance of the ELVC
task, which is contrary to the results in Table 2. This result may be
attributed to the fact that the Mel and SSL features were generated
independently in two separate operations. Consequently, a simple combination
of Mel and SSL features without further refinement would not yield better
results. Using cross-domain features in the vocoder to improve ELVC
performance will be the focus of our future work.

Table 4: Comparison of ELVC systems with SSL-only features and cross-domain features. In both ELVC systems, the vocoder used the SSL features. Model | MCD | F0 RMSE | F0 CORR
---|---|---|---
VAE(SSL) | 7.39 | 50.01 | 0.015
CDVAE(SSL+Mel) | 7.03 | 51.21 | 0.041

####  3.2.4 Investigating the need for cross-domain features in the ELVC task

In this section, we examine whether it is necessary to exploit cross-domain
features and investigate whether using only SSL features can achieve the best
performance in the ELVC task. Since the CDVAE model requires at least two
types of input features, we used a variational autoencoder (VAE) model for
NLVC pre-training when only the SSL features were used. The other training
steps are the same for CDVAE- and VAE-based systems. As shown in Table 4, we
observed that better performance using cross-domain features compared to using
only the SSL features. The two types of features (Mel and SSL) complemented
each other, and the experimental results confirmed that utilizing cross-domain
features improved the performance for the ELVC task.

####  3.2.5 Subjective listening test

Finally, listening tests were conducted to further validate the effectiveness
of the proposed ELVC system. For intelligibility, the evaluation criteria are
as follows: 5 means that every word in the sentence can be understood; 4 means
that a few words in the sentence cannot be understood, but it does not affect
the understanding of the sentence; 3 means that nearly half of the words in
the sentence can be understood, and the content of the sentence can be roughly
judged; 2 means that only a few words in the sentence can be understood, but
not the whole sentence; and 1 means that the sentence cannot be understood at
all. The mean opinion score (MOS) was used to assess speech quality on a scale
from 1 to 5, with 1 being the worst and 5 being the best. From Table 5, Our
findings confirm that the CDVAE model with SSL+Mel features achieved better
performance in terms of intelligibility compared with using the original
SP+Mel features. By fine-tuning the vocoder, we obtained audio signals with
higher intelligibility and quality scores.

##  4 Conclusions

In this paper, we proposed a novel ELVC system that uses cross-domain
features, that is, a combination of spectral features and SSL representations.
We first demonstrated that, by using cross-domain features, a vocoder could be
trained to achieve better results for the NLVC task. Next, we confirmed that
the cross-domain features could improve the conversion model, leading to
better performance for the ELVC task. Finally, we further improved the overall
performance by fine-tuning the vocoder to match the output of the conversion
model in terms of objective and subjective evaluations. In the future, we will
explore the effectiveness of cross-domain features by using different
conversion and vocoder model architectures. Moreover, we will investigate the
use of SSL features in multimodal ELVC tasks.

Table 5: Subjective evaluation results of ELVC systems with different feature combinations in terms of intelligibility and quality. CDVAE | Vocoder | Intelligibility | Quality
---|---|---|---
SP+Mel | Mel | 1.7 | 3.81
SSL+Mel | SSL | 2.5 | 2.8
SSL+Mel | SSL(FT) | 2.6 | 3.27

## References

  * [1] H. Kameoka, T. Kaneko, K. Tanaka, and N. Hojo, ``Stargan-vc: Non-parallel many-to-many voice conversion using star generative adversarial networks,'' in _Proc. SLT_ , 2018, pp. 266–273.
  * [2] K. Qian, Y. Zhang, S. Chang, X. Yang _et al._ , ``Autovc: Zero-shot voice style transfer with only autoencoder loss,'' in _Proc. ICML_ , 2019, pp. 5210–5219.
  * [3] B. Sisman, J. Yamagishi, S. King, and H. Li, ``An overview of voice conversion and its challenges: From statistical modeling to deep learning,'' _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 29, pp. 132–157, 2020.
  * [4] K. Nakamura, T. Toda, H. Saruwatari, and K. Shikano, ``Speaking-aid systems using gmm-based voice conversion for electrolaryngeal speech,'' _Speech communication_ , vol. 54, no. 1, pp. 134–146, 2012.
  * [5] M. Li, L. Wang, Z. Xu, and D. Cai, ``Mandarin electrolaryngeal voice conversion with combination of gaussian mixture model and non-negative matrix factorization,'' in _2017 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC)_. IEEE, 2017, pp. 1360–1363.
  * [6] P. Malathi, G. Sureshw, and M. Moorthi, ``Enhancement of electrolaryngeal speech using frequency auditory masking and gmm based voice conversion,'' in _2018 Fourth International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB)_. IEEE, 2018, pp. 1–4.
  * [7] K. Kobayashi and T. Toda, ``Electrolaryngeal speech enhancement with statistical voice conversion based on cldnn,'' in _Proc. EUSIPCO_ , 2018, pp. 2115–2119.
  * [8] K. Kobayashi and T. Toda, ``Implementation of low-latency electrolaryngeal speech enhancement based on multi-task cldnn,'' in _2020 28th European Signal Processing Conference (EUSIPCO)_ , 2021, pp. 396–400.
  * [9] T. Dinh, A. Kain, R. Samlan, B. Cao _et al._ , ``Increasing the intelligibility and naturalness of alaryngeal speech using voice conversion and synthetic fundamental frequency.'' in _Proc. Interspeech_ , 2020, pp. 4781–4785.
  * [10] M.-C. Yen, W.-C. Huang, K. Kobayashi, Y.-H. Peng _et al._ , ``Mandarin electrolaryngeal speech voice conversion with sequence-to-sequence modeling,'' in _Proc. ASRU_ , 2021, pp. 650–657.
  * [11] A. Baevski, Y. Zhou, A. Mohamed, and M. Auli, ``wav2vec 2.0: A framework for self-supervised learning of speech representations,'' _in Proc. NIPS_ , vol. 33, pp. 12 449–12 460, 2020.
  * [12] W.-N. Hsu, B. Bolte, Y.-H. H. Tsai, K. Lakhotia _et al._ , ``Hubert: Self-supervised speech representation learning by masked prediction of hidden units,'' _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 29, pp. 3451–3460, 2021.
  * [13] S. Chen, C. Wang, Z. Chen, Y. Wu _et al._ , ``Wavlm: Large-scale self-supervised pre-training for full stack speech processing,'' _IEEE Journal of Selected Topics in Signal Processing_ , vol. 16, no. 6, pp. 1505–1518, 2022.
  * [14] J.-h. Lin, Y. Y. Lin, C.-M. Chien, and H.-y. Lee, ``S2vc: a framework for any-to-any voice conversion with self-supervised pretrained representations,'' in _Proc. Interspeech_ , 2021, pp. 836–840.
  * [15] W.-C. Huang, Y.-C. Wu, and T. Hayashi, ``Any-to-one sequence-to-sequence voice conversion using self-supervised discrete speech representations,'' in _Proc. ICASSP_ , 2021, pp. 5944–5948.
  * [16] A. Hernandez, P. A., Pérez-Toro, and E. Nöth, Orozco-Arroyave _et al._ , ``Cross-lingual self-supervised speech representations for improved dysarthric speech recognition,'' in _Proc. Interspeech_ , 2022, pp. 51–55.
  * [17] L. P. Violeta, W.-C. Huang, and T. Toda, ``Investigating self-supervised pretraining frameworks for pathological speech recognition,'' in _Proc. Interspeech_ , 2022, pp. 41–45.
  * [18] W.-C. Huang, H. Luo, H.-T. Hwang, C.-C. Lo _et al._ , ``Unsupervised representation disentanglement using cross domain features and adversarial learning in variational autoencoder based voice conversion,'' _IEEE Transactions on Emerging Topics in Computational Intelligence_ , vol. 4, no. 4, pp. 468–479, 2020.
  * [19] ``links to pre-trained wavlm model,'' https://github.com/microsoft/unilm/tree/master/wavlm.
  * [20] R. Yamamoto, E. Song, and J.-M. Kim, ``Parallel wavegan: A fast waveform generation model based on generative adversarial networks with multi-resolution spectrogram,'' in _Proc. ICASSP_ , 2020, pp. 6199–6203.
  * [21] M. Huang, ``Development of taiwan mandarin hearing in noise test,'' _Department of speech language pathology and audiology, National Taipei University of Nursing and Health science_ , 2005.
  * [22] M. Morise, F. Yokomori, and K. Ozawa, ``World: a vocoder-based high-quality speech synthesis system for real-time applications,'' _IEICE TRANSACTIONS on Information and Systems_ , vol. 99, no. 7, pp. 1877–1884, 2016\.