# Some formulas for fully degenerate Bernoulli numbers and polynomials

Taekyun Kim  Department of Mathematics, Kwangwoon University, Seoul 139-701,
Republic of Korea<EMAIL_ADDRESS>and  DAE SAN KIM  Department of Mathematics,
Sogang University, Seoul 121-742, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The aim of this paper is to study the fully degenerate Bernoulli polynomials
and numbers, which are a degenerate version of Bernoulli polynomials and
numbers and arise naturally from the Volkenborn integral of the degenerate
exponential functions on $\mathbb{Z}_{p}$. We find some explicit expressions
for the fully degenerate Bernoulli polynomials and numbers in terms of the
degenerate Stirling numbers of the second kind, the degenerate $r$-Stirling
numbers of the second kind and of the degenerate Stirling polynomials. We also
consider the degenerate poly-Bernoulli polynomials and derive explicit
representations for them in terms of the same degenerate Stirling numbers and
polynomials.

###### Key words and phrases:

fully degenerate Bernoulli polynomials; degenerate poly-Bernoulli polynomials;
degenerate Stirling numbers of the second kind; degenerate $r$-Stirling
numbers of the second kind; degenerate Stirling polynomials

###### 2010 Mathematics Subject Classification:

11B68; 11B73; 11B83

##  1\. Introduction

In recent years, studies on degenerate versions of special polynomials and
numbers, which started with the paper of Carlitz in [4], have regained
interests of some mathematicians. They have been done with various tools,
including combinatorial methods, generating functions, umbral calculus,
$p$-adic analysis, probability theory, differential equations, operator
theory, analytic number theory and special functions.

The aim of this paper is to study the fully degenerate Bernoulli polynomials
and numbers (see (3)), which are a degenerate version of Bernoulli polynomials
and numbers. They are different from Carlitz degenerate Bernoulli polynomials
and numbers (see (2)) and arise naturally from the Volkenborn integral of the
degenerate exponential functions on $\mathbb{Z}_{p}$ (see (4)). We find some
explicit expressions for the fully degenerate Bernoulli polynomials and
numbers in terms of the degenerate Stirling numbers of the second kind, the
degenerate $r$-Stirling numbers of the second kind and of the degenerate
Stirling polynomials (see (5), (6), (34)). We also consider the degenerate
poly-Bernoulli polynomials and derive explicit representations for them in
terms of the same degenerate Stirling numbers and polynomials. The degenerate
two variable Fubini polynomials (see (10)) are degenerate versions of the two
variable Fubini polynomials. We note here that the generating function of the
fully degenerate Bernoulli polynomials is the integral over the unit interval
of the generating function of the degenerate two variable Fubini polynomials.
This fact is used repeatedly throughout this paper. For the rest of this
section, we recall some necessary facts that will be used throughout this
paper.

For any $\lambda\in\mathbb{R}$, the degenerate exponential functions are
defined by

(1)
$e_{\lambda}^{x}(t)=\sum_{k=0}^{\infty}\frac{(x)_{k,\lambda}}{k!}t^{k},\quad(\mathrm{see}\
[11,15,22]).$

where $(x)_{0,\lambda}=1,\
(x)_{n,\lambda}=x(x-\lambda)\cdots(x-(n-1)\lambda),\ (n\geq 1)$. Note that
$\lim_{\lambda\rightarrow 0}e_{\lambda}^{x}(t)=e^{xt}$.

For $x=1$, we use notation $e_{\lambda}(t)=e_{\lambda}^{1}(t)$.

In [4], Carlitz introduced the degenerate Bernoulli polynomials given by

(2)
$\frac{t}{e_{\lambda}(t)-1}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}\beta_{n}(x|\lambda)\frac{t^{n}}{n!}.$

When $x=0$, $\beta_{n}(\lambda)=\beta_{n}(0|\lambda)$ are called the
degenerate Bernoulli numbers.

In 2016, a variant of the Carlitz degenerate Bernoulli polynomials, called the
fully degenerate Bernoulli polynomials, is defined by

(3) $\frac{\log(1+\lambda
t)}{\lambda(e_{\lambda}(t)-1)}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}\beta_{n,\lambda}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[12,19]).$

When $x=0$, $\beta_{n,\lambda}=\beta_{n,\lambda}(0)$ are called the fully
degenerate numbers.

Note that $\displaystyle\lim_{\lambda\rightarrow
0}\beta_{n,\lambda}(x)=B_{n}(x),\ (n\geq 0),\displaystyle$ where $B_{n}(x)$
are the ordinary Bernoulli polynomials given by

$\frac{t}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[1-23]).$

Before proceeding further, we would like to explain that
$\beta_{n,\lambda}(x)$ arise naturally as the Volkenborn integral of the
degenerate exponential in (1) on $\mathbb{Z}_{p}$. Let $p$ be a fixed prime
number. Let $\mathbb{Z}_{p},\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ denote the
ring of $p$-adic integers, the field of $p$-adic rational numbers and the
completion of the algebraic closure of $\mathbb{Q}_{p}$, respectively. Here
the $p$-adic norm $|\cdot|_{p}$ is normalized as $|p|_{p}=\frac{1}{p}$.

For any $\mathbb{C}_{p}$-valued uniformly differential function $f$ on
$\mathbb{Z}_{p}$, the Volkenborn integral of $f$ on $\mathbb{Z}_{p}$ is
defined by

$\displaystyle\int_{\mathbb{Z}_{p}}f(x)d\mu(x)=\lim_{n\rightarrow\infty}\frac{1}{p^{n}}\sum_{x=0}^{p^{n}-1}f(x),$

which satisfies the fundamental property

$\displaystyle\int_{\mathbb{Z}_{p}}f(x+1)d\mu(x)-\int_{\mathbb{Z}_{p}}f(x)d\mu(x)=f^{\prime}(0).$

By using this property, for $\lambda,t\in\mathbb{C}_{p}$ with $|\lambda
t|_{p}<p^{-\frac{1}{p-1}}$, we have

(4)
$\displaystyle\int_{\mathbb{Z}_{p}}e_{\lambda}^{x+y}(t)d\mu(y)=\frac{\log(1+\lambda
t)}{\lambda(e_{\lambda}(t)-1)}e_{\lambda}^{x}(t).$

Then we get from (1) and (4) the following Witt type identity for
$\beta_{n,\lambda}(x)$:

$\displaystyle\beta_{n,\lambda}(x)=\int_{\mathbb{Z}_{p}}(x+y)_{n,\lambda}d\mu(y),$

This shows that the fully degenerate Bernoulli polynomials and numbers indeed
arise naturally as $p$-adic integrals on $\mathbb{Z}_{p}$.

It is known that the degenerate Stirling numbers of the second kind are
defined by

(5) $(x)_{n,\lambda}=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(x)_{k},\quad(n\geq
0),\quad(\mathrm{see}\ [12,13,19]),$

where $(x)_{0}=1,\ (x)_{n}=x(x-1)\cdots(x-n+1),\ (n\geq 1)$.

For $r\in\mathbb{Z}$ with $r\geq 0$, the degenerate $r$-Stirling numbers of
the second kind are defined by

(6) $(x+r)_{n,\lambda}=\sum_{k=0}^{n}S_{2,\lambda}^{(r)}(n+r,k+r)(x)_{k},\ \
(n\geq 0),\quad(\mathrm{see}\ [18,20]).$

From (5) and (6), we note that

(7)
$\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}=\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[11]),$

and

(8)
$e_{\lambda}^{r}(t)\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}=\sum_{n=k}^{\infty}S_{2,\lambda}^{(r)}(n+r,k+r)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[11,20]).$

Note that $\displaystyle\lim_{\lambda\rightarrow
0}S_{2,\lambda}(n,k)=S_{2}(n,k)\displaystyle$ and
$\displaystyle\lim_{\lambda\rightarrow
0}S_{2,\lambda}^{(r)}(n,k)=S_{2}^{(r)}(n,k)\displaystyle$, where $S_{2}(n,k)$
and $S_{2}^{(r)}(n+r,k+r)$ are respectively the Stirling numbers of the second
kind and the $r$-Stirling numbers of the second kind. It is well known that
two variable Fubini polynomials are defined by

(9)
$\frac{1}{1-x(e^{t}-1)}e^{yt}=\sum_{n=0}^{\infty}F_{n}(x|y)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[8,9]).$

When $y=0$, $F_{n}(x|0)=F_{n}(x)$ are called the Fubini polynomials,.

From (9), we note that

$F_{n}(x|r)=\sum_{k=0}^{n}x^{k}k!S_{2}(n+r,k+r),\quad(\mathrm{see}\ [8,9]),$

where $n,r\in\mathbb{Z}$ with $n,r\geq 0$.

In [10], the degenerate two variable Fubini polynomials are defined by

(10)
$\frac{1}{1-x(e_{\lambda}(t)-1)}e_{\lambda}^{y}(t)=\sum_{n=0}^{\infty}F_{n,\lambda}(x|y)\frac{t^{n}}{n!}.$

Note that

(11)
$F_{n,\lambda}(x|y)=\sum_{m=0}^{n}\sum_{k=0}^{m}\binom{n}{m}S_{2,\lambda}(m,k)k!x^{k}(y)_{n-m,\lambda},\quad(\mathrm{see}\
[10,16]),$

where $n$ is a nonnegative integer.

When $y=0$, $F_{n,\lambda}(x)=F_{n,\lambda}(x|0)$ are called the degenerate
Fubini polynomials (see [16]).

##  2\. Some formulas for fully degenerate Bernoulli numbers and polynomials

From (7), we have

(12) $\displaystyle\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}e_{\lambda}^{j}(t)$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(j)_{n,\lambda}\bigg{)}\frac{t^{n}}{n!}.$

Thus, we get

(13)
$S_{2,\lambda}(n,k)=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(j)_{n,\lambda},\quad(n,k\geq
0).$

From (8), we note that

(14)
$\sum_{n=k}^{\infty}S_{2,\lambda}^{(r)}(n+r,k+r)\frac{t^{n}}{n!}=e_{\lambda}^{r}(t)\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}=\sum_{n=0}^{\infty}\bigg{(}\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(r+j)_{n,\lambda}\bigg{)}\frac{t^{n}}{n!}.$

Comparing the coefficients on both sides of (14), we have

(15)
$S_{2,\lambda}^{(r)}(n+r,k+r)=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(r+j)_{n,\lambda},\quad(n,k\geq
0).$

By (10), we get

(16)
$\frac{1}{1+y(e_{\lambda}(t)-1)}=\sum_{n=0}^{\infty}F_{n,\lambda}(-y)\frac{t^{n}}{n!}.$

Thus, we have

(17) $F_{n,\lambda}(-y)=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(-1)^{k}k!y^{k},$

and

(18)
$\int_{0}^{x}F_{n,\lambda}(-y)dy=\sum_{k=0}^{n}S_{2,\lambda}(n,k)(-1)^{k}\frac{k!}{k+1}x^{k+1}.$

Now, we observe that

(19)
$\displaystyle\frac{1}{e_{\lambda}(t)-1}\log\Big{(}1+x(e_{\lambda}(t)-1)\Big{)}$
$\displaystyle=\sum_{n=0}^{\infty}\int_{0}^{x}F_{n,\lambda}(-y)dy\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}S_{2,\lambda}(n,k)\frac{k!}{k+1}x^{k+1}(-1)^{k}\bigg{)}\frac{t^{n}}{n!}.$

Let us take $x=1$ in (19). Then we have

(20) $\frac{1}{\lambda(e_{\lambda}(t)-1)}\log(1+\lambda
t)=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}S_{2,\lambda}(n,k)\frac{k!}{k+1}(-1)^{k}\bigg{)}\frac{t^{n}}{n!}.$

By (3) and (20), we get

(21)
$\beta_{n,\lambda}=\sum_{k=0}^{n}\frac{k!}{k+1}S_{2,\lambda}(n,k)(-1)^{k},\quad(n\geq
0).$

From (13) and (21), we have the following theorem.

######  Theorem 1.

For $n\geq 0$, we have

$\beta_{n,\lambda}=\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}k!S_{2,\lambda}(n,k)=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}(j)_{n,\lambda}.$

Let $\triangle$ be the difference operator with $\triangle f(x)=f(x+1)-f(x)$.
Then we have

(22) $\triangle^{k}f(x)=\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}f(j+x),\quad(k\geq
0).$

Let us take $f(x)=(x)_{n,\lambda},\ (n\geq 0)$. Then we have

(23)
$\displaystyle\triangle^{k}(0)_{n,\lambda}=\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(j)_{n,\lambda}=k!S_{2,\lambda}(n,k).$

Therefore, by Theorem 1 and (23), we obtain the following corollary.

######  Corollary 2.

For $n\geq 0$, we have

$\beta_{n,\lambda}=\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}\triangle^{k}(0)_{n,\lambda}.$

Let $r$ be a nonnegative integer. Then, by (10), we get

(24)
$\frac{e_{\lambda}^{r}(t)}{e_{\lambda}(t)-1}\log\Big{(}1+x(e_{\lambda}(t)-1)\Big{)}=\sum_{n=0}^{\infty}\int_{0}^{x}F_{n,\lambda}(-y|r)dy\frac{t^{n}}{n!}.$

From (8) and (10), we note that

(25) $\displaystyle\sum_{n=0}^{\infty}F_{n,\lambda}(-y|r)\frac{t^{n}}{n!}$
$\displaystyle=\frac{1}{1+y(e_{\lambda}(t)-1)}e_{\lambda}^{r}(t)$
$\displaystyle=\sum_{k=0}^{\infty}(-1)^{k}y^{k}k!\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}e_{\lambda}^{r}(t)$
$\displaystyle=\sum_{k=0}^{\infty}(-1)^{k}y^{k}k!\sum_{n=k}^{\infty}S_{2,\lambda}^{(r)}(n+r,k+r)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(-1)^{k}y^{k}k!S_{2,\lambda}^{(r)}(n+r,k+r)\bigg{)}\frac{t^{n}}{n!}.$

By comparing the coefficients on both sides of (25), we get

(26)
$F_{n,\lambda}(-y|r)=\sum_{k=0}^{n}(-1)^{k}y^{k}k!S_{2,\lambda}^{(r)}(n+r,k+r).$

From (24) and (26), we can derive the following:

(27)
$\displaystyle\frac{e_{\lambda}^{r}(t)}{e_{\lambda}(t)-1}\log\Big{(}1+x(e_{\lambda}(t)-1)\Big{)}=\sum_{n=0}^{\infty}\int_{0}^{x}F_{n,\lambda}(-y|r)dy\frac{t^{n}}{n}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(-1)^{k}\frac{x^{k+1}}{k+1}k!S_{2,\lambda}^{(r)}(n+r,k+r)\bigg{)}\frac{t^{n}}{n!}.$

Let us take $x=1$ in (27). Then, by (3), (15) and (27), we get

(28) $\displaystyle\sum_{n=0}^{\infty}\beta_{n,\lambda}(r)\frac{t^{n}}{n!}$
$\displaystyle=\frac{\log(1+\lambda t)}{\lambda(e_{\lambda}(t)-1)}\cdot
e_{\lambda}^{r}(t)$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(-1)^{k}\frac{1}{k+1}k!S_{2,\lambda}^{(r)}(n+r,k+r)\bigg{)}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(-1)^{k}\frac{k!}{k+1}\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}(r+j)_{n,\lambda}\bigg{)}\frac{t^{n}}{n!}.$

Now, the next theorem follows from (28).

######  Theorem 3.

For $n\geq 0$, we have

$\displaystyle\beta_{n,\lambda}(r)=\sum_{k=0}^{n}(-1)^{k}\frac{1}{k+1}k!S_{2,\lambda}^{(r)}(n+r,k+r)=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}(r+j)_{n,\lambda}.$

Let us take $f(x)=(x)_{n,\lambda},\ (n\geq 0),$ in (22). Then we have

(29)
$\triangle^{k}(r)_{n,\lambda}=\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(j+r)_{n,\lambda},\quad(n\geq
0).$

Therefore, by (29), we obtain the following corollary.

######  Corollary 4.

For $n\geq 0$, we have

$\beta_{n,\lambda}(r)=\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}\triangle^{k}(r)_{n,\lambda}.$

Now, we observe from (14) that

(30)
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(j+r)_{n,\lambda}\frac{t^{n}}{n!}$
$\displaystyle=e_{\lambda}^{r}(t)\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}$
$\displaystyle=\sum_{m=0}^{\infty}(r)_{m,\lambda}\frac{t^{m}}{m!}\sum_{l=k}^{\infty}S_{2,\lambda}(l,k)\frac{t^{l}}{l!}$
$\displaystyle=\sum_{n=k}^{\infty}\bigg{(}\sum_{l=k}^{n}\binom{n}{l}S_{2,\lambda}(l,k)(r)_{n-l,\lambda}\bigg{)}\frac{t^{n}}{n!}.$

By comparing the coefficients on both sides of (30), we obtain the following
theorem.

######  Theorem 5.

For $k,n\geq 0$, we have

$\displaystyle\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}(j+r)_{n,\lambda}$
$\displaystyle=(-1)^{k}k!\sum_{l=k}^{n}\binom{n}{l}S_{2,\lambda}(l,k)(r)_{n-l,\lambda}$
$\displaystyle=(-1)^{k}\triangle^{k}(r)_{n,\lambda}.$

From (1), (7) and (8), we note that

(31)
$S_{2,\lambda}^{(r)}(n+r,k+r)=\sum_{l=k}^{n}\binom{n}{l}S_{2,\lambda}(l,k)(r)_{n-l,\lambda},\quad(n,k\geq
0),$

and

(32)
$(x+y)_{n,\lambda}=\sum_{l=0}^{n}\binom{n}{l}(x)_{l,\lambda}(y)_{n-l,\lambda},\quad(n\geq
0).$

By (32), we get

$\displaystyle\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}(r+j)_{n,\lambda}$
$\displaystyle=\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}\sum_{l=0}^{n}\binom{n}{l}(r)_{n-l,\lambda}(j)_{l,\lambda}$
(33)
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}(r)_{n-l,\lambda}\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}(j)_{l,\lambda}$
$\displaystyle=\sum_{l=0}^{n}\binom{n}{l}(r)_{n-l,\lambda}S_{2,\lambda}(l,k)k!(-1)^{k}.$

Thus, for $n,k\geq 0$ with $n\geq k$, we have

$\sum_{j=0}^{k}\binom{k}{j}(-1)^{j}(r+j)_{n,\lambda}=\sum_{l=k}^{n}\binom{n}{l}(r)_{n-l,\lambda}(-1)^{k}k!S_{2,\lambda}(l,k).$

The degenerate Stirling polynomials are defined by

(34)
$\displaystyle\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}e_{\lambda}^{x}(t)=\sum_{n=k}^{\infty}S_{2,\lambda}(n,k|x)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[14])$

where $k$ is nonnegative integer.

Then, we easily get

(35) $\displaystyle S_{2,\lambda}(n,k|x)$
$\displaystyle=\sum_{l=k}^{n}\binom{n}{l}S_{2,\lambda}(l,k)(x)_{n-l,\lambda},\quad(n\geq
0)$
$\displaystyle=\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}(l+x)_{n,\lambda},\quad(\mathrm{see}\
[14]).$

From (25), we note that

(36) $F_{n,\lambda}(-y|x)=\sum_{k=0}^{n}(-1)^{k}y^{k}k!S_{2,\lambda}(n,k|x).$

Thus, by (36), we get

(37)
$\displaystyle\frac{e_{\lambda}^{x}(t)}{e_{\lambda}(t)-1}\log\Big{(}1+z(e_{\lambda}(t)-1)\Big{)}$
$\displaystyle=\sum_{n=0}^{\infty}\int_{0}^{z}F_{n,\lambda}(-y|x)dy\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(-1)^{k}\frac{z^{k+1}}{k+1}k!S_{2,\lambda}(n,k|x)\bigg{)}\frac{t^{n}}{n!}.$

Let us take $z=1$ in (37). Then we have

(38) $\displaystyle\sum_{n=0}^{\infty}\beta_{n,\lambda}(x)\frac{t^{n}}{n!}$
$\displaystyle=\frac{\log(1+\lambda
t)}{\lambda(e_{\lambda}(t)-1)}e_{\lambda}^{x}(t)=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}k!S_{2,\lambda}(n,k|x)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}(-1)^{k}\frac{k!}{k+1}\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}(l+x)_{n,\lambda}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (38), we obtain the following theorem.

######  Theorem 6.

For $n\geq 0$, we have

$\displaystyle\beta_{n,\lambda}(x)=\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}k!S_{2,\lambda}(n,k|x)=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}(l+x)_{n,\lambda}.$

Note that

(39)
$\triangle^{k}(x)_{n,\lambda}=\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}(j+x)_{n,\lambda}.$

Hence, by (39), we get

$\beta_{n,\lambda}(x)=\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}\triangle^{k}(x)_{n,\lambda},\quad(n\geq
0).$

In [13,17,19], the degenerate polylogarithm function of index $k$ is defined
by

(40)
$\mathrm{Li}_{k,\lambda}(x)=\sum_{n=1}^{\infty}\frac{(-\lambda)^{n-1}(1)_{n,1/\lambda}}{(n-1)!n^{k}}x^{n},\quad(k\in\mathbb{Z}).$

Let $\log_{\lambda}t$ be the compositional inverse of $e_{\lambda}(t)$. Then
we have

(41)
$\log_{\lambda}(1+t)=\sum_{n=1}^{\infty}\lambda^{n-1}(1)_{n,1/\lambda}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[11]).$

By (40) and (41), we get

(42) $\mathrm{Li}_{1,\lambda}(t)=-\log_{\lambda}(1-t).$

The degenerate poly-Bernoulli polynomials of index $k$ are defined by

(43)
$\frac{\mathrm{Li}_{k,\lambda}(1-e_{\lambda}(-t))}{1-e_{\lambda}(-t)}e_{\lambda}^{-x}(-t)=\sum_{n=0}^{\infty}\beta_{n,\lambda}^{(k)}(x)\frac{t^{n}}{n!}.$

Note that

(44)
$\displaystyle\frac{\mathrm{Li}_{1,\lambda}(1-e_{\lambda}(-t))}{1-e_{\lambda}(-t)}e_{\lambda}^{-x}(-t)$
$\displaystyle=\frac{t}{1-e_{\lambda}(-t)}e_{\lambda}^{-x}(-t)$
$\displaystyle=\frac{t}{e_{-\lambda}(t)-1}e_{-\lambda}^{(x+1)}(t)=\sum_{n=0}^{\infty}\beta_{n}(x+1|-\lambda)\frac{t^{n}}{n!},$

where $\beta_{n}(x|\lambda)$ are the Carlitz degenerate Bernoulli polynomials
in (2).

From (43) and (44), we have

(45) $\beta_{n,\lambda}^{(1)}(x)=\beta_{n}(x+1|-\lambda),\quad(n\geq 0).$

Now, we observe that

(46)
$\displaystyle\frac{\mathrm{Li}_{p,\lambda}(1-e_{\lambda}(-t))}{1-e_{\lambda}(-t)}e_{\lambda}^{-x}(-t)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}\frac{1}{k!}\Big{(}e_{-\lambda}(t)-1\Big{)}^{k}e_{-\lambda}^{x-k}(t)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}\sum_{n=k}^{\infty}S_{2,-\lambda}(n,k|x-k)\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{k=0}^{n}\frac{(-\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}S_{2,-\lambda}(n,k|x-k)\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (43), and (46), we obtain the following theorem.

######  Theorem 7.

For $p\in\mathbb{Z}$ and $n\geq 0$, we have

$\beta_{n,\lambda}^{(p)}(x)=\sum_{k=0}^{n}\frac{(-\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}S_{2,-\lambda}(n,k|x-k).$

From (35), we note that

(47) $\displaystyle S_{2,-\lambda}(n,k|x-k)$
$\displaystyle=\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}(l+x-k)_{n,-\lambda}$
$\displaystyle=\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}(x-l)_{n,-\lambda}.$

Therefore, by Theorem 7 and (47), we obtain the following theorem.

######  Theorem 8.

For $p\in\mathbb{Z}$ and $n\geq 0$, we have

$\beta_{n,\lambda}^{(p)}(x)=\sum_{k=0}^{n}\frac{(-\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}(x-l)_{n,-\lambda}.$

Note that

(48) $\displaystyle\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}(x-l)_{n,-\lambda}$
$\displaystyle=\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}\sum_{j=0}^{n}\binom{n}{j}(-1)^{j}(x)_{n-j,-\lambda}(l)_{j,\lambda}$
$\displaystyle=\sum_{j=0}^{n}\binom{n}{j}(-1)^{j}(x)_{n-j,-\lambda}\sum_{l=0}^{k}\binom{k}{l}(-1)^{l}(l)_{j,\lambda}$
$\displaystyle=\sum_{j=0}^{n}\binom{n}{j}(-1)^{j}(x)_{n-j,-\lambda}k!(-1)^{k}S_{2,\lambda}(j,k).$

Therefore, by Theorem 8 and (48), we obtain the following theorem.

######  Theorem 9.

For $n\geq 0$ and $p\in\mathbb{Z}$, we have

$\beta_{n,\lambda}^{(p)}(x)=\sum_{k=0}^{n}\frac{(\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}\sum_{j=0}^{n}\binom{n}{j}(-1)^{j}S_{2,\lambda}(j,k)(x)_{n-j,-\lambda}.$

From (15), we note that

(49)
$\displaystyle\sum_{n=k}^{\infty}S_{2,\lambda}^{(r)}(n+r,k+r)\frac{t^{n}}{n!}$
$\displaystyle=e_{\lambda}^{r}(t)\frac{1}{k!}\Big{(}e_{\lambda}(t)-1\Big{)}^{k}$
$\displaystyle=\sum_{n=k}^{\infty}\bigg{(}\sum_{j=k}^{n}S_{2,\lambda}(j,k)(r)_{n-j,\lambda}\binom{n}{j}\bigg{)}\frac{t^{n}}{n!}.$

For $r\in\mathbb{N}\cup\\{0\\}$, by Theorem 9 and (49), we get

(50) $\displaystyle\beta_{n,\lambda}^{(p)}(-r)$
$\displaystyle=\sum_{k=0}^{n}\frac{(\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}\sum_{j=0}^{n}\binom{n}{j}(-1)^{j}S_{2,\lambda}(j,k)(-r)_{n-j,-\lambda}$
$\displaystyle=(-1)^{n}\sum_{k=0}^{n}\frac{(\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}\sum_{j=0}^{n}\binom{n}{j}S_{2,\lambda}(j,k)(r)_{n-j,\lambda}$
$\displaystyle=(-1)^{n}\sum_{k=0}^{n}\frac{(\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}S_{2,\lambda}^{(r)}(n+r,k+r).$

Therefore, by (50), we obtain the following theorem.

######  Theorem 10.

For $r,p\in\mathbb{Z}$ with $r\geq 0$, we have

$\displaystyle\beta_{n,\lambda}^{(p)}(-r)$
$\displaystyle=(-1)^{n}\sum_{k=0}^{n}\frac{(\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}\sum_{j=0}^{n}\binom{n}{j}S_{2,\lambda}(j,k)(r)_{n-j,\lambda}$
$\displaystyle=(-1)^{n}\sum_{k=0}^{n}\frac{(\lambda)^{k}(1)_{k+1,1/\lambda}}{(k+1)^{p}}S_{2,\lambda}^{(r)}(n+r,k+r).$

##  3\. Conclusion

In recent years, a lot of mathematicians have explored degenerate versions of
many special numbers and polynomials, which were initiated by Carlitz. Their
interests were not only in combinatorial and arithmetical properties but also
in applications to differential equations, identities of symmetry and
probability theory. These degenerate versions include the degenerate Stirling
numbers of the first and second kinds, degenerate Bernoulli numbers of the
second kind and degenerate Bell numbers and polynomials.

In this paper, we studied the fully degenerate Bernoulli polynomials and
numbers that are a degenerate version of Bernoulli polynomials and numbers,
and different from the Carlitz degenerate Bernoulli polynomials and numbers.
We found several explicit expressions for the fully degenerate Bernoulli
polynomials and numbers in terms of the degenerate Stirling numbers of the
second, the degenerate $r$-Stirling numbers of the second kind and of the
degenerate Stirling polynomials. We also investigated the degenerate poly-
Bernoulli polynomials and derived explicit representations in terms of the
same degenerate Stirling numbers and polynomials.

It is one of our future research projects to continue to study various
degenerate versions of many special polynomials and numbers with applications
to physics, science and engineering in mind.

## References

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# Quadrupole Moment of a Magnetically Confined Mountain on an Accreting
Neutron Star in General Relativity

Pedro H. B. Rossetto School of Physics, University of Melbourne, Parkville,
VIC 3010, Australia  ARC Centre of Excellence for Gravitational Wave Discovery
(OzGrav), University of Melbourne, Parkville, VIC 3010, Australia  Jörg
Frauendiener Department of Mathematics & Statistics, University of Otago, 730
Cumberland Street, Dunedin 9016, New Zealand  Andrew Melatos School of
Physics, University of Melbourne, Parkville, VIC 3010, Australia  ARC Centre
of Excellence for Gravitational Wave Discovery (OzGrav), University of
Melbourne, Parkville, VIC 3010, Australia

###### Abstract

General relativistic corrections are calculated for the quadrupole moment of a
magnetically confined mountain on an accreting neutron star. The hydromagnetic
structure of the mountain satisfies the general relativistic Grad-Shafranov
equation supplemented by the flux-freezing condition of ideal
magnetohydrodynamics, as in previous calculations of the magnetic dipole
moment. It is found that the ellipticity and hence the gravitational wave
strain are $4\%$ to $12\%$ smaller than in the analogous Newtonian system. The
direct contribution of the magnetic field to the nonaxisymmetric component of
the stress-energy tensor is shown to be negligible in accreting systems such
as low-mass X-ray binaries.

Accretion — General relativity — Gravitational waves — Gravitational wave
sources — LMXBs — Magnetic fields — Magnetohydrodynamics — Neutron stars —
Relativistic fluid dynamics

##  1 Introduction

Fast-spinning deformed neutron stars are prime candidates for the emission of
continuous gravitational waves (CWs) (Riles, 2023). The Laser Interferometer
Gravitational-Wave Observatory (LIGO), Virgo, and Kamioka Gravitational Wave
Detector (KAGRA) routinely conduct targeted narrowband searches for CWs from
known pulsars in the LIGO Scientific Collaboration data (Abbott, 2022; Abbott
et al., 2022a), directed searches in supernova remnants (Abbott et al., 2021)
and all-sky searches for unknown CW sources (Abbott et al., 2022b). No CW
signal has been found so far. It is important to model from first principles
the formation mechanisms of non-axisymmetric stellar deformations, such as
‘mountains’, which are static in the star’s rotating frame, in order to
predict the detectability of the associated CW signal. Proposed mechanisms
include elastic stresses in the crust (Ushomirsky et al., 2000; Gittins et
al., 2020; Gittins & Andersson, 2021), single and repeated crustal fracture
(Giliberti et al., 2019; Giliberti & Cambiotti, 2022; Kerin & Melatos, 2022),
the Lorentz force in the magnetized interior (Bonazzola & Gourgoulhon, 1996;
Haskell et al., 2008) and the Lorentz force in the surface layers of polar
mountains confined magnetically on accreting neutron stars (Melatos & Phinney,
2001; Payne & Melatos, 2004; Melatos & Payne, 2005; Rossetto et al., 2023),
the latter of which are the focus of this paper.

Magnetically confined mountains are formed by the accretion of conducting
plasma onto the magnetic polar cap of a neutron star in systems like low-mass
X-ray binaries (LMXBs). Several features have been incorporated previously
into the theoretical model, including mountain stability (Vigelius & Melatos,
2008; Mukherjee & Bhattacharya, 2012; Mukherjee et al., 2013), Ohmic and
thermal relaxation (Vigelius & Melatos, 2009; Suvorov & Melatos, 2019), core
superconductivity (Passamonti & Lander, 2014; Sur & Haskell, 2021), mountain
sinking (Choudhuri & Konar, 2002; Wette et al., 2010), various equations of
state (Priymak et al., 2011; Mukherjee, 2017), triaxial configurations (Singh
et al., 2020), higher magnetic multipole moments and toroidal fields (Suvorov
& Melatos, 2020; Fujisawa et al., 2022). Recently, Rossetto et al. (2023)
formulated the problem of magnetically confined mountains on neutron stars in
general relativity. They found that relativistic corrections change the
hydromagnetic equilibrium of the accreted matter, diminishing the screening of
the magnetic moment three-fold when compared to the Newtonian formulation.

In this paper, we discuss how general relativistic corrections modify the mass
quadrupole moment of a magnetically confined mountain and hence affect the
emission of gravitational radiation. In Section 2, we summarize the
relativistic model of the system and the numerical method. In Section 3, we
calculate the ellipticity of the star, breaking down the contributions from
various physical effects. We contrast our findings with the respective
Newtonian results. Finally, in Section 4, we estimate the amplitude of the
associated gravitational radiation and discuss its detectability.

##  2 Hydromagnetic equilibrium of a polar mountain

We employ the same theoretical framework presented by Rossetto et al. (2023).
In this section, we review the equations that describe the hydromagnetic
structure of a magnetically confined mountain produced by magnetic burial in
general relativity. We assume that the neutron star generates a background
Schwarzschild metric and that the accreted matter and magnetic field adjust to
this curved spacetime. In the usual $(t,r,\theta,\phi)$ coordinates, the
metric is given by111In this paper we use geometrized units, i.e. $G=c=1$. For
electromagnetic quantities, we use Gaussian units, and we renormalise the
magnetic field by a factor of $\sqrt{4\pi}$.

$\differential{s}^{2}=-e^{2\Phi}\differential{t}^{2}+e^{-2\Phi}\differential{r}^{2}+r^{2}\differential{\theta}^{2}+r^{2}\sin^{2}\theta\differential{\phi}^{2},$
(1)

with

$\Phi(r)=\frac{1}{2}\ln(1-\frac{2M_{*}}{r}),$ (2)

where $M_{*}$ is the gravitational mass of the neutron star. We assume the
fiducial value $M_{*}=1.4M_{\odot}$.

The hydromagnetic equilibrium structure of the mountain is determined by the
balance between the Lorentz force, thermal pressure gradient, and gravity.
That is, we look for motionless, steady-state configurations satisfying
conservation of the energy-momentum tensor, $\nabla_{a}T^{ab}=0$. The tensor
$T^{ab}$ has contributions from the accreted mass $M_{a}$, assumed to be a
perfect fluid and a perfect conductor, and the magnetic field of the star
$B^{a}=F^{ab}u_{b}$, where $F_{ab}$ is the Faraday tensor and $u^{b}$ is the
fluid’s four-velocity, with

$T^{ab}=(e+B^{2})u^{a}u^{b}+\frac{1}{2}g^{ab}B^{2}+(g^{ab}+u^{a}u^{b})p-B^{a}B^{b},$
(3)

where $g^{ab}$ is the inverse metric, $e$ is the energy density of the fluid,
$p$ is the pressure, and one has $B^{2}=B_{a}B^{a}$.

Assuming that the magnetic field of the star has no toroidal component and
that the whole system is axisymmetric, the magnetic field can be described by
a scalar function $\psi=\psi(r,\theta)$ according to

$B^{a}=\frac{1}{r^{2}\sin^{2}\theta}\epsilon^{abcd}u_{d}\nabla_{b}\phi\nabla_{c}\psi,$
(4)

where $\epsilon^{abcd}$ is the completely anti-symmetric tensor. We also
assume an isothermal equation of state, given by $p=c_{\rm s}^{2}e$
(Yabushita, 1973; Chavanis, 2008), with $c_{\rm s}$ being the sound speed.
Additionally, we are interested in a static fluid configuration, in which the
four-velocity of the fluid takes the form $u^{a}=e^{-\Phi}\partial^{a}_{t}$.
Under these assumptions, $\nabla_{a}T^{ab}=0$ reduces to the relativistic
isothermal Grad-Shafranov equation

$\Delta^{*}\psi=-F^{\prime}(\psi)\exp[-(1+c_{\rm s}^{-2})(\Phi-\Phi_{0})],$
(5)

where $\Delta^{*}$ is the relativistic Grad-Shafranov operator given by

$\Delta^{*}\psi=\frac{1}{r^{2}\sin^{2}\theta}\left\\{\partialderivative{r}\left[\left(1-\frac{2M_{*}}{r}\right)\partialderivative{\psi}{r}\right]+\frac{\sin\theta}{r^{2}}\partialderivative{\theta}\left(\frac{1}{\sin\theta}\partialderivative{\psi}{\theta}\right)\right\\},$
(6)

and $F(\psi)$ is a function related to the fluid pressure at the altitude of
the reference potential $\Phi_{0}$.

Payne & Melatos (2004) developed a recipe to calculate the function $F(\psi)$
uniquely in the Newtonian theory, by connecting the pre- and post-accretion
states through the flux freezing condition of ideal magnetohydrodynamics.
Rossetto et al. (2023) adapted the recipe to general relativity, obtaining

$F(\psi)=\left(\derivative{M}{\psi}\right)^{1+c_{\rm
s}^{2}}\left(\frac{2\pi}{c_{\rm
s}^{2}}\int_{C}r\sin\theta|\nabla\psi|^{-1}e^{-\left(\Phi-\Phi_{0}\right)/c_{\rm
s}^{2}}\differential{s}\right)^{-(1+c_{\rm s}^{2})}.$ (7)

In (7), $\differential{M}/\differential{\psi}$ specifies how accreted mass is
distributed along flux tubes labelled uniquely by $\psi$. Its functional form
is arbitrary and depends on the unknown details of the accretion process at
the disk-magnetosphere boundary. Qualitatively, however, it should express the
fact, that accreting matter is funnelled preferentially onto field lines
centered on the magnetic pole. One phenomenological functional form with this
property is (Payne & Melatos, 2004; Priymak et al., 2011)

$M(\psi)=\frac{M_{a}}{2}\frac{1-e^{-\psi/\psi_{a}}}{1-e^{-\psi_{*}/\psi_{a}}},$
(8)

with $\psi_{*}=\psi(R_{*},0)$, where $R_{*}$ is the radius of the neutron star
and $\psi_{a}$ is the flux surface that touches the inner edge of the
accretion disk.

Numerical solutions to equations (5) and (7) are obtained iteratively.
Initially, we choose the dipole $\psi_{\rm d}$ as a solution and numerically
compute the integral in (7). With this value of $F(\psi)$, and thus
$F^{\prime}(\psi)$, we compute the source term of equation (5) and solve for
$\psi$ via a successive over-relaxation scheme. We recompute the integral in
(7) using the updated solution for $\psi$, and the scheme proceeds iteratively
until the difference between the solutions $\psi^{n}$ and $\psi^{n+1}$
obtained in two successive steps is less than a specified threshold, viz.
$(\psi^{n}-\psi^{n+1})/\psi^{n}<10^{-5}$ over the grid. The iterative nature
of the numerical scheme resembles qualitatively, but is not the same as, the
true temporal evolution of the system (Mouschovias, 1974), in which an initial
dipole evolves to a final distorted equilibrium configuration. Figure 1 shows
an example of a hydromagnetic equilibrium generated by the solver, for $M_{\rm
a}=10^{-5}M_{\odot}$ and $b=\psi_{*}/\psi_{\rm a}=10$. Panel 1(a) shows the
general relativistic equilibrium state for the magnetic field lines ($\psi$
level curves) as solid curves, and the Newtonian equivalent as dashed curves.
The system reaches a deformed equilibrium state for both theories, but the
deformation in the relativistic case is smaller. Panel 1(b) confirms that the
numerical $\psi$ residuals decrease with iteration number in a controlled
fashion; that is, the numerical solution converges.

Figure 1: General relativistic versus Newtonian hydromagnetic equilibria for
$M_{a}=10^{-5}M_{\odot}$ and $b=\psi_{*}/\psi_{a}=10$. (a) Magnetic field
lines (contours of $\psi$). Solid lines are used for the relativistic solution
and dashed for the Newtonian. (b) Residuals of the numerical solver. The
dashed red line indicates a tolerance of $10^{-5}$.

##  3 Quadrupole moment and ellipticity

In the linear regime for the generation of gravitational waves (Maggiore,
2007), the mass quadrupole source term is proportional to the second time
derivative of the mass quadrupole moment, defined by

$Q_{ij}=\int_{V^{\prime}}T^{00}\left({x^{\prime}}_{i}{x^{\prime}}_{j}-\frac{1}{3}{x^{\prime}}^{k}{x^{\prime}}_{k}\delta_{ij}\right)\differential[3]x^{\prime},$
(9)

where $x^{\prime i}$ is a spatial position vector (assuming a small source),
$V^{\prime}$ is the volume of the source and $\delta_{ij}$ is the identity
tensor. Note that equation (9) is usually termed the mass quadrupole moment to
distinguish it from the current quadrupole moment, which is proportional to
the fluid velocity and generates current quadrupole gravitational radiation.
However, it is important to recognise that $T^{00}$ in equation (9) includes
contributions from the energy density in the magnetic field as well as the
mass density of the fluid. We discuss this in Section 3.2. From equation (3),
$T^{00}$ is given by

$T^{00}=e+\frac{B^{2}}{2}.$ (10)

Furthermore, $e$ can be decomposed as $e=\rho(1+\varepsilon)$ where $\rho$ is
the rest-mass density of the fluid and $\varepsilon$ is its specific internal
energy. We keep both terms in (10) for the sake of generality and compare
their magnitudes below.

In the triaxial model of a rigidly rotating neutron star, the gravitational
wave amplitude is given by (Maggiore, 2007)

$h_{0}=\frac{16\pi^{2}I_{0}f^{2}\epsilon}{D},$ (11)

where $I_{0}=(2/5)M_{*}{R_{*}}^{2}$ is the moment of inertia of the star
before accretion, $f$ is the rotation frequency, $D$ is the distance to the
source and $\epsilon$ is the (mass-energy) ellipticity of the star given by

$\epsilon=\frac{\pi}{I_{0}}\int_{V^{\prime}}T^{00}r^{4}\sin\theta(3\cos^{2}\theta-1)\differential{r}\differential{\theta}.$
(12)

We also define the material and magnetic ellipticities $\epsilon_{e}$ and
$\epsilon_{B}$ in terms of the same integral as in (12) but replacing $T^{00}$
with $e$ and $B^{2}/2$ respectively to give
$\epsilon=\epsilon_{e}+\epsilon_{B}$. In equilibrium, $e$ and $B^{2}$ can be
related to the flux-function $\psi$ output from the numerical solver, with

$\displaystyle e=\frac{F(\psi)}{c_{\rm s}^{2}}\exp[-(1+c_{\rm
s}^{-2})(\Phi-\Phi_{0})],$ (13) $\displaystyle
B^{2}=\frac{1}{r^{2}\sin^{2}\theta}\left[\frac{1}{r^{2}}\left(\partialderivative{\psi}{\theta}\right)^{2}+e^{2\Phi}\left(\partialderivative{\psi}{r}\right)^{2}\right].$
(14)

Figure 2: Hydromagnetic structure of the general relativistic equilibrium
obtained for $M_{a}=10^{-5}M_{\odot}$ and $b=\psi_{*}/\psi_{a}=10$. (a) Rest-
mass density $\rho$. (b) Magnitude of the magnetic field $B=\sqrt{B^{2}}$.

###  3.1 Total ellipticity

Figure 2 shows the equilibrium density and magnitude of the magnetic field of
the system, plotted in meridional cross-section. We see that the mass
distribution is centered at the pole of the star, as expected. In contrast,
the magnitude of the magnetic field is highest around the colatitude
$25^{\circ}$. This happens because of the flux freezing condition. The frozen-
in magnetic field is dragged equatorward with the fluid, as the accreted
material spreads under its own weight (Payne & Melatos, 2004; Rossetto et al.,
2023). Both these configurations indicate a non-zero ellipticity of the star,
as per equations (10)–(14).

Figure 3 displays calculations of the total ellipticity $\epsilon$ of the
star. Panel 3(a) shows the convergence achieved by the numerical solver in
both the relativistic and Newtonian scenarios. The ellipticity is plotted as a
function of iteration number, not as a function of time; the reader is
reminded that the Grad-Shafranov equation applies in the steady state. We
observe that $\epsilon$ relaxes over $\sim 50$ iterations for both scenarios,
decreasing by $\approx 4\%$ and $\approx 2\%$ during the iterative process in
the Newtonian and relativistic cases respectively.

Panel 3(b) shows the converged ellipticity for different values of accreted
mass in the range $1\times 10^{-6}\leq M_{\rm a}/M_{\odot}\leq 3\times
10^{-5}$. The upper limit of this mass range is set by the convergence of the
numerical solver. For $M_{\rm a}\gtrsim 10^{-5}M_{\odot}$, closed magnetic
bubbles form, whose field lines do not connect to the stellar surface, and
this phenomenon hinders the convergence of the solution. In the Newtonian
regime, this happens for $M_{\rm a}/M_{\odot}\gtrsim 10^{-4}b$ (Payne &
Melatos, 2004) and, in the relativistic regime, for $M_{\rm
a}/M_{\odot}\gtrsim 3\times 10^{-4}b$. This difference is consistent with
smaller magnetic deformation in the relativistic case. For $1\times
10^{-6}\leq M_{\rm a}/M_{\odot}\leq 3\times 10^{-5}$, a simple linear
regression on the curves shown in Panel 3(b) yields:

$\displaystyle\epsilon_{\rm N}=0.57M_{\rm a}/M_{\odot}+3\times 10^{-7},$ (15)
$\displaystyle\epsilon_{\rm GR}=0.54M_{\rm a}/M_{\odot}+1\times 10^{-7},$ (16)

where $\epsilon_{\rm N}$ is the Newtonian ellipticity222This agrees with
Newtonian values in the literature, cf. Figure 2(a) in Melatos & Payne
(2005)., displayed in blue in Figure 3, and $\epsilon_{\rm GR}$ is the
relativistic ellipticitiy, displayed in orange. Equations (15) and (16)
reflect the reduction of the ellipticity in the general relativistic case. The
simple regressions (15) and (16) must be modified for $M_{\rm a}\geq 3\times
10^{-5}M_{\odot}$, when the magnetic burial process saturates; cf. Equation
(8) of Melatos & Payne (2005). In fractional terms, the relativistic result is
$\sim 4\%$ to $\sim 12\%$ smaller than its Newtonian counterpart.

Figure 3: (a) Numerical convergence for $M_{\rm a}=10^{-5}M_{\odot}$ and
$b=\psi_{*}/\psi_{a}=3$: total ellipticity versus iteration number of the
numerical scheme. (b) Total ellipticity versus accreted mass in the range
$10^{-6}\leq M_{\rm a}/M_{\odot}\leq 3\times 10^{-5}$, for
$b=\psi_{*}/\psi_{a}=3$, in the Newtonian and relativistic scenarios. In both
panels, the blue and orange curves correspond to the Newtonian and
relativistic calculations respectively.

###  3.2 Purely magnetic ellipticity

The magnetic field contributes directly to the quadrupole moment (9) and the
ellipticity (12) of the star through the $B^{2}$ term in equation (10). This
contribution is distinct from the magnetically induced nonaxisymmetry in the
accreted matter, which enters through the term $\rho$ in (10). In the case of
magnetically confined mountains, the accreted matter deforms the magnetic
field and pushes it equatorwards (Payne & Melatos, 2004; Rossetto et al.,
2023), increasing the magnetic pressure $B^{2}/2$ locally. Figure 4 displays
how the magnetic ellipticity increases with $M_{\rm a}$. We find that the
ellipticity generated directly by the magnetic field is of the order
$10^{-11}$, which is $\sim 10^{6}$ times smaller than the ellipticity
generated by the accreted mass (cf. Figure 3).

Recent studies (Hacyan, 2017; Nazari & Roshan, 2020; Contopoulos et al., 2023)
investigate the gravitational radiation emitted by a purely dipolar bar magnet
rotating in vacuo about an axis inclined to its magnetic axis. They find that
the $B^{2}$ contribution is generally low but may be detectable by future-
generation gravitational-wave detectors in the case of highly-magnetised fast-
spinning neutron stars. In our simulations, we can recover the magnetic dipole
solution in vacuo by taking the accreted mass to be low ($M_{\rm a}\sim
10^{-9}M_{\odot}$). In the latter regime, we obtain $\epsilon_{B}\sim
10^{-15}$. This value is consistent with the literature, cf. equation (13) of
Hacyan (2017), equation (46) of Nazari & Roshan (2020) for $B\sim
10^{12}\,\rm{G}$, and equation (57) of Contopoulos et al. (2023).

Figure 4: Direct contribution of the magnetic field to the ellipticity of the
star. Magnetic ellipticity $\epsilon_{B}$ versus accreted mass in the range
$10^{-6}\leq M_{\rm a}/M_{\odot}\leq 3\times 10^{-5}$, for
$b=\psi_{*}/\psi_{a}=3$, in the Newtonian (blue curve) and relativistic
(orange curve) scenarios.

###  3.3 Compactness

The relativistic corrections to the mountain structure depend on the
compactness $M_{*}/R_{*}$ of the star. These corrections are greater for more
compact objects. Figure 5 shows the effect of the compactness of the star for
$M_{\rm a}=10^{-5}M_{\odot}$ and $b=10$. Panel 5(a) shows the effects of
varying $M_{*}$ in the range $1\leq M_{*}/M_{\odot}\leq 2.5$ with
$R_{*}=10\,\mathrm{km}$ fixed, i.e. compactness in the range $0.15\leq
M_{*}/R_{*}\leq 0.37$. We can see from panel 5(a) that as the mass and
compactness grow, the Newtonian and relativistic results deviate. The
percentage difference reaches $\sim 30\%$ at $M_{*}/R_{*}=2.50$.

Panel 5(b) shows the effects of varying $R_{*}$ in the range
$10\,\mathrm{km}\leq R_{*}\leq 15\,\mathrm{km}$ with $M_{*}=1.4M_{\odot}$
fixed, i.e. compactness in the range $0.20\leq M_{*}/R_{*}\leq 0.31$. Panel
5(b) shows that the difference between the Newtonian and relativistic results
($\sim 4\%$ in fractional terms) stays practically unchanged for different
values of compactness. This result indicates, that the compactness influences
$\epsilon$ mainly through $M_{*}$.

Figure 5: Effect of the compactness on the star’s ellipticity for $M_{\rm
a}=10^{-5}M_{\odot}$ and $b=3$. (a) Ellipticity versus neutron star mass in
the range $1\leq M_{*}/M_{\odot}\leq 2.5$ for $R_{*}=10\,\mathrm{km}$ fixed.
(b) Ellipticity versus neutron star radius in the range $10\,\mathrm{km}\leq
R_{*}\leq 15\,\mathrm{km}$ for $M_{*}=1.4M\odot$ fixed. In both panels, the
blue and orange curves correspond to the Newtonian and relativistic
calculations respectively.

##  4 Discussion

The present work focuses on the modifications arising from including general
relativity in models of the formation of magnetically confined mountains and
hence the emission of continuous gravitational waves by accreting neutron
stars. The baseline for comparison is the Newtonian model calculated by
Melatos & Payne (2005) for an isothermal equation of state. The modifications
brought by general relativity to the hydromagnetic structure of the mountain
are discussed in detail by Rossetto et al. (2023). The main effect is to
reduce the ellipticity in the relativistic case, as can be seen in equations
(15) and (16).

The characteristic gravitational wave strain $h_{0}$ from a star with a static
mountain with ellipticity $\epsilon$ can be written as

$h_{0}=4.7\times
10^{-26}\left(\frac{\epsilon}{10^{-6}}\right)\left(\frac{I_{0}}{1.1\times
10^{38}\mathrm{kg\,m^{2}}}\right)\left(\frac{f}{100\,\rm{Hz}}\right)^{2}\left(\frac{1\,\rm{kpc}}{r}\right).$
(17)

From the ellipticity results shown in Figure 3 we obtain strains between $\sim
10^{-26}$ and $\sim 10^{-25}$. The relativistic results are $4\%$ to $12\%$
smaller than the Newtonian ones, depending on $M_{\rm a}$. The purely magnetic
contribution from $\epsilon_{B}$ is negligible, generating a wave strain of
the order $10^{-32}$.

Several important effects already modeled in the Newtonian literature must
also be incorporated into the relativistic model. A summary of these effects
can be found in Section 1. The unmodeled effect that probably affects the
system the most is the choice of the equation of state. In the Newtonian
literature (Priymak et al., 2011), a polytropic or piecewise-polytropic
equation of state reduced the quadrupole between three and four orders of
magnitude compared to an isothermal equation of state. Furthermore, dynamical
effects also need to be included. The Grad-Shafranov equation cannot model
these effects, as it is an equilibrium equation. Dynamical effects are
responsible for the stability of the accreted mountain (Payne & Melatos, 2007;
Vigelius & Melatos, 2008; Mukherjee et al., 2013; Mukherjee, 2017).

## Acknowledgements

This work received financial support from the Australian Research Council
Centre of Excellence for Gravitational Wave Discovery (OzGrav), through
project number CE170100004. This work was partially supported financially by
the Catalyst Fund provided by the New Zealand Ministry of Business, Innovation
and Employment and administered by the Royal Society Te Apārangi; and the
Division of Science of the University of Otago, New Zealand.

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# Mechanism of intermetallic charge transfer and bond disproportionation
in BiNiO3 and PbNiO3 revealed by hard X-ray photoemission spectroscopy

Tatsuya Yamaguchi  Department of Physics and Electronics, Graduate School of
Engineering, Osaka Metropolitan University 1-1 Gakuen-cho, Nakaku, Sakai,
Osaka 599-8531, Japan.  Mizuki Furo  Department of Physics and Electronics,
Graduate School of Engineering, Osaka Metropolitan University 1-1 Gakuen-cho,
Nakaku, Sakai, Osaka 599-8531, Japan.  Yuki Sakai  Kanagawa Institute of
Industrial Science and Technology (KISTEC), Ebina 243-0435, Japan  Materials
and Structures Laboratory, Institute of Innovative Research, Tokyo Institute
of Technology 4259 Nagatsuta, Midori, Yokohama, 226-8501, Japan  Takumi
Nishikubo  Kanagawa Institute of Industrial Science and Technology (KISTEC),
Ebina 243-0435, Japan  Materials and Structures Laboratory, Institute of
Innovative Research, Tokyo Institute of Technology 4259 Nagatsuta, Midori,
Yokohama, 226-8501, Japan  Hajime Hojo  Materials and Structures Laboratory,
Tokyo Institute of Technology 4259 Nagatsuta, Midori, Yokohama, 226-8501,
Japan.  Masaki Azuma  Materials and Structures Laboratory, Institute of
Innovative Research, Tokyo Institute of Technology 4259 Nagatsuta, Midori,
Yokohama, 226-8501, Japan  Kanagawa Academy of Science and Technology, KSP,
3-2-1 Sakado, Takatsu-ku, Kawasaki City, Kanagawa, 213-0012, Japan.  Kengo Oka
Department of Applied Chemistry, Faculty of Science and Engineering, Chuo
University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan.  Department of
Applied Chemistry, Faculty of Science and Engineering, Kindai University,
3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan.  Daisuke Mori  Department
of Chemistry, Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima-
ku, Tokyo 171-8588, Japan.  Department of Applied Chemistry, Graduate School
of Engineering, Mie University, Tsu, Mie 514-8507, Japan.  Yoshiyuki Inaguma
Department of Chemistry, Faculty of Science, Gakushuin University, 1-5-1
Mejiro, Toshima-ku, Tokyo 171-8588, Japan.  Masaichiro Mizumaki  Institute of
Industrial Nanomaterials, Kumamoto University 2-39-1 Kurokami Chuo-ku Kumamoto
860-8555, Japan.  Kento Yamamoto  Department of Applied Physics, Waseda
University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan.  Jan Kuneš
Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria  Department of
Condensed Matter Physics, Faculty of Science, Masaryk University, Kotlářská 2,
611 37 Brno, Czechia  Takashi Mizokawa  Department of Applied Physics, Waseda
University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan.  Atsushi Hariki
Department of Physics and Electronics, Graduate School of Engineering, Osaka
Metropolitan University 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan.

###### Abstract

Perovskites with Bi or Pb on the A-site host a number of interesting and yet
to be understood phenomena such as negative thermal expansion in BiNiO3. We
employ hard X-ray photoemission spectroscopy of Ni 2$p$ core-level as well as
valence band to probe the electronic structure of BiNiO3 and PbNiO3. The
experimental results supported by theoretical calculations using dynamical
mean-field theory reveal essentially identical electronic structure of the
Ni–O subsystem typical of Ni2+ charge-transfer insulators. The two materials
are distinguished by filling of the Bi(Pb)–O antibonding states in the
vicinity of the Fermi level, which is responsible for the Bi
disproportionation in BiNiO3 at ambient pressure and absence of similar
behavior in PbNiO3. The present experiments provide evidence for this
conclusion by revealing the presence/absence of Bi/Pb $6s$ states at the top
of the valence band in the two materials.

##  I Introduction

The A-sites of typical AMO3 perovskite transition-metal oxides (TMOs) are
occupied by divalent alkaline earth metal or trivalent Y or rare-earth ions,
which act as electron donors and form ionic bonds with oxygen. Perovskites
with Pb or Bi on the A-site are different. The Pb or Bi ions play a more
active role thanks to their ability to form covalent bonds with oxygen. Both
Pb and Bi belong to so-called valence skippers, ions that prefer to adopt the
6$s^{0}$ (Pb4+, Bi5+) and 6$s^{2}$ (Pb2+, Bi3+) valence configurations while
avoiding the 6$s^{1}$ (Pb3+, Bi4+) configuration. This behavior can be
modelled with an attractive on-site interaction, negative $U$, within the 6$s$
shell Matsuura  _et al._ (2022). The origin of the negative $U$ has been
attributed to covalent bonding plus electron-lattice coupling Anderson (1975)
or non-linear non-local screening Varma (1988).

Electronic structure calculations Paul  _et al._ (2019); Hariki  _et al._
(2021) have shown that valence skipping in perovskite TMOs is caused by
Pb(Bi)–O hybridization, which leads to formation of a relatively narrow anti-
bonding band. Its energy, determined by bonding-antibonding splitting, is
sensitive to small changes of the Pb(Bi)–O bond lengths and therefore strongly
coupled to the lattice An and Pickett (2001). Formal 6$s^{0}$ and 6$s^{2}$
valence configurations correspond to situation where the antibonding band is
located above and below the Fermi level $E_{F}$, respectively. The $6s^{1}$ or
partially filled configurations would amount to pinning of the antibonding
band at $E_{F}$. Such states/structures are often unstable towards small
changes of the Pb(Bi)–O lengths, which cause the antibonding states to move
away from $E_{F}$. This is accompanied by transfer of electrons/holes to other
orbitals, e.g., TM $3d$ ones. Alternatively, Pb(Bi)–O bond-lengths
disproportionation splits the antibonding band away from $E_{F}$ and leads to
redistribution of charge among inequivalent Pb(Bi) ions. While the negative
$U$ picture also leads to valence skipping it does not account for the
geometrical constraints and correlations between Pb(Bi)–O bond lengths due to
the A-sites sharing their oxygen neighbors and vice versa. This affects
possible ordering patterns of 6$s^{2}$ and 6$s^{0}$ sites on the lattice Paul
_et al._ (2019); Hariki  _et al._ (2021) as well as dynamics of the 6$s$
electrons.

The Pb(Bi)–O hybridization picture can be probed by studying the Pb(Bi) $6s$
spectra. To this end, we have measured the valence and core-level X-ray
photoemission spectroscopy (XPS) spectra of closely related PbNiO3 and BiNiO3
and analyzed them with the help of electronic structure calculations using
local-density approximation (LDA) combined with dynamical mean-field theory
(DMFT) method. We employ hard X-ray photoemission spectroscopy (HAXPES) which
has large photoionization cross sections (CSs) for the Pb(Bi) 6s orbitals.

At ambient pressure BiNiO3 is a good insulator Ishiwata  _et al._ (2005) while
PbNiO3 shows a semiconducting behavior Inaguma  _et al._ (2011). BiNiO3
undergoes a pressure-driven metal-insulator transition (MIT) at the critical
pressure 3.5 GPa Azuma  _et al._ (2011, 2007), which has been identified as a
new type of MIT driven by electron transfer from Bi to Ni and interpreted
using the negative $U$ picture Naka  _et al._ (2016); Kojima  _et al._ (2016).
At ambient pressure, BiNiO3 is an insulator with two distinct Bi sites and
formal valence Bi${}^{3+}_{0.5}$Bi${}^{5+}_{0.5}$Ni2+O3 Ishiwata  _et al._
(2002). At high pressure, it is a metal with formal valence Bi3+Ni3+O3 Azuma
_et al._ (2011, 2007). PbNiO3 takes formal valence Pb4+Ni2+O3.

##  II Methods

###  A Experimental method

Polly-crystalline samples of BiNiO3 and PbNiO3 were prepared under high
pressure Inaguma  _et al._ (2011). HAXPES measurements for valence band and Ni
2$p$ core-level were performed at 300 K with photon energy of 7939.8 eV at the
undulator beamlines BL47XU of SPring-8. For these measurements, hemispherical
photoelectron analyzers (SES2002/R-4000) were used. The polycrystalline
samples were fractured in-situ for the XPS measurements. The binding energies
were calibrated using Au 4f7/2 peak (84.0 eV) and Fermi edge of gold reference
samples. The total energy resolution was 280 meV.

###  B Computational method

The valence-band and core-level XPS spectra are simulated using the LDA+DMFT
method Metzner and Vollhardt (1989); Georges  _et al._ (1996); Kotliar  _et
al._ (2006). The computation starts with a standard LDA calculation using
Wien2k Blaha  _et al._ (2018) for the experimental crystal structures of
PbNiO3 ($Pbn2_{1}$ space group) Sakai  _et al._ (2019) and BiNiO3
($P\overline{1}$ space group) Ishiwata  _et al._ (2002). In BiNiO3, two
inequivalent Bi sites Ishiwata  _et al._ (2002) are present as shown in Fig.
1. The two Bi sites, with the shortest Bi–O bonds of 2.21 Å and 2.03 Å, are
referred to as the “lone” and “pair” Bi ions hereafter.

Next, the LDA bands are mapped onto a tight-binding model spanning the Bi (Pb)
5$d$, 6$s$, 6$p$, Ni 3$d$, 4$s$, O 2$s$ and 2$p$ bands Kuneš  _et al._ (2010);
Mostofi  _et al._ (2014). Although the low-energy electronic structures of the
two compounds are governed by the Ni 3$d$, O 2$p$ and Bi (Pb) 6$s$ states, the
other orbitals can be observed in the HAXPES spectra due to hybridization. The
tight-binding model is augmented with a local electron-electron interaction
within the Ni 3$d$ shell. We employ Hubbard $U$ and Hund’s $J$ parameters of
($U$, $J$) = (7.0 eV, 1.1 eV) Anisimov  _et al._ (1991). The double-counting
correction $\mu_{\rm dc}$ Kotliar  _et al._ (2006); Karolak  _et al._ (2010)
is set to reproduce the experimental valence-band and core-level spectra as in
Refs. Hariki  _et al._ (2017); Higashi  _et al._ (2021); Hariki  _et al._
(2022). The details can be found in the Supplementary Material (SM) sm . The
strong-coupling continuous-time quantum Monte Carlo impurity solver Werner
_et al._ (2006); Boehnke  _et al._ (2011); Hafermann  _et al._ (2012) of the
Anderson impurity model (AIM) is used to compute Ni 3$d$ self-energies
$\Sigma(i\omega_{n})$. The valence-band spectra and hybridization densities
$\Delta(\omega)$ on the real frequency axis are calculated from the local
self-energy $\Sigma(\omega)$ analytically continued to the real-frequency axis
using maximum entropy method Wang  _et al._ (2009); Jarrell and Gubernatis
(1996).

The Ni 2$p$ core-level XPS spectra are calculated using a configuration-
interaction impurity solver Hariki  _et al._ (2017); Ghiasi  _et al._ (2019);
Winder  _et al._ (2020) from the DMFT AIM augmented with the Ni 2$p$ core
orbitals and their interaction with the valence electrons. The hybridization
densities $\Delta(\omega)$ are represented by 25 discrete levels (per spin and
orbital).

Figure 1: Crystal structure of BiNiO3 Ishiwata  _et al._ (2002) visualized by
VESTA Momma and Izumi (2011).

##  III Results and Discussion

Figure 2: The Ni 2$p$ HAXPES experimental data (middle) and the LDA+DMFT AIM
calculations (bottom) for BiNiO3 in the AFI phase (blue) and PbNiO3 (red) in
the PMI phase. The Ni 2$p_{3/2}$ (2$p_{1/2}$) main line (ML) and satellite
(ST) are indicated. The spectral broadening is considered using Gaussian of
300 meV and Lorentian of 300 meV (HWHM) in the calculated results. The Ni 2$p$
XPS data of NiO Taguchi  _et al._ (2008) is shown for comparison (top). Figure
3: The LDA+DMFT Ni 2$p$ XPS spectra calculated for BiNiO3 (top) and PbNiO3
(bottom) in the PMI (red) and AFI (blue) phases.

###  A Core-level XPS

We start our discussion with Ni 2$p$ core-level XPS spectra. Figure 2 shows
the Ni 2$p$ XPS spectra of BiNiO3 and PbNiO3 measured at room temperature.
Overall spectral shapes are similar to those of typical Ni2+ oxides, such as
NiO Taguchi  _et al._ (2008) and La2NiO4 Eisaki  _et al._ (1992). Both the
2$p_{1/2}$ (884 – 868 eV) and 2$p_{3/2}$ (868 – 852 eV) contributions consist
of a main line (ML) and a satellite (ST). The Ni 2$p_{3/2}$ ML in BiNiO3
exhibits a shoulder on the higher binding energy side, while it is suppressed
in PbNiO3. The difference in the spectra for the two materials is well
captured by the calculations. Given that the Néel temperatures of 300 K in
BiNiO3 and 225 K in PbNiO3 Ishiwata  _et al._ (2002); Inaguma  _et al._
(2011), we show the simulations for the antiferromagnetic insulating (AFI)
phase for BiNiO3 and the paramagnetic insulating (PMI) phase for PbNiO3 in
Fig. 2. Figure 3 shows that the spectra calculated within the same phase (i.e.
AFI or PMI) are very similar in the two compounds, suggesting essentially
identical electronic structures of Ni–O subsystems in BiNiO3 and PbNiO3.

We briefly discuss the physical origin of the observed core-level spectra.
Sudden creation of a localized core hole triggers a dynamical charge response
in the XPS final states, which amounts to charge transfer (CT) from
surrounding ligands as well as distant Ni ions to the excited Ni site,
conventionally refereed to as CT screening. The two components in the ML
originate from two CT screening processes: one from nearest-neighboring
ligands and the other, at lower binding energy, from distant Ni ions, often
called local and nonlocal screening, respectively van Veenendaal and Sawatzky
(1993); Hariki  _et al._ (2013); Taguchi  _et al._ (2008). The nonlocal CT
screening in Ni2+ oxides Hariki  _et al._ (2013, 2017) is less efficient in
PMI than in AFI due to Pauli blocking, which explains the suppression of low-
energy peak in PMI.

Figure 4: The LDA+DMFT valence-band spectral intensities of (a) BiNiO3 in the
AFI phase and (b) PbNiO3 in the PMI phase. The experimental valence-band
spectra (blue) of BiNiO3 Wadati  _et al._ (2007); Nishikubo  _et al._ (2019)
and PbNiO3 are shown for comparison. (c) The difference of the experimental
HAXPES intensities for BiNiO3 and PbNiO3.

###  B Valance-band XPS

Figure 4 shows the experimental and LDA+DMFT valence-band spectra of BiNiO3
Nishikubo  _et al._ (2019) and PbNiO3. Soft X-ray XPS (SXPS) spectrum of
Bi0.95La0.05NiO3 measured at photon energy of 600 eV Wadati  _et al._ (2007)
is shown for comparison. The difference between the spectra obtained with hard
and soft X-rays reflects the different photon-energy dependencies of the
photoionization CSs for the constituent orbitals. The empirical CS values
summarized in Table. 1 show that while Ni 3$d$ orbitals have large CS at 500
eV, at 7940 eV the Bi (Pb) CSs are dominant. This makes the hard X-rays well
suited to study the Bi and Pb $6s$ states.

We identify features labeled as $A$, $B$, $C$, $D$ in the HAXPES spectra of
BiNiO3 and PbNiO3. In addition, a feature $E$, enhanced in soft X-ray spectra,
is attributed to the Ni 3$d$ orbital contributions and matches well the
position of the Ni 3$d$ peak in the LDA+DMFT result. The peak $A$ corresponds
to Bi(Pb) 6$s$ states – bonding state in the language of Bi(Pb)–O
hybridization. In order to identify the origin of features $B$, $C$, $D$, we
weight the LDA+DMFT valence-band intensities with the CSs at 7940 eV in Fig.
5, where underestimation ($\sim 0.5$ eV) of the binding energy in the LDA+DMFT
spectra was corrected. Although the Bi (Pb) 5$d$ and 6$p$ states are
positioned far below and above $E_{F}$, see SM sm , their CSs are extremely
large in the hard X-ray regime, Table. 1. This together with a tiny
hybridization with the valence band makes them apparent in the HAXPES spectra
in the valence region. The simulated spectra in Fig. 5 show that the Bi or Pb
6$p$ states are important for the feature $B$, while the $5d$ states give rise
to the feature $C$. The shallower Pb 5$d$ states could explain the enhanced
intensity of the feature $C$ in PbNiO3 compared to that in BiNiO3.

|  | 8000 eV |  | 500 eV
---|---|---|---|---
| $\beta$ |  $\sigma$ [b] |  $d\sigma/d\Omega$ [b/sr] |  $\sigma$ [b] |  $d\sigma/d\Omega$ [b/sr]
Ni 3$d$ | 0.322 | 0.741 | 0.078 | 19.969 | 1.589
Ni 4$s$ | 1.960 | 6.855 | 1.615 | 2.569 | 0.204
Bi 5$d$ | 1.198 | 68.690 | 12.014 | 14.282 | 1.137
Bi 6$s$ | 1.890 | 25.050 | 5.761 | 2.658 | 0.212
Bi 6$p$ | 1.731 | 14.340 | 3.116 | 1.569 | 0.125
Pb 5$d$ | 1.184 | 59.350 | 10.317 | 13.116 | 1.044
Pb 6$s$ | 1.898 | 21.495 | 4.957 | 2.338 | 0.186
Pb 6$p$ | 1.653 | 13.475 | 2.845 | 0.983 | 0.078
O 2$s$ | 1.923 | 6.795 | 1.581 | 11.510 | 0.916
O 2$p$ | 0.053 | 0.080 | 0.007 | 2.214 | 0.176
Table 1: Subshell photoionization cross section $\sigma$ at 8000 eV (hard
X-ray) and 500 eV (soft X-ray) for relevant orbitals Trzhaskovskaya and
Yarzhemsky (2018); Trzhaskovskaya  _et al._ (2001, 2002). The differential
cross section $d\sigma/d\Omega$ at 8000 eV is calculated taking the dipole
parameter $\beta$ for the angular distribution into account. To simulate the
experimental geometry in the previous SXPS study in Ref. Wadati  _et al._
(2007), $\beta$ is not considered for $d\sigma/d\Omega$ at 500 eV. Figure 5:
The LDA+DMFT valence-band spectral intensities of (a) BiNiO3 and (b) PbNiO3
weighted with the CSs listed in Table. 1. The filled curves show the
contributions of the Bi (Pb) 5$d$, 6$s$ and 6$p$ states. The simulated
spectral intensities are shifted by $-$0.5 eV for both compounds. The
experimental HAXPES data (blue) of BiNiO3 Nishikubo  _et al._ (2019) and
PbNiO3 are shown for comparison.  Figure 6: The valence-band spectra in (a)
BiO3 and (b) PbO3 sublattice computed by removing the Ni 3$d$ orbitals in the
LDA+DMFT valence-band models in Figs. 4ab. The Bi and Pb $6s$ spectral weights
are magnified by the factor of 3. The solid (dashed) line in BiNiO3 represents
lone (pair) Bi site contribution.

The main qualitative difference between the HAXPES spectra of BiNiO3 and
PbNiO3 is the enhanced spectral weight at the top of the valence band. The
analysis of Fig. 5 links it to the presence of Bi $6s$ orbital character,
while Pb $6s$ states are missing at the valence band top in PbNiO3. To isolate
the physics of Bi(Pb) $6s$–O 2$p$ bonding, we present, in Fig. 6, the Bi(Pb)
6$s$ and O 2$p$ spectral densities obtained with the Ni–O and Ni–Bi(Pb)
hopping artificially switched off in the LDA+DMFT valence-band models in Fig.
4. Besides the quantitative differences consisting in shallower $6s$ states
for lighter Pb and the resulting somewhat stronger covalent character of the
Pb–O bond, there is a qualitative difference reflecting the disproportionation
of the Bi sites. This leads to splitting of the anti-bonding band into states
associated with the lone Bi site at lower and pair Bi sites at higher
energies, consistent with the previous GGA+$U$ study Paul  _et al._ (2019). No
splitting of the anti-bonding band takes place in PbNiO3. This picture,
necessarily with broader and richer structures in the $6s$ spectra, is
preserved when Ni is fully included. The CT gap of Ni–O subsystem overlaps
with lone–pair anti-bonding gap in BiNiO3 and O $2p$–anti-bonding gap in
PbNiO3 in Fig. 4.

##  IV Conclusions

We have presented a comparative study of two A-site active nickelates BiNiO3
and PbNiO3. The valence and Ni 2$p$ core-level photoemission experiments
supported by LDA+DMFT calculations reveal essentially identical electronic
structures of Ni–O subsystems in the two compounds, characteristic of Ni2+ in
charge-transfer system such as NiO Sawatzky and Allen (1984); Fujimori  _et
al._ (1984); Kuneš  _et al._ (2007). The main difference between the two
compounds is derived from the Pb(Bi) $6s$ orbitals. The calculations find
strong Pb(Bi)–O hybridization leading to bonding–antibonding splitting of
almost 10 eV, with the anti-bonding states located within the charge-transfer
gap. In PbNiO3 electron counting places the Fermi level below the anti-bonding
band, while in BiNiO3 the Fermi level falls in the middle of the anti-bonding
band. This situation in the latter is resolved by disproportionation of the
Bi–O bonds, which gives rise to two distinct Bi sites, with antibonding states
above and below the Fermi level.

Our main experimental result is the observation of the Bi–O antibonding states
at the top of the valence band, which confirms the above mechanism of Bi
disproportionation. We conclude our presentation by commenting on the
relationship of the present description to the negative $U$ model of valence
skippers. The negative $U$ model qualitatively explains presence (absence) of
A-site disproportionation in BiNiO3 (PbNiO3). It can be viewed as an effective
model for the Pb(Bi)–O antibonding states obtained by the process of
integrating out the O states. However, in this process also non-local
interaction terms are expected to appear, e.g., due to O site having several
Pb(Bi) neighbors. Such terms determine the pattern of disproportionated Bi
ions on the lattice.

A question also arises about the negative $U$ treatment of possible
superconductivity in materials with valence skippers. Strong covalent bonds in
systems with antibonding states at the Fermi level result in a strong
electron-phonon coupling An and Pickett (2001). As in other such materials
superconductivity is competing with a structural instability. The capability
of the negative $U$ model to capture this competition appears questionable and
we find the that full treatment of Pb(Bi)–O bonding without the effective $6s$
interaction $U$ preferable.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers 21K13884, 21H01003,
23K03324, 23H03816, 23H03817 (A.H.), by JP19H0562 and JSTCREST (JPMJCR22O1)
(M.A.), by the project No. CZ.02.01.01/00/22_008/0004572 of the Programme
Johannes Amos Commenius (J.K.). The synchrotron-radiation experiments were
performed at SPring-8 with the approval of the Japan Synchrotron Radiation
Research Institute (2017B1721). The authors would like to thank Dr. Shigenori
Ueda for the contribution to HAXPES measurements at BL15XU, SPring8, in the
early stage of this work.

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