# The $(l,r)$-Stirling numbers: a combinatorial approach.

Belbachir Hacène  USTHB, Faculty of Mathematics, RECITS Laboratory
P.O. Box 32
El Alia, 16111, Bab Ezzouar, Algiers
Algeria<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Djemmada Yahia
USTHB, Faculty of Mathematics, RECITS Laboratory
P.O. Box 32
El Alia, 16111, Bab Ezzouar, Algiers
Algeria<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This work deals with a new generalization of $r$-Stirling numbers using
$l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of
both kinds. We study various properties of these numbers using combinatorial
interpretations and symmetric functions. Also, we give a limit representation
of the multiple zeta function using $(l,r)$-Stirling of the first kind.

###### Key words and phrases:

Permutations, Set partitions, Stirling numbers, Symmetric functions,
$r$-Stirling numbers.

###### 2010 Mathematics Subject Classification:

Primary 11B73, 11B83; Secondary 05A05, 05A18, 05E05.

##  1\. Introduction

Let $\sigma$ be a permutation of the set $[n]=\\{1,2,\dots,n\\}$ having $k$
cycles $c_{1},c_{2},\dots,c_{k}$. A cycle leaders set of $\sigma$, denoted
$cl(\sigma)$, is the set of the smallest elements on their cycles, i. e.

$cl(\sigma)=\\{\min c_{1},\min c_{2},\dots,\min c_{k}\\}.$

As the same way, let $\pi$ be a partition of the set $[n]=\\{1,2,\dots,n\\}$
into $k$ blocks $b_{1},b_{2},\dots,b_{k}$. A block leaders set of $\pi$,
denoted $bl(\pi)$, is the set of the smallest elements on their blocks, i. e.

$bl(\pi)=\\{\min b_{1},\min b_{2},\dots,\min b_{k}\\}.$

######  Example.

  *   * •

For $n=6$, the permutation $\sigma=(13)(245)(6)$ have the set of cycle leaders
$cl(\sigma)=\\{1,2,6\\}$.

  * •

For $n=7$, the partition $\pi=1,2,4|3,5,7|6$ have the set of block leaders
$bl(\pi)=\\{1,3,6\\}$.

It is well known that the Stirling numbers of the first kind, denoted
${n\brack k}$, count the number of all permutations of $[n]$ having exactly
$k$ cycles, and Stirling numbers of the second kind, denoted ${n\brace k}$,
count the number of all partitions of $[n]$ having exactly $k$ blocks.

One of the most interesting generalization of Stirling numbers was the
$r$-Stirling numbers of both kind introduced By Broder [6]. Analogously to the
classical Stirling numbers of both kinds, the author considered that
$r$-Stirling numbers of the first kind ${n\brack k}_{r}$ (resp. the second
kind ${n\brace k}_{r}$) counts the number of permutations $\sigma$ (resp.
partitions $\pi$) having exactly $k$ cycles (resp. $k$ blocks) such that the
$r$ first elements $1,2,\dots,r$ lead.

Dumont, in [8], gives the first interpretation for the ”central factorial”
numbers of the second kind $U(n,k)$ given by the recurrence

(1.1) $U(n,k)=U(n-1,k-1)+k^{2}U(n-1,k),\qquad\text{for }0<k\leq n;$

where $U(n,k)=T(2n,2k)$.

Then, using the notion of quasi-permutations, Foata and Han [9] , showed that
$U(n,k)$ counts the number of pair $(\pi_{1},\pi_{2})$-partitions of $[n]$
into $k$ blocks such that $bl(\pi_{1})=bl(\pi_{2})$.

In this work, we give an extension of the $r$-Stirling numbers of both kinds
with considering $l$-tuple partitions (and permutations) of Dumont’s partition
model [8, 9].

This paper is organized as follows. In Section 2 and Section 4, we introduce
the $(l,r)$-Stirling numbers of both kinds. Some properties are given as
recurrences, orthogonality, generating functions, a relation between
$(l,r)$-Stirling numbers and Bernoulli polynomials via Faulhaber sums and
symmetric functions. In Section 7, we show the relations between multiple-zeta
function and the $(l,r)$-Stirling numbers. Finally, in Section 8, we discuss
some remarks which connect this numbers to the rooks polynomials [3].

##  2\. The $(l,r)$-Stirling numbers of both kinds

Let us consider the following generalization,

######  Definition 2.1.

The $(l,r)$-Stirling number of the first kind ${n\brack k}_{r}^{(l)}$ counts
the number of $l$-tuple of permutations
$(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$ of $[n]$ having exactly $k$ cycles
such that $1,2,\dots,r$ first elements lead, and

$cl(\sigma_{1})=cl(\sigma_{2})=\dots=cl(\sigma_{l}).$

######  Definition 2.2.

The $(l,r)$-Stirling number of the second kind ${n\brace k}_{r}^{(l)}$ counts
the number of $l$-tuple of partitions $(\pi_{1},\pi_{2},\dots,\pi_{l})$ of
$[n]$ having exactly $k$ blocks such that $1,2,\dots,r$ first elements lead,
and

$bl(\pi_{1})=bl(\pi_{2})=\dots=bl(\pi_{l}).$

######  Theorem 2.3.

The $(l,r)$-Stirling numbers of the first satisfy the following recurrences

(2.1) ${n\brack k}_{r}^{(l)}={n-1\brack k-1}_{r}^{(l)}+(n-1)^{l}{n-1\brack
k}_{r}^{(l)},\qquad\text{for }n>r$

and

(2.2) ${n\brack k}_{r}^{(l)}=\frac{1}{(r-1)^{l}}\left({n\brack
k-1}_{r-1}^{(l)}-{n\brack k-1}_{r}^{(l)}\right),\qquad\text{for }n\geq r>1.$

with boundary conditions ${n\brack k}_{r}^{(l)}=0,$ for $n<r$; and ${n\brack
k}_{r}^{(l)}=\delta_{k,r}$, for $n=r$.

###### Proof.

The $(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations of the set $[n]$
having $k$ cycles such that $1,2,\dots,r$ first elements are in distinct
cycles and $cl(\sigma_{1})=cl(\sigma_{2})=\dots=cl(\sigma_{l})$ is either
obtained from:

  * •

Inserting the $nth$ elements after any element in each permutation of
$(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations of the set $[n-1]$
having $k$ cycles such that $1,2,\dots,r$ first elements are in distinct
cycles and $cl(\sigma_{1})=cl(\sigma_{2})=\dots=cl(\sigma_{l})$, hence there
are $(n-1)^{l}{n-1\brack k}_{r}^{(l)}$ choices.

  * •

The $nth$ element forms a cycle in each permutation of
$(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations, the remaining $[n-1]$
have to be $(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permuted in $(k-1)$
cycles under the preceding conditions, hence there are ${n-1\brack
k-1}_{r}^{(l)}$.

This correspondence yields the first recurrence.

For the second recurrence, we use the double counting principle. Let us count
the numbers of $(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations of the
set $[n]$ having $(k-1)$ cycles such that $1,\dots,r-1$ are cycle leaders but
$r$ is not, with $cl(\sigma_{1})=cl(\sigma_{2})=\dots=cl(\sigma_{l})$, this is
either obtained from:

  * •

We count the $(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations of the
set $[n]$ having $(k-1)$ cycles such that $1,\dots,r-1$ are cycle leaders then
we exclude from them the
$(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations having $r$ as cycle
leader. That gives

${n\brack k-1}_{r-1}^{(l)}-{n\brack k-1}_{r}^{(l)},$

  * •

Or we count the $(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations of the
set $[n]$ having $k$ cycles such that $1,\dots,r$ are cycle leaders then we
appending the cycle having $r$ as leader at the end of a cycle having a
smaller leader. We have $(r-1)$ choices to do in each permutation. That gives

$(r-1)^{l}{n\brack k}_{r}^{(l)},$

from the two ways of counting we get the result. ∎

######  Theorem 2.4.

The $(l,r)$-Stirling numbers of the second satisfy the following recurrences

(2.3) ${n\brace k}_{r}^{(l)}={n-1\brace k-1}_{r}^{(l)}+k^{l}{n-1\brace
k}_{r}^{(l)},\qquad\text{for }n>r$

and

(2.4) ${n\brace k}_{r}^{(l)}={n\brace k}_{r-1}^{(l)}-(r-1)^{l}{n-1\brace
k}_{r-1}^{(l)},\qquad\text{for }n\geq r>1.$

with boundary conditions ${n\brace k}_{r}^{(l)}=0$, for $n<r$; and ${n\brace
k}_{r}^{(l)}=\delta_{k,r}$, for $n=r$.

###### Proof.

As in Theorem 2.3, the $(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of the set
$[n]$ into $k$ blocks such that $1,2,\dots,r$ first elements are in distinct
blocks and $bl(\pi_{1})=bl(\pi_{2})=\dots=bl(\pi_{l})$ is either obtained
from:

  * •

Inserting the $nth$ elements in a block of each partition of
$(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of the set $[n-1]$ into $k$
blocks such that $1,2,\dots,r$ first elements are in distinct blocks and
$bl(\pi_{1})=bl(\pi_{2})=\dots=bl(\pi_{l})$, hence there are $k^{l}{n-1\brace
k}_{r}^{(l)}$ choices (the position of the $nth$ element in a block doesn’t
matter).

  * •

The $nth$ element forms a block in each partition of
$(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions, the remaining $[n-1]$ have to be
$(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitioned into $(k-1)$ blocks under the
preceding conditions, hence there are ${n-1\brace k-1}_{r}^{(l)}$.

For the Identity (2.4), we use the double counting principle to count the
numbers of $(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of $[n]$ into $k$
blocks such that $1,2,\dots,(r-1)$ are block leaders but $r$ is not, with
$bl(\pi_{1})=bl(\pi_{2})=\dots=bl(\pi_{l})$, this is either obtained from:

  * •

We count the $(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of the set $[n]$
into $k$ blocks such that $1,\dots,r-1$ are block leaders then we exclude from
them the $(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions having $r$ as block
leader, with $bl(\pi_{1})=bl(\pi_{2})=\dots=bl(\pi_{l})$. That gives

${n\brace k}_{r-1}^{(l)}-{n\brace k}_{r}^{(l)},$

  * •

Or we count the $(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of the set
$[n]$\$\\{r\\}$ into $k$ blocks such that $1,\dots,r-1$ are block leaders then
we include the element $\\{r\\}$ in any block having a smaller leader then
$r$. We have $(r-1)$ choices to do in each partition of
$(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions, that gives

$(r-1)^{l}{n-1\brace k}_{r-1}^{(l)},$

from the two ways of counting we get the result.

∎

######  Remark 2.5.

Using the previous recurrences it is easy to get the following special cases

(2.5) $\displaystyle{n\brack
r}_{r}^{(l)}=r^{l}(r+1)^{l}\cdots(n-2)^{l}(n-1)^{l}=(r^{\overline{n-r}})^{l},\qquad\text{for
}n\geq r$ and (2.6) $\displaystyle{n\brace
r}_{r}^{(l)}=r^{l(n-r)},\qquad\text{for }n\geq r.$

##  3\. Orthogonality of $(l,r)$-Stirling numbers pair

######  Theorem 3.1.

For $n\geq k\geq 0$, for all positive integer $l$, we have the two
orthogonality relations bellow

(3.1) $\sum_{j}{n\brack j}_{r}^{(l)}{j\brace
k}_{r}^{(l)}(-1)^{j}=\left\\{\begin{array}[]{l
r}(-1)^{n}\delta_{n,k},&\qquad\text{for }n\geq r;\\\ &\\\ 0,&\qquad\text{for
}n<r\end{array}\right.$

and

(3.2) $\sum_{j}{j\brack n}_{r}^{(l)}{k\brace
j}_{r}^{(l)}(-1)^{j}=\left\\{\begin{array}[]{l
r}(-1)^{n}\delta_{n,k},&\qquad\text{for }n\geq r;\\\ &\\\ 0,&\qquad\text{for
}n<r.\end{array}\right.$

###### Proof.

Let us start by Identity (3.1). The proof goes by induction on $n$

  * •

For $n<r$ the assertion is obvious.

  * •

For $n=r$,

$\begin{split}\sum_{j}{r\brack j}_{r}^{(l)}{j\brace
k}_{r}^{(l)}(-1)^{j}&={r\brace
k}_{r}^{(l)}(-1)^{r}=(-1)^{r}\delta_{k,r}.\end{split}$

  * •

For $n>r$, Theorem 2.3 and the induction hypothesis implies that

$\begin{split}\sum_{j}{n\brack j}_{r}^{(l)}{j\brace
k}_{r}^{(l)}(-1)^{j}&=\sum_{j}\left({n-1\brack
j-1}_{r}^{(l)}+(n-1)^{l}{n-1\brack j}_{r}^{(l)}\right){j\brace
k}_{r}^{(l)}(-1)^{j}\\\ &=(n-1)^{l}(-1)^{n-1}\delta_{n-1,k}+\sum_{j}{n-1\brack
j-1}_{r}^{(l)}{j\brace k}_{r}^{(l)}(-1)^{j},\\\ \end{split}$

and from Theorem 2.4, we get

$\begin{split}\sum_{j}{n\brack j}_{r}^{(l)}{j\brace
k}_{r}^{(l)}(-1)^{j}&=(n-1)^{l}(-1)^{n-1}\delta_{n-1,k}-(-1)^{n-1}\delta_{n-1,k-1}-(k)^{l}(-1)^{n-1}\delta_{n-1,k}\\\
&=(-1)^{n}\delta_{n,k}.\end{split}$

For the Identity (3.2), we go by induction on $k$ as same as the previous
proof. ∎

##  4\. Properties via symmetric functions

Let $x_{1},x_{2},\dots,x_{n}$ be $n$ random variables. We denote,
respectively, by
$e_{k}(x_{1},x_{2},\dots,x_{n})$ and $h_{k}(x_{1},x_{2},\dots,x_{n})$ the
elementary symmetric function and the complete homogeneous symmetric function
of degree $k$ in $n$-variables given for $n\geq k\geq 1$, by

(4.1) $e_{k}(x_{1},x_{2},\dots,x_{n})=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq
n}x_{i_{1}}\cdots x_{i_{k}}$

and

(4.2) $h_{k}(x_{1},x_{2},\dots,x_{n})=\sum_{1\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{k}\leq n}x_{i_{1}}\cdots x_{i_{k}}.$

In particular
$e_{0}(x_{1},x_{2},\dots,x_{n})=h_{0}(x_{1},x_{2},\dots,x_{n})=\delta_{0,n}$.

The generating functions of the symmetric functions are given by

(4.3) $E(t)=\sum_{k\geq
0}e_{k}(x_{1},x_{2},\cdots,x_{n})t^{k}=\prod_{i=1}^{n}(1+x_{i}t)$

and

(4.4) $H(t)=\sum_{k\geq
0}h_{k}(x_{1},x_{2},\cdots,x_{n})t^{k}=\prod_{i=1}^{n}(1-x_{i}t)^{-1}.$

For more details about symmetric functions we refer readers to [2, 13, 15] and
the references therein.

Let us now give some results linked to the symmetric functions and their
generating functions.

######  Theorem 4.1.

The $(l,r)$-Stirling of the first kind and the elementary symmetric function
are linked as

(4.5) ${n+1\brack n+1-k}_{r}^{(l)}=e_{k}(r^{l},\dots,n^{l}),$

equivalently

(4.6) ${n\brack k}_{r}^{(l)}=e_{n-k}(r^{l},\dots,(n-1)^{l}).$

###### Proof.

It is clear that in each
$(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutation having $(n-k)$ cycles
with $\\{1,\dots,r\\}$ lead, we have $\\{1,2,\dots,r,y_{r+1},\dots,y_{n-k}\\}$
lead a cycle and $\\{x_{1},x_{2},\dots,x_{k}\\}$ elements don’t lead where
$r<y_{r+1}<y_{n-k}<\dots\leq n$ and $r<x_{1}<x_{2}<\dots\leq n$.

To construct all $(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations
having $(n-k)$ cycles where $\\{1,\dots,r\\}$ lead, we proceed as follows

  * •

Construct $(n-k)$ cycles having only one element from $\\{1,2,\dots,r,$
$y_{r+1},\dots,{y_{n-k}}\\}$, i. e.

$\sigma=(1)(2)\dots(r)(y_{r+1})\dots(y_{n-k}),$

  * •

Insert $x_{1}$ after an element of cycles smaller than $x_{1}$, we have
$(x_{1}-1)$ ways of inserting $x_{1}$. Then Insert $x_{2}$ after an element of
cycles smaller than $x_{2}$, we have $(x_{2}-1)$ choices, and so on. We have
$(x_{1}-1)(x_{2}-1)\cdots(x_{k}-1)$ ways to construct a permutation.

  * •

Repeat the process with each permutation
$\sigma\in\\{\sigma_{1},\dots,\sigma_{l}\\}$, so we have
$(x_{1}-1)^{l}(x_{2}-1)^{l}\cdots(x_{k}-1)^{l}$ ways of construction.

  * •

Summing over all possible set of numbers $\\{x_{1},x_{2},\dots,x_{k}\\}$,
hence the total number of ways to construct
$(\sigma_{1},\sigma_{2},\dots,\sigma_{l})$-permutations having $(n-k)$ cycles
with $\\{1,\dots,r\\}$ lead is

$\begin{split}{n\brack n-k}_{r}^{(l)}&=\sum_{r<x_{1}<x_{2}<\cdots\leq
n}(x_{1}-1)^{l}(x_{2}-1)^{l}\cdots(x_{k}-1)^{l}\\\ &=\sum_{r\leq
x_{1}<x_{2}<\cdots<n}x_{1}^{l}x_{2}^{l}\cdots x_{k}^{l}\\\
&=e_{k}(r^{l},\dots,(n-1)^{l}).\end{split}$

∎

######  Theorem 4.2.

The $(l,r)$-Stirling of the first kind and the complete homogeneous symmetric
function are linked as

(4.7) ${n+k\brace n}_{r}^{(l)}=h_{k}(r^{l},\dots,n^{l}),$

###### Proof.

Let us count the number of $(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of
$[n+k]$ into $n$ blocks with $\\{1,2,\dots,r\\}$ are leaders. First, we
denote, $\\{y_{1},y_{2},\dots,y_{k}\\}$ the elements that are not leaders
where $y_{1}<y_{2}<\dots<y_{k}$. Let $x_{i}$ be the number of leaders smaller
than $y_{i}$, $i\in\\{1,\dots,k\\}$, it is clear that $r\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{k}\leq n$.

The construction of such partition goes as follows

  * •

Construct a partition of $n$ blocks with
$[n+k]$\$\\{y_{1},y_{2},\dots,y_{k}\\}$ where ${1,2,\dots,r}$ are leaders, i.
e.

$\\{1\\}\\{2\\}\dots\\{r\\}\\{z_{r+1}\\}\dots\\{z_{n}\\}.$

  * •

Insert the $\\{y_{1},y_{2},\dots,y_{k}\\}$ elements to the $n$ blocks. It is
clear that $y_{i}$ can belong only to a block having a leader smaller than
$y_{i}$, we have $x_{1}\cdot x_{2}\cdots x_{k}$ ways to do.

  * •

Repeat the process with each partition $\pi\in\\{\pi_{1},\dots,\pi_{l}\\}$, so
we have $(x_{1})^{l}(x_{2})^{l}\dots(x_{k})^{l}$ ways of construction.

  * •

Summing over all possible set of numbers $\\{x_{1},x_{2},\dots,x_{k}\\}$,
hence the total number of ways to construct
$(\pi_{1},\pi_{2},\dots,\pi_{l})$-partitions of $[n+k]$ having $n$ blocks with
$\\{1,\dots,r\\}$ lead is

$\begin{split}{n+k\brace n}_{r}^{(l)}&=\sum_{r\leq x_{1}\leq
x_{2}\leq\cdots\leq n}x_{1}^{l}x_{2}^{l}\cdots x_{k}^{l}\\\
&=h_{k}(r^{l},\dots,n^{l}).\end{split}$

∎

##  5\. Generating functions

Now, we can use the symmetric functions to construct the generating functions
for the $(l,r)$-Stirling of both kinds.

######  Theorem 5.1.

The generating function for the $(l,r)$-Stirling numbers of the first kind is

(5.1) $\sum_{k}{n\brack
k}_{r}^{(l)}z^{k}=z^{r}\prod_{i=r}^{n-1}\left(z+i^{l}\right)=z^{r}\left(z+r^{l}\right)\left(z+(r+1)^{l}\right)\cdots\left(z+(n-1)^{l}\right),$

###### Proof.

From Theorem 4.1 and the generating function (4.3) we obtain

(5.2) $\begin{split}\sum_{k}{n\brack
k}_{r}^{(l)}z^{k}&=z^{n}\sum_{k}e_{k}(r^{l},\dots,(n-1)^{l})(z^{-1})^{k}\\\
&=z^{n}\prod_{i=r}^{n-1}\left(1+\frac{i^{l}}{z}\right)\\\
&=z^{r}\prod_{i=r}^{n-1}(z+i^{l}).\end{split}$

∎

######  Theorem 5.2.

The generating function for the $(l,r)$-Stirling numbers of the second kind is

(5.3) $\sum_{n=k}{n\brace
k}_{r}^{(l)}z^{n}=z^{k}\left(\prod_{i=r}^{k}(1-zi^{l})\right)^{-1}=\frac{z^{k}}{(1-zr^{l})(1-z(r+1)^{l})(1-zk^{l})}.$

###### Proof.

From Theorem 4.2 and the generating function of homogeneous symmetric function
(4.4), we obtain

$\begin{split}\sum_{n\geq k}{n\brace k}_{r}^{(l)}z^{n}&=\sum_{j\geq
0}{k+j\brace k}z^{k+j}=z^{k}\sum_{j\geq
0}h_{j}(r^{l},\dots,k^{l})z^{j}=z^{k}\left(\prod_{i=r}^{k}(1-zi^{l})\right)^{-1}.\end{split}$

∎

In the following theorem we investigate the symmetric functions to obtain a
convolution formula for the $(l,r)$-Stirling numbers of both kinds.

######  Theorem 5.3.

For all positive integers $l$, $n$, $k$ and $r$ with $(n\geq k\geq r)$, we
have

(5.4) $\sum_{\begin{array}[]{c}i_{0}+2i_{1}\cdots+2^{l}i_{l}=k\\\
i_{0},\cdots,i_{l}\geq 0\end{array}}{n+i_{l}\brace
n}_{r}^{(2^{l})}\prod_{s=0}^{l-1}{n+1\brack
n+1-i_{s}}_{r}^{(2^{s})}={n+k\brace n}_{r}.$

###### Proof.

Let us consider the generating function of the complete homogeneous symmetric
function (4.4). From that we have

$\begin{split}\sum_{k\geq
0}h_{k}(x_{1},\dots,x_{n})z^{k}&=\prod_{i=1}^{n}\frac{1}{(1-x_{i}z)}\\\
&=\prod_{i=1}^{n}\frac{1}{(1-x_{i}z)}\prod_{s=0}^{l-1}\left(\frac{1+x_{i}^{2^{s}}z^{2^{s}}}{1+x_{i}^{2^{s}}z^{2^{s}}}\right)\\\
&=\prod_{i=1}^{n}\frac{1}{(1-x_{i}^{2^{l}}z^{2^{l}})}\prod_{s=0}^{l-1}\left(1+x_{i}^{2^{s}}z^{2^{s}}\right)\\\
&=\sum_{k\geq
0}h_{k}(x_{1}^{2^{l}},\dots,x_{n}^{2^{l}})z^{2^{l}k}\prod_{s=0}^{l-1}\sum_{k\geq
0}e_{k}(x_{1}^{2^{s}},\dots,x_{n}^{2^{s}})z^{2^{s}k}\\\ &=\sum_{k\geq
0}h_{k}(x_{1}^{2^{l}},\dots,x_{n}^{2^{l}})z^{2^{l}k}\sum_{k\geq
0}e_{k}(x_{1},\dots,x_{n})z^{k}\sum_{k\geq
0}e_{k}(x_{1}^{2},\dots,x_{n}^{2})z^{2k}\cdots\sum_{k\geq
0}e_{k}(x_{1}^{2^{l-1}},\dots,x_{n}^{2^{l-1}})z^{2^{l-1}k}\\\ &=\sum_{k\geq
0}\left(\sum_{\begin{array}[]{c}i_{0}+2i_{1}+\dots+2^{l}i_{l}=k;\\\
i_{0},\dots,i_{l}\geq
0.\end{array}}h_{i_{l}}(x_{1}^{2^{l}},\dots,x_{n}^{2^{l}})\prod_{s=0}^{l-1}e_{i_{s}}(x_{1}^{2^{s}},\dots,x_{n}^{2^{s}})\right)z^{k}.\end{split}$

From Theorem 4.1 and Theorem 4.2 and by comparing the coefficients of $z^{k}$
of the two sides the result holds true. ∎

The simplest case of the previous theorem is the corollary bellow which
generalize the result of Broder [6].

######  Corollary 5.4.

For $l=1$, we have

(5.5) $\sum_{i=0}^{\lfloor k/2\rfloor}{n+i\brace n}_{r}^{(2)}{n+1\brack
n+1+2i-k}_{r}={n+k\brace n}_{r}.$

##  6\. The $(l,r)$-Stirling numbers, the sum powers and Bernoulli polynomials

Recall, for every integer $n\geq 0$, the Bernoulli polynomials, denoted
$B_{n}(x)$, are defined by

(6.1) $\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n}=\frac{te^{xt}}{e^{t}-1}.$

The sum of the powers of natural numbers is closely related to the Bernoulli
polynomials $B_{n}(x)$. Jacobi [12, 16] gives the following identity using the
sum of powers and Bernoulli polynomials

(6.2) $\sum_{j=1}^{n}j^{m}=\frac{B_{m+1}(n+1)-B_{m+1}(0)}{m+1}.$

The following theorem gives the relation between $(l,r)$-Stirling of both
kinds and Bernoulli polynomials.

######  Theorem 6.1.

For all positive integers $n$, $k$ and $l$, we have

(6.3) $\sum_{j=0}^{k}(-1)^{j}(j+1){n+1\brack n-j}^{(l)}{n+k-j\brace
n}^{(l)}=\frac{B_{lk+l+1}(n+1)-B_{lk+l+1}(0)}{lk+l+1},$

###### Proof.

In the first hand we have Jacobi’s Identity (6.2)

(6.4) $\sum_{j=1}^{n}(j^{l})^{k}=\frac{B_{lk+1}(n+1)-B_{lk+1}(0)}{lk+1},$

in the second hand, we have

$H(t)=\sum_{k\geq
0}h_{k}(1^{l},2^{l},\dots,n^{l})t^{k}=\prod_{j=1}^{n}\frac{1}{(1-j^{s}t)}$

and

$E(t)=\sum_{k\geq
0}e_{k}(1^{l},2^{l},\dots,n^{l})t^{k}=\prod_{j=1}^{n}(1+j^{s}t),$

from the obvious observation that $H(t)=1/E(-t)$, we obtain

(6.5) $\frac{d}{dt}\ln{H(t)}=\frac{H^{\prime}(t)}{H(t)}=H(t)E^{\prime}(-t)$

but

(6.6)
$\frac{d}{dt}\ln{H(t)}=\sum_{j=1}^{n}\frac{j^{l}}{(1-j^{l}t)}=\sum_{k\geq
0}\sum_{j=1}^{n}j^{s(k+1)}t^{k}.$

Then from equations (6.5) and (6.6), we get

(6.7) $\begin{split}\sum_{k\geq
0}\sum_{j=1}^{n}j^{s(k+1)}t^{k}&=H(t)E^{\prime}(-t)\\\ &=\left(\sum_{k\geq
0}h_{k}(1^{l},\dots,n^{l})t^{k}\right)\left(\sum_{k\geq
1}k(-1)^{k-1}e_{k}(1^{l},\dots,n^{l})t^{k-1}\right).\\\ \end{split}$

Cauchy product and equating coefficient of $t^{k}$ gives

(6.8) $\begin{split}\sum_{j=1}^{n}j^{s(k+1)}&=\sum_{j\geq
1}^{n}(j+1)(-1)^{j}e_{j+1}(1^{l},\dots,n^{l})h_{k-j}(1^{l},\dots,n^{l}),\end{split}$

replacing symmetric functions by stirling numbers from Theorem 4.1 and Theorem
4.2, and comparing with Equation (6.4) we get the result.

∎

##  7\. Multiple zeta function and $(l,r)$-Stirling numbers of the first kind

For any ordered sequence of positive integers $i_{1},i_{2},\dots,i_{k}$, the
multiple zeta function is introduced by Hoffman [11] and independently Zagier
[17] by the following infinite sums

(7.1)
$\zeta(i_{1},i_{2},\dots,i_{k})=\sum_{0<j_{1}<j_{2}<\cdots<j_{k}}\frac{1}{j_{1}^{i_{1}}j_{2}^{i_{2}}\cdots
j_{k}^{i_{k}}}.$

Recently, the multiple zeta function has been studied quite intensively by
many authors in various fields of mathematics and physics (see [4, 5, 7, 11,
17, 19]). Here we give a relation between $(l,r)$-Stirling numbers of the
first kind and the multiple zeta function.

######  Theorem 7.1.

For all positive integers $n$, $k$, $l$ and $r$ with $(n\geq k\geq r)$, we
have

(7.2) $\begin{split}{n+1\brack
k+1}_{r}^{(l)}&=\left(\frac{n!}{(r-1)!}\right)^{l}\sum_{j_{k}=k}^{n}\sum_{j_{k-1}=k-1}^{j_{k}-1}\cdots\sum_{j_{r}=r}^{j_{(r+1)}-1}\frac{1}{\left(j_{r}j_{2}\cdots
j_{k}\right)^{l}}\\\
&=\left(\frac{n!}{(r-1)!}\right)^{l}\sum_{r-1<j_{1}<j_{2}<\cdots<j_{k}\leq
n}\frac{1}{(j_{1}j_{2}\cdots j_{k})^{l}}.\end{split}$

###### Proof.

Since ${n\brack k}_{r}^{(l)}={n-1\brack k-1}_{r}^{(l)}+(n-1)^{l}{n-1\brack
k}_{r}^{(l)}$ from Theorem 2.3. If we proceed iteratively, we obtain that

(7.3) ${n\brack
k}_{r}^{(l)}=\left((n-1)!\right)^{l}\sum_{j=k-1}^{n-1}\frac{1}{(j!)^{l}}{j\brack
k-1}_{r}^{(l)}.$

For $k={r}$, from (2.5) and (7.3) we obtain

(7.4) ${n\brack
r}_{r}^{(l)}=(r^{\overline{n-r}})^{l}=\left(\frac{(n-1)!}{(r-1)!}\right)^{l}.$

For $k={r+1}$, from (7.3) and (7.4) we obtain

(7.5) $\begin{split}{n\brack
r+1}_{r}^{(l)}&=\left((n-1)!\right)^{l}\sum_{j=r}^{n-1}\frac{1}{(j!)^{l}}{j\brack
r}_{r}^{(l)}\\\
&=\left(\frac{(n-1)!}{(r-1)!}\right)^{l}\sum_{j=r}^{n-1}\left(\frac{(j-1)!}{j!}\right)^{l}\\\
&=\left(\frac{(n-1)!}{(r-1)!}\right)^{l}\sum_{j=r}^{n-1}\frac{1}{j^{l}}.\end{split}$

For $k=r+2$, from (7.4) and (7.5) we obtain

(7.6) ${n\brack
r+2}_{r}^{(l)}=\left(\frac{(n-1)!}{(r-1)!}\right)^{l}\sum_{j=r+1}^{n-1}\sum_{i=r}^{j-1}\frac{1}{(ij)^{l}},$

iterating the process with $k\in\\{r+3,r+4,\dots\\}$ and so on, then yields
the result. ∎

######  Proposition 7.2.

For $r=1$, we have

(7.7) $\lim_{n\to\infty}\frac{1}{\left(n!\right)^{l}}{\displaystyle{n+1\brack
k+1}}^{(l)}=\zeta(\\{l\\}_{k}),$

where $\\{l\\}_{n}=(\underbrace{l,l,\dots,l}_{n\text{ times}}).$

###### Proof.

The proposition follows immediately from the definition of multiple zeta
function (7.1) as an infinity sums and Theorem 7.1 for $r=1$. ∎

######  Corollary 7.3.

For $k\geq 1$, we have

  * •

For $l=2$

(7.8) $\lim_{n\to\infty}\frac{1}{\left(n!\right)^{2}}\displaystyle{n+1\brack
k+1}^{(2)}=\frac{\pi^{2k}}{(2k+1)!}.$

  * •

For $l=4$

(7.9) $\lim_{n\to\infty}\frac{1}{\left(n!\right)^{4}}\displaystyle{n+1\brack
k+1}^{(4)}=\frac{4(2\pi)^{4k}}{(4k+2)!}\left(\frac{1}{2}\right)^{2k+1}.$

  * •

For $l=6$

(7.10) $\lim_{n\to\infty}\frac{1}{\left(n!\right)^{6}}\displaystyle{n+1\brack
k+1}^{(6)}=\frac{6(2\pi)^{6k}}{(6k+3)!}.$

  * •

For $l=8$

(7.11) $\lim_{n\to\infty}\frac{1}{\left(n!\right)^{8}}\displaystyle{n+1\brack
k+1}^{(8)}=\frac{\pi^{8k}}{(8k+4)!}2^{8k+3}\left(\left(1+\frac{1}{\sqrt{2}}\right)^{4k+2}+\left(1-\frac{1}{\sqrt{2}}\right)^{4k+2}\right).$

###### Proof.

Authors in [7] give the following special values of multiple zeta function

$\displaystyle\zeta(\\{2\\}_{n})$ $\displaystyle=$
$\displaystyle\frac{\pi^{2n}}{(2n+1)!},$ $\displaystyle\zeta(\\{4\\}_{n})$
$\displaystyle=$
$\displaystyle\frac{4(2\pi)^{4n}}{(4n+2)!}\left(\frac{1}{2}\right)^{2n+1},$
$\displaystyle\zeta(\\{6\\}_{n})$ $\displaystyle=$
$\displaystyle\frac{6(2\pi)^{6n}}{(6n+3)!},$ $\displaystyle\zeta(\\{8\\}_{n})$
$\displaystyle=$
$\displaystyle\frac{\pi^{8n}}{(8n+4)!}2^{8n+3}\left(\left(1+\frac{1}{\sqrt{2}}\right)^{4n+2}+\left(1-\frac{1}{\sqrt{2}}\right)^{4n+2}\right),$

the corollary is a consequence of the previous special cases and Proposition
7.2. ∎

##  8\. Remarks

  * •

The $(l,r)$-Stirling gives another graphical view of Rooks polynomials of
higher dimensions in triangle boards [3, 18] using set partitions.

  * •

In this work we gives a limit representation of multiple zeta function using
$(l,r)$-Stirling numbers.

  * •

We can obtain the well-known Euler identity $\zeta(2)=\frac{\pi^{2}}{6}$ from
Equation (7.8) for $k=1$.

### Acknowledgment

We would like to thank the anonymous reviewers for their suggestions and
comments which improved the quality of the present paper. The paper was
partially supported by the DGRSDT grant №:C0656701.

## References

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  * [5] Bradley, D. M. (2005). Partition identities for the multiple zeta function. In Zeta functions, topology and quantum physics (pp. 19-29). Springer, Boston, MA.
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  * [10] Gelineau, Y., and Zeng, J. (2010). Combinatorial interpretations of the Jacobi-Stirling numbers. the electronic journal of combinatorics, 17(R70), 1.
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  * [16] Srivastava, H. M. (2000, July). Some formulas for the Bernoulli and Euler polynomials at rational arguments. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 129, No. 1, pp. 77-84).
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# Escape probability of particle from Kerr-Sen black hole

Ming Zhang<EMAIL_ADDRESS>Department of Physics, Jiangxi Normal
University, Nanchang 330022, China  Jie Jiang  Corresponding author,
<EMAIL_ADDRESS>Department of Physics, Jiangxi Normal University,
Nanchang 330022, China  Department of Physics, Beijing Normal University,
Beijing 100875, China

###### Abstract

Assuming that a particle source is at rest in a locally non-rotating frame on
the equatorial plane of the Kerr-Sen black hole, we investigate the escape of
the massless particle and massive particle from the black hole to spatial
infinity. We calculate the escape probabilities of the emitted particles. We
find that the angular momentum of the Kerr-Sen black hole boosts the escape
probabilities near the horizon of the extreme Kerr-Sen black hole; however the
angular momentum of the Kerr-Sen black hole suppresses the escape
probabilities near the horizon of the non-extreme Kerr-Sen black hole. We also
discover that the horizon limit of the Kerr-Sen black hole is not the most
difficult position to escape for a particle in certain conditions.

##  I Introduction

The Penrose process Penrose and Floyd (1971), which is well-known as an energy
extraction process from ergosphere of the rotating black hole by one emitted
particle originating from an initial ingoing one which breaks up, has been
developed into the collisional Penrose process Piran  _et al._ (1975) for
astrophysical considerations, as it was pointed out that the process is
unplausible in astrophysics Bardeen  _et al._ (1972). Generally, there are
three kinds of collisional Penrose process. The first case Piran and Shaham
(1977) is that two (massive or massless) particles fall into the ergosphere of
the rotating black hole from spatial infinity at rest and then collide with
each other, producing a particle that can escape to infinity. The second case
Schnittman (2014); Mukherjee (2018) is that one particle with an impact
parameter greater than the critical value falls into the black hole from
spatial infinity and then turns around, colliding with the other ingoing
particle. The third case, which is named as the super Penrose process Berti
_et al._ (2015); Zaslavskii (2015a); Zhang  _et al._ (2018), is that two
particles take a head-on collision in the ergosphere of the black hole. In
this process, one of the particles comes from infinity and the other is a
consequence of previous scattering events (for more discussions on the
origination of this outgoing particle, see Refs. Zaslavskii (2018);
Leiderschneider and Piran (2015); Zaslavskii (2015b, 2016)). The efficiencies
of these processes are quite different, as

$\eta_{\text{penrose}}\approx
121\%<\eta_{\text{first}}<\eta_{\text{second}}<\eta_{\text{third}}\to\infty.$
(1)

Besides the (Collisional) Penrose process, another high-energy event is the
Bañados-Silk-West mechanism, which states that the center-of-mass energy of
two test particles can be arbitrarily high in the condition that the collision
between them happens near the event horizon of the extremal black hole and the
angular momentum and the energy of one particle follow a critical relation
Banados  _et al._ (2009); Sadeghi and Pourhassan (2012); Sadeghi  _et al._
(2014); Pourhassan and Debnath (2019); An  _et al._ (2018); Guo and Gao
(2016); Jiang and Gao (2019). What is more, the observation of the shadow
Akiyama  _et al._ (2019a, b, c) of the black hole depends on the photons that
escape from the black hole.

The observations of the high-energy astrophysical events and the black hole
shadow depend on the massive particle and photons we can observe. Of
particular interest is the escape probability of the particle in the strong
gravitational field. Particularly, the escape probability of the photon
determines the brightness of the halo around the black hole we observe. The
escape probability of the produced particles in the super Penrose process for
the extreme Kerr black hole was studied in Ogasawara  _et al._ (2017). It was
shown that the resulting massless particles with extremely high energy in the
super Penrose process can escape to infinity with a probability of $5/12$ and
an impact parameter close to $2M$, where $M$ is the mass of the Kerr black
hole. It motivates our research in this paper. We are interested in not only
the escape probability of the massless photon but also that of the massive
particle. We will study the observability of the massless and massive
particles emitted from the Kerr-Sen black hole.

The remaining parts of this paper are arranged as follows. In section II, we
will show the four-momentum of a particle in the Kerr-Sen spacetime using a
locally non-rotating frame (LNRF). In section III, we will calculate the
escaping probability of the massless particle from the extreme Kerr-Sen black
hole. In section IV, we will evaluate the escape probability of the massive
particle from the extreme Kerr-Sen black hole. In section V, we will extend
our study of the emission of the massless and massive particle from the
extreme Kerr-Sen black hole to the non-extreme Kerr-Sen black hole case.
Section VI will be employed to discuss the escape probability variation of the
particle with respect to the position of the particle source. Section VII will
be devoted to our conclusions and discussions.

##  II Four-momentum of the particle in the Kerr-Sen spacetime

The line element of the Kerr-Sen black hole is Sen (1992)

$\displaystyle ds^{2}=$
$\displaystyle-\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}$ (2)
$\displaystyle+\frac{\Xi\sin^{2}\theta}{\Sigma}d\phi^{2}-\frac{4Mra\sin^{2}\theta}{\Sigma}dtd\phi,$

where

$\displaystyle\Sigma$ $\displaystyle=r(r+2c)+a^{2}\cos^{2}\theta,$
$\displaystyle\Delta$ $\displaystyle=r(r+2c)-2Mr+a^{2},$ $\displaystyle\Xi$
$\displaystyle=[a^{2}+r(2c+r)]^{2}-\Delta a^{2}\sin^{2}\theta,$ $\displaystyle
c$ $\displaystyle=Q^{2}/2M.$

The inner and outer horizons of the black hole are given as

$\displaystyle r_{\text{i}}$
$\displaystyle=M-c-\sqrt{\left(M-c\right)^{2}-a^{2}},$ (3) $\displaystyle
r_{\text{h}}$ $\displaystyle=M-c+\sqrt{\left(M-c\right)^{2}-a^{2}}.$ (4)

When $|a|=\left|M-c\right|$, the Kerr-Sen black hole becomes extreme.

There are two Killing vectors, $\xi^{t}\equiv\delta^{t}_{\mu}$ and
$\xi^{\phi}\equiv\delta^{\phi}_{\mu}$, for the stationary Kerr-Sen black hole.
Correspondingly, the conserved quantities, namely conserved energy and angular
momentum of a particle, read

$\displaystyle e$ $\displaystyle=-g_{t\nu}\xi^{t}p^{\nu},$ (5) $\displaystyle
l$ $\displaystyle=g_{\phi\nu}\xi^{\phi}p^{\nu},$ (6)

where $p^{\mu}$ is the four-momentum of the particle with mass $m$.
$p^{\mu}p_{\mu}=-m^{2}$, $m=0$ for the massless particle (photon) and $m>0$
for the massive particle. Not loss of generality, by setting $\theta=\pi/2$,
meaning that the particle moves on the equatorial plane, we obtain the
particle’s four-momentum

$\displaystyle p^{t}$
$\displaystyle=\frac{2alMr\Sigma-e\Xi\Sigma}{a^{2}\left(\Xi-4M^{2}r^{2}\right)-\Delta\Xi},$
(7) $\displaystyle p^{r}$ $\displaystyle=\sigma\sqrt{V(r)},$ (8)
$\displaystyle p^{\theta}$ $\displaystyle=0,$ (9) $\displaystyle p^{\phi}$
$\displaystyle=\frac{\Sigma\left(-a^{2}l+2aeMr+\Delta
l\right)}{\Delta\Xi-a^{2}\left(\Xi-4M^{2}r^{2}\right)},$ (10)

where $\sigma=\pm 1$ correspond respectively to the particles that move
radially outwards and inwards,

$V(r)=\frac{-a^{2}\Delta p_{t}^{2}+4a\Delta Mp_{t}p_{\phi}r-\Delta
m^{2}\Sigma+\Delta^{2}p_{t}^{2}-\Delta\Xi p_{\phi}^{2}}{\Sigma^{2}}$ (11)

can be viewed as the radial effective potential of the particle.

To describe the particle moving in the rotating Kerr-Sen spacetime, we resort
to an orthonormal tetrad accompanying with a locally non-rotating observer,
namely LNRF Bardeen  _et al._ (1972), which can be written as

$\displaystyle e^{(t)}_{\mu}$
$\displaystyle=\left(\sqrt{\frac{\Delta\Sigma}{\Xi}},0,0,0\right),$ (12)
$\displaystyle e^{(r)}_{\mu}$
$\displaystyle=\left(0,\sqrt{\frac{\Sigma}{\Delta}},0,0\right),$
$\displaystyle e^{(\theta)}_{\mu}$ $\displaystyle=(0,0,\sqrt{\Sigma},0),$
$\displaystyle e^{(\phi)}_{\mu}$
$\displaystyle=\left(-\frac{2aMr}{\sqrt{\Xi\Sigma}},0,0,\sqrt{\frac{\Xi}{\Sigma}}\right).$

The tetrad basis is related to the metric by the transformation

$g_{\mu\nu}=\eta_{(a)(b)}e^{(a)}_{\mu}e^{(b)}_{\nu},\quad\eta_{(a)(b)}=\text{diag}(-1,1,1,1).$
(13)

Then by the translation

$p^{(\alpha)}=e^{(\alpha)}_{\mu}p^{\mu},$ (14)

we can obtain the four-momentum of the particle in the LNRF.

Figure 1: Emission angle of the particle in the LNRF. $p^{(a)}$ is the four-
momentum of the particle.

##  III Escaping probability of the massless particle from extreme Kerr-Sen
black hole at horizon limit

We will now evaluate the escaping probability of the massless particle from
the extreme Kerr-Sen black hole produced by a source at rest in LNRF at the
horizon limit. The four-momentum of the massless particle in the LNRF can be
obtained as

$\displaystyle p^{(t)}=$
$\displaystyle\frac{eZ\left(2M(M+r)+Q^{2}\right)+4lM^{3}\left(Q^{2}-2M^{2}\right)}{\left(2M(r-M)+Q^{2}\right)\sqrt{Z\left(Mr+Q^{2}\right)}},$
(15) $\displaystyle p^{(r)}=$
$\displaystyle\frac{2\sigma\sqrt{V(r)}\sqrt{Mr\left(Mr+Q^{2}\right)}}{-2M^{2}+2Mr+Q^{2}},$
(16) $\displaystyle p^{(\theta)}=$ $\displaystyle 0,$ (17) $\displaystyle
p^{(\phi)}=$
$\displaystyle\frac{2lM\left(Mr+Q^{2}\right)}{\sqrt{Z\left(Mr+Q^{2}\right)}},$
(18)

where

$\displaystyle Z=$
$\displaystyle\left[4M^{4}-2M^{3}r+2M^{2}\left(r^{2}-2Q^{2}\right)+3MQ^{2}r+Q^{4}\right]$
$\displaystyle\times\left(2M(M+r)+Q^{2}\right).$

We consider a particle source at rest in the LNRF on the equatorial plane of
the Kerr-Sen black hole at the position $r_{*}$. We can introduce the emission
angle $\alpha$ with respect to the LNRF tetrad (see Fig. 1), which can be
determined by the relations among the components of $p^{(a)}$ for the massless
particle, as

$\displaystyle\sin\alpha=$
$\displaystyle\frac{p^{(\phi)}}{\sqrt{\left(p^{(r)}\right)^{2}+\left(p^{(\phi)}\right)^{2}}}=\frac{p^{(\phi)}}{p^{(t)}}$
(19) $\displaystyle=$
$\displaystyle\frac{2bM\left(Mr_{*}+Q^{2}\right)\left(2M(r_{*}-M)+Q^{2}\right)}{4bM^{3}\left(Q^{2}-2M^{2}\right)+Z(r_{*})},$
$\displaystyle\cos\alpha=$
$\displaystyle\frac{p^{(r)}}{\sqrt{\left(p^{(r)}\right)^{2}+\left(p^{(\phi)}\right)^{2}}}=\frac{p^{(r)}}{p^{(t)}}$
(20) $\displaystyle=$
$\displaystyle\frac{\sqrt{\xi}\sqrt{Z(r_{*})}\left(2M(M-r_{*})-Q^{2}\right)}{\sqrt{M}\left(4bM^{3}\left(Q^{2}-2M^{2}\right)+Z(r_{*})\right)},$

where

$\displaystyle\xi=$ $\displaystyle
b^{2}M\left(2M^{2}-Mr_{*}-Q^{2}\right)+2bM^{2}\left(Q^{2}-2M^{2}\right)$
$\displaystyle+M\left(M-\frac{Q^{2}}{2M}\right)^{2}\left[M(2M+r_{*})+Q^{2}\right]$
$\displaystyle+r_{*}\left(Mr_{*}+Q^{2}\right)^{2},$

with the impact parameter defined by $b\equiv l/e$.

Figure 2: A schematic diagram of the impact parameter $b$ of the particle.

Letting $V(r)=0$, we can obtain the impact parameters

$\displaystyle b_{+}(r)$ $\displaystyle:=$
$\displaystyle\frac{Q^{2}}{2M}+M+r,$ (21) $\displaystyle b_{-}(r)$
$\displaystyle:=$
$\displaystyle-\frac{2M\left(2M^{2}-Q^{2}\right)}{M(r-2M)+Q^{2}}-\frac{Q^{2}}{2M}-M-r.$
(22)

Then we can know particles with the impact parameter

$2M<b<b_{+}(r_{*})$ (23)

can escape from the extreme Kerr-Sen black hole to spatial infinity for any
sign of initial radial velocity; for the particle with the impact parameter

$b_{e}<b<2M,$ (24)

with

$b_{e}=\frac{-4M\sqrt{4M^{2}-2Q^{2}}-6M^{2}+Q^{2}}{2M}$ (25)

the extreme point of $b_{-}(r)$, it cannot escape to spatial infinity if its
initial radial velocity is inward. Then according to the schematic diagram in
Fig. 2, critical angles related to the critical impact parameters that the
massless particle can escape to infinity can be defined as

$\displaystyle\alpha_{\rm I}$
$\displaystyle\equiv\alpha[\sigma_{3}=-1,\;b=2M],$ (26)
$\displaystyle\alpha_{\rm II}$
$\displaystyle\equiv\alpha[\sigma_{3}=-1,\;b=b_{+}(r_{*})],$ (27)
$\displaystyle\alpha_{\rm III}$
$\displaystyle\equiv\alpha[\sigma_{3}=1,\;b=b_{+}(r_{*})],$ (28)
$\displaystyle\alpha_{\rm IV}$
$\displaystyle\equiv\alpha[\sigma_{3}=1,\;b=b_{e}].$ (29)

As

$\sin\alpha_{\rm II}=\sin\alpha_{\rm III}=1$

and

$\displaystyle\sin\alpha_{\rm I}>0,\;\cos\alpha_{\rm I}<0,\;\sin\alpha_{\rm
IV}<0,\;\cos\alpha_{\rm IV}>0,$

we have

$\alpha_{\rm IV}<\alpha_{\rm III}=\alpha_{\rm II}=\frac{\pi}{2}<\alpha_{\rm
I},$

which indicates that the massless particle that escapes from the extreme Kerr-
Sen black hole to spatial infinity must has an angle

$\alpha\in(\alpha_{\rm IV},\alpha_{\rm I}).$ (30)

To obtain concrete values of the characterized critical angles, we can set
that the collision takes place at a near-horizon point of the extreme Kerr-Sen
black hole

$r_{*}=\frac{M-Q^{2}/(2M)}{1-\epsilon},$

with $0<\epsilon\ll 1$. Then the critical angles can be directly calculated as

$\displaystyle\alpha_{\rm I}$
$\displaystyle=\pi-\arcsin\left(\frac{2M^{2}+Q^{2}}{4M^{2}}\right)+\mathcal{O}(\epsilon^{0}),$
(31) $\displaystyle\alpha_{\rm II}$ $\displaystyle=\alpha_{\rm
III}=\frac{\pi}{2},$ (32) $\displaystyle\alpha_{\rm IV}$
$\displaystyle=-\frac{\epsilon\left(2M^{2}+Q^{2}\right)\varsigma}{4M^{2}\left(4M^{3}+\varsigma\right)}+\mathcal{O}(\epsilon),$
(33)

where

$\varsigma=6M^{3}+4\sqrt{4M^{6}-2M^{4}Q^{2}}-MQ^{2}.$

When $Q\to\sqrt{2}M$, we have

$\displaystyle\alpha_{\rm I}=\frac{\pi}{2},\quad\alpha_{\rm
IV}=-\frac{\epsilon}{2}.$ (34)

The escape probability depends on the escape cone from which the photon can
escape to spatial infinity. If viewing that the particle on the equatorial
plane emitted isotropically, we can calculate the escape probability of the
massless particle produced by the particle source near the event horizon of
the extreme Kerr-Sen black hole by comparing the escape cone encircled by
$\alpha_{i}$ ($i={\rm I,II,III,IV}$) with $2\pi$ Ogasawara  _et al._ (2017),

$\displaystyle P\equiv\frac{\alpha_{\rm I}-\alpha_{\rm IV}}{2\pi},$ (35)

which gives escape probability of the massless particle as

$\displaystyle
P=\frac{1}{2}-\frac{1}{2\pi}\arcsin\left(\frac{1}{2}+\frac{Q^{2}}{4M^{2}}\right).$
(36)

It is evident that the escape probability of the particle decreases with the
increasing electric charge of the extreme Kerr-Sen black hole. Especially, in
the limits $\epsilon\to 0,Q\to\sqrt{2}M$, the escape probability becomes

$P\to\frac{1}{4},$ (37)

and for $Q\to 0$, we have $P\to 5/12$, which is just the result of extreme
Kerr black hole obtained in Ogasawara  _et al._ (2017).

##  IV Escaping probability of massive particle from extreme Kerr-Sen black
hole at horizon limit

For convenience of calculations, we set $M=m=1$ for what follows in this
passage. For the massive particle, we can introduce the emission angle $\beta$
from the particle source with respect to the LNRF tetrad, which can be
determined by the relations among the components of $p^{(a)}$ for the massive
particle as

$\displaystyle\sin\beta=\frac{p^{(\phi)}}{\sqrt{\left(p^{(r)}\right)^{2}+\left(p^{(\phi)}\right)^{2}}},$
(38)
$\displaystyle\cos\beta=\frac{p^{(r)}}{\sqrt{\left(p^{(r)}\right)^{2}+\left(p^{(\phi)}\right)^{2}}}.$
(39)

$V(r)=0$ gives the impact parameters

$\bar{b}_{+}(r)=\frac{2Q^{2}-4+\sqrt{\mathcal{X}_{1}\left(Q^{2}+r\right)}\left(Q^{2}+2r-2\right)/e}{2\left(Q^{2}+r-2\right)},$
(40)
$\bar{b}_{-}(r)=\frac{2Q^{2}-4-\sqrt{\mathcal{X}_{1}\left(Q^{2}+r\right)}\left(Q^{2}+2r-2\right)/e}{2\left(Q^{2}+r-2\right)},$
(41)

where $\mathcal{X}_{1}=e^{2}\left(Q^{2}+r\right)-\left(Q^{2}+r-2\right)$.
Letting $r\to r_{h}$ for the extreme Kerr-Sen black hole, we can obtain
$\bar{b}_{+}\to 2$. Denoting the extreme value of $\bar{b}_{-}$ as
$\bar{b}_{e}$, we can know that for $2<\bar{b}<\bar{b}_{+}(r_{*})$, the
massive particle can escape from the extreme Kerr-Sen black hole to spatial
infinity irrespective of the initial sign of its radial velocity and for
$\bar{b}_{e}<\bar{b}<2$, the escape can only happen for the particle with
outward initial radial velocity.

Table 1: The critical escape angle $\beta_{I}$ for a massive particle from the extreme Kerr-Sen black hole at the event horizon limit. $Q~{}~{}~{}$ |  0 |  0.1 |  0.3 |  0.5 |  0.7 |  0.9 |  1.1 |  1.3 |  $\sqrt{2}$
---|---|---|---|---|---|---|---|---|---
$\sin\beta_{I}~{}~{}~{}$ |  $1/\sqrt{3}$ |  0.58 |  0.60 |  0.65 |  0.71 |  0.79 |  0.87 |  0.96 |  1
$\cos\beta_{I}~{}~{}~{}$ |  $-\sqrt{2/3}$ |  -0.81 |  -0.80 |  -0.76 |  -0.70 |  -0.61 |  -0.48 |  -0.29 |  0
Figure 3: Escape probabilities of the massive and massless particles from the
extreme Kerr-Sen black hole at the event horizon limit with $M=1$.

Like the massless case, the critical angles related to the critical impact
parameters that the massive particle can escape to infinity can be defined as

$\displaystyle\beta_{\rm I}$
$\displaystyle\equiv\beta[\sigma_{3}=-1,\;\bar{b}=2],$ (42)
$\displaystyle\beta_{\rm II}$
$\displaystyle\equiv\beta[\sigma_{3}=-1,\;\bar{b}=\bar{b}_{+}(r_{*})],$ (43)
$\displaystyle\beta_{\rm III}$
$\displaystyle\equiv\beta[\sigma_{3}=1,\;\bar{b}=\bar{b}_{+}(r_{*})],$ (44)
$\displaystyle\beta_{\rm IV}$
$\displaystyle\equiv\beta[\sigma_{3}=1,\;\bar{b}=\bar{b}_{e}].$ (45)

Setting the conserved energy of the massive particle as $e=1$ (so that the
particle can escape from the black hole just to an observer at spatial
infinity) gives

$\displaystyle\sin\beta_{\rm II}$ $\displaystyle=\sin\beta_{\rm III}=1,$ (46)
$\displaystyle\cos\beta_{\rm II}$ $\displaystyle=\cos\beta_{\rm III}=0,$ (47)

which means

$\beta_{\rm II}=\beta_{\rm III}=\frac{\pi}{2}.$ (48)

The critical compact parameter of $\bar{b}_{-}$ is

$\bar{b}_{e}=\frac{-2\sqrt{2-Q^{2}}\left(Q^{2}+\mathcal{X}_{2}\right)+Q^{2}-2}{Q^{2}+\mathcal{X}_{2}-2},$
(49)

where

$\mathcal{X}_{2}=\frac{1}{2}\left(-3Q^{2}+4\sqrt{2-Q^{2}}+6\right).$

Then we have

$\sin\beta_{\text{IV}}=0,\,\cos\beta_{\text{IV}}=1,$ (50)

which gives

$\beta_{\text{IV}}=0.$ (51)

The critical angles $\beta_{I}$ can be obtained numerically and we have listed
them in Table 1. The escape probability of the massive particle in the extreme
Kerr-Sen black hole spacetime background is

$\bar{P}=\frac{\beta_{\text{I}}-\beta_{\text{IV}}}{2\pi}.$ (52)

We show the escape probability of the massive particle together with that of
the massless particle at the event horizon limit of the extreme Kerr-Sen black
hole in Fig. 3 according to Eqs. (36) and (52). From the figure, we can see
that, for both the two kinds of particles, the escape probabilities decrease
with the increasing electric charge of the extreme Kerr-Sen black hole. This
signifies that the rotation of the black hole promotes the escape of both the
massless and massive particles. Besides, we can see that the escape
probability of the massless particle is always greater than that of the
massive particle at the event horizon limit. The two branches intersect at
$Q=\sqrt{2}$ where the angular momentum of the extreme black hole vanishes. At
that point, both of the two kinds of particles own an escaping probability
1/4.

Figure 4: Escape probabilities of the massive and massless particles from the
non-extreme Kerr-Sen black hole at the domain of outer communication region
with $K=0.9$. We choose the position of the particle source at $r_{*}=4$.

Figure 5: Top panel: variations of the escape probabilities of the massive
particle in the domain of outer communication region of extreme Kerr-Sen black
hole with respect to the position of the particle source. Bottom panel:
variations of the escape probabilities of the massive particle in the domain
of outer communication region of the non-extreme Kerr-Sen black hole with
respect to the position of the particle source for K=0.9.

##  V Escape probabilities of particles from a source near the horizon of the
non-extreme Kerr-Sen black hole

We will probe the escape probabilities of the massless and massive particles
escaping from a particle source near the horizon of the non-extreme Kerr-Sen
black hole. The regularity condition of the non-extreme Kerr-Sen black hole
gives $a+Q^{2}/2<1$. We now investigate the escape probability of the massless
and massive particle from the non-extreme Kerr-Sen black hole, with
$a+Q^{2}/2=K<1$ and $a\geqslant 0$ (so that $Q\leqslant\sqrt{2K}$). We will
choose $K=0.9$ for instance.

In the previous extreme cases for the massless and massive particles, the
effective potential $V(r)$ in (8) is fulfilled at and near the event horizon.
In the non-extreme case, we should care about the demand of $V(r)\geqslant 0$.
We will not consider the event horizon limit as $V(r)<0$ there; instead, we
will choose a position where the effective potentials of the massless and
massive particles are fulfilled.

It is evident that the impact parameters at the position $r_{*}$ for the
particles make their critical escape angles to be

$\alpha_{\rm II}=\alpha_{\rm III}=\beta_{\rm II}=\beta_{\rm
III}=\frac{\pi}{2}.$ (53)

For the extreme Kerr-Sen black hole, we have
$\beta_{\text{IV}}\sim\bar{\beta}_{\text{IV}}\sim 0$ (even not at the event
horizon limit). However, for the non-extreme Kerr-Sen black hole, this is not
the case. For $K=0.9$,

$-\frac{\pi}{2}<\beta_{\text{IV}}<0,\quad-\frac{\pi}{2}<\bar{\beta}_{\text{IV}}<0.$
(54)

When $Q\to 0$, $\beta_{\text{IV}}\sim\bar{\beta}_{\text{IV}}\to-\pi/2$.

For both of the two kinds of particles, we can obtain the same impact
parameter at the event horizon of the non-extreme Kerr-Sen black hole as

$b_{+}=\bar{b}_{+}=\frac{2\left(Q^{2}-2K\right)}{2\sqrt{(K-1)\left(-K+Q^{2}-1\right)}+Q^{2}-2},$
(55)

with which we have

$\frac{\pi}{2}<\beta_{\text{I}}<\pi,\quad\frac{\pi}{2}<\bar{\beta}_{\text{I}}<\pi,$
(56)

for $K=0.9$. When $Q\to\sqrt{2K}$,
$\beta_{\text{I}}\sim\bar{\beta}_{\text{I}}\to\pi/2$.

We show the escape probabilities for both of the two kinds of particles in
Fig. 4. (Note that one branch of the impact parameters for the particle around
the non-extreme Kerr-Sen black hole at the horizon is not minimal and the
minimal value can be got at a radial position beyond the horizon. We have
considered this fact when we calculate the escape probability.) We can see
that the escape probabilities increase with the increasing electric charge of
the black hole, which is qualitatively different from the extreme Kerr-Sen
black hole case.

##  VI Variation of escape probability with respect to position of particle
source

In the previous sections, we investigated the escape probability of the
particle from the source near the Kerr-Sen black hole and explored the effect
of the black hole parameters on it. We will now study the escape probability
variation of the particle with respect to the position of the particle. As
seen in above, there is no qualitative difference between the massless
particle case and the massive particle case, we will now choose the massive
particle case as an example to illustrate the characteristics of the escape
probability variation in terms of the particle position.

We find that, as shown in the upper diagram of Fig. 5, outside the extreme
Kerr-Sen black hole, for the massive particle, if $Q<0.82$, the escape
probability first decreases and then increases with the increasing distance
apart from the horizon; or else, the escape probability increases
monotonically. The reason is that, with the increasing of the electric charge
of the extreme Kerr-Sen black hole, the extreme points of the escape
probabilities move nearer and nearer to the event horizons; and if $Q>0.82$,
the extreme points are inside the event horizons. (We also find that, the
massless particle case is similar to the massive particle case, and the
critical electric charge becomes $Q=0.69$. We do not show the figure of this
escape probability.)

In the non-extreme Kerr-Sen black hole case, as seen in the bottom diagram of
Fig. 5, we can know that the escape probabilities of the particles first
decrease and then increase monotonically with the increasing of the position
of the particle source. There is a cusp point on each curve. This is quite
different from the extreme Kerr-Sen black hole case.

##  VII Conclusions and Discussions

We considered the escape probabilities of the massless and massive particles
produced by a particle source which is at rest in a LNRF on the equatorial
plane of the Kerr-Sen black hole. Supposing that the particles escape from the
black hole to spatial infinity isotropically, we found that the escape
probability of the particle decreases with the electric charge (or increases
with the angular momentum) for the extreme Kerr-Sen black hole at the horizon
limit but increases with the electric charge (or decreases with the angular
momentum) for the non-extreme Kerr-Sen black hole near the horizon. This
illustrates that the angular momentum of the Kerr-Sen black hole plays a role
in promoting (or suppressing) the escape probability of the particle. This
implies that the rapidly rotating extreme Kerr-Sen black hole or the slowly
rotating non-extreme Kerr-Sen black hole can be easier to be observed by a
distant observer. In this regard, our results are quite different from those
in Ogasawara  _et al._ (2019); Igata  _et al._ (2020).

We also studied the escape probability variation of the particle with respect
to the radial position of the particle source. We found that there is a
critical electric charge for the extreme Kerr-Sen black hole, beyond which the
escape probability of the massless or massive particle increases monotonically
with the increasing distance apart from the event horizon, and below which the
probability decreases first and then increases. In the non-extreme Kerr-Sen
black hole background, we found that the escape probability of the particle
first decreases and then increases with the radial position of the particle
source, which is quite against common sense. The results shown in Fig. 5
indicate a fact, that is, it can be more difficult for a particle at a
position beyond the horizon than for a particle at a position near the horizon
of the Kerr-Sen black hole in certain conditions.

## Acknowledgements

Jie Jiang is supported by the National Natural Science Foundation of China
(Grant Nos. 11775022 and 11873044). M. Zhang is supported by the Initial
Research Foundation of Jiangxi Normal University with Grant No. 12020023.

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