# Results on Strong multiplicity one theorem

Chandrasheel Bhagwat and Gunja Sachdeva  Indian Institute of Science Education
and Research, Dr. Homi Bhabha Road, Pashan, Pune 411008, India.
<EMAIL_ADDRESS>BITS Pilani, K.K. BIRLA GOA Campus, NH 17B, Bypass
Road, Zuarinagar, Sancoale, Goa 403726, India<EMAIL_ADDRESS>
###### Abstract.

We prove an analogue of the strong multiplicity one theorem in the context of
$\tau_{n}$-spherical representations of the group $G={\rm SO}(2,1)^{\circ}$
appearing in $L^{2}(\Gamma_{i}\backslash G)$ for uniform torsion-free lattices
$\Gamma_{i},i=1,2$ in $G$. This is a generalisation of a previous result by
the first author and C. S. Rajan in [1] for the case of $G={\rm
SO}(2,1)^{\circ}$.

###### 2020 Mathematics Subject Classification:

22E40, 22E45,53C35

Keywords: Lie groups, Representation theory, Symmetric spaces, Spectral theory

##  1\. Introduction

It is well known that the representation theory of semisimple Lie groups and
the spectral theory of differential operators on associated symmetric and
locally symmetric spaces are intimately related. In the context of discrete
subgroup $\Gamma$ of a Lie group $G$ with compact quotient, the spectra are
discrete, and hence it makes sense to study the multiplicity function for
eigenvalues of Laplacian operator as well as the irreducible representations
appearing in function spaces on $\Gamma\backslash G$.

In [1], C.S. Rajan and the first author established a strong multiplicity one
property for $L^{2}(\Gamma\backslash G)$. In the same article, they have also
studied the rank-1 semisimple Lie groups and established a strong multiplicity
one property for the $K$-spherical representations of rank-1 semisimple Lie
group $G$, equivalently for Laplacian eigenvalues for
$\mathcal{C}^{\infty}(\Gamma\backslash G/K)$.

The aim of this short note is to prove an analogue of the above results for
the $\tau$-spherical representations of ${\rm SO}(2,1)^{\circ}$ for
irreducible representations $\tau$ of its maximal compact subgroup $K\cong{\rm
SO}(2)$. We use the classification of irreducible representations of ${\rm
SL}(2,\mathbb{R})$ and ${\rm SO}(2,1)^{\circ}$, their $K$-types, some
properties of the function spaces on ${\rm SO}(2,1)^{\circ}$ with respect to
the regular action of $K\times K$, and Selberg trace formula (as in [1]).

Our main theorem is

######  Theorem.

Let $G={\rm SO}(2,1)^{\circ}$, $\tau$ be an irreducible representation of $K$,
and $\widehat{G}_{\tau}$ be the set of isomorphism classes of irreducible
unitary representations $(\pi,V)$ of $G$ such that the $\tau$-isotypic
component $V^{\tau}\neq(0)$. Suppose $\Gamma_{1}$ and $\Gamma_{2}$ are uniform
torsion-free lattices in $G$ such that the multiplicities of $\pi$ in
$L^{2}(\Gamma_{i}\backslash G)$ for $i=1,2$ satisfy

$m(\pi,\Gamma_{1})=m(\pi,\Gamma_{2})\quad\text{for all but finitely many
representations}\ \pi\in\widehat{G}_{\tau}.$ $\text{Then,}\quad
m(\pi,\Gamma_{1})=m(\pi,\Gamma_{2})\quad\text{for all representations}\
\pi\in\widehat{G}_{\tau}.$

We also establish the relation for the multiplicities of
$\pi\in\widehat{G}_{\tau}$ in $L^{2}(\Gamma\backslash G)$ and multiplicities
of eigenvalues of Laplacian in $\mathcal{C}^{\infty}(\Gamma\backslash G,\tau)$
in Prop. 4.3. We note that the Laplacian eigenvalues corresponding to the non-
trivial discrete series representations are negative.

It is well-known that these eigenvalues are exactly the eigenvalues of the
Laplacian operator acting on the smooth sections of the homogeneous vector
bundles on $G/K$ defined by $\tau$, and hence all these eigenvalues must be
positive. This bodes well with the conjecture that the non-trivial discrete
series representations do not appear in $L^{2}(\Gamma\backslash G)$.

##  2\. Preliminaries

###  2.1. Lie groups and Lie algebras

Consider the real Lie group ${\rm SO}(2,1)(\mathbb{R})$ defined by

${\rm SO}(2,1)(\mathbb{R})=\Bigg{\\{}g\in{\rm SL}_{3}(\mathbb{R})\ :\
g^{T}\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right]g=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right]\Bigg{\\}}.$

Let $G={\rm SO}(2,1)^{\circ}$ be the connected component of $1_{G}$ in ${\rm
SO}(2,1)(\mathbb{R})$. An Iwasawa decomposition of $G$ is given by $G=ANK$
where

$\begin{split}A&=\Bigg{\\{}a_{t}=\left[\begin{array}[]{ccc}\cosh t&0&\sinh
t\\\ 0&1&0\\\ \sinh t&0&\cosh t\end{array}\right]\ :\
t\in\mathbb{R}\Bigg{\\}},\\\
N&=\Bigg{\\{}n_{u}=\left[\begin{array}[]{ccc}1-\dfrac{u^{2}}{2}&u&\dfrac{u^{2}}{2}\\\
-u&1&u\\\ -\dfrac{u^{2}}{2}&u&1+\dfrac{u^{2}}{2}\end{array}\right]\ :\
u\in\mathbb{R}\Bigg{\\}},\\\
K&=\Bigg{\\{}k_{\theta}=\left[\begin{array}[]{ccc}\cos\theta&-\sin\theta&0\\\
\sin\theta&\cos\theta&0\\\ 0&0&1\end{array}\right]\ :\
\theta\in[0,2\pi]\Bigg{\\}}=\Bigg{\\{}\left[\begin{array}[]{cc}B&\\\
&1\end{array}\right]\ :\ B\in{\rm SO}(2)\Bigg{\\}}.\end{split}$

The Lie algebra $\mathfrak{g}$ of $G$ is given by

$\begin{split}\mathfrak{g}&=\Bigg{\\{}X\in\mathfrak{s}\mathfrak{l}_{3}(\mathbb{R})\
:\ X^{\intercal}\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right]+\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right]X=0\Bigg{\\}}\\\
&\quad=\Bigg{\\{}\left[\begin{array}[]{ccc}0&a&b\\\ a&0&c\\\
b&-c&0\end{array}\right]\ :\ a,b,c\in\mathbb{R}\Bigg{\\}},\end{split}$

which is generated by the three matrices

$v_{1}=\left[\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&0\end{array}\right],\quad v_{2}=\left[\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\end{array}\right],\quad
W=v_{3}=\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\ 0&-1&0\end{array}\right].$

The Lie algebras of $A,N,$ and $K$ are given by

$\begin{split}\mathfrak{a}&=\Bigg{\\{}\left[\begin{array}[]{ccc}0&0&t\\\
0&0&0\\\ t&0&0\end{array}\right]\ :\
t\in\mathbb{R}\Bigg{\\}}=\mathbb{R}v_{2},\quad\mathfrak{n}=\Bigg{\\{}\left[\begin{array}[]{ccc}0&x&0\\\
x&0&-x\\\ 0&x&0\end{array}\right]\ :\
x\in\mathbb{R}\Bigg{\\}}=\mathbb{R}(v_{1}-v_{3}),\\\
\mathfrak{k}&=\Bigg{\\{}\left[\begin{array}[]{ccc}0&0&0\\\ 0&0&-\theta\\\
0&\theta&0\end{array}\right]\ :\
\theta\in\mathbb{R}\Bigg{\\}}=\mathbb{R}v_{3}.\end{split}$

The map ${\rm ad}~{}W$ under the adjoint representation on
$\mathfrak{g}_{\mathbb{C}}$ is given by ${\rm ad}~{}W:X\longmapsto[W,X].$ One
can verify that

${\rm ad}~{}W(v_{1})=[W,v_{1}]=-v_{2},\text{ and }{\rm
ad}~{}W(v_{2})=[W,v_{2}]=v_{1}.$

Using these two Lie bracket relations, we get

$[W,v_{1}+{{\mathbf{i}}}v_{2}]={{\mathbf{i}}}(v_{1}+iv_{2}),\quad[W,v_{1}-{\bf
i}v_{2}]=-{\mathbf{i}}(v_{1}-{\mathbf{i}}v_{2}),$

which tells that $v_{1}\pm{\mathbf{i}}v_{2}$ are eigenvectors of ${\rm
ad}~{}W$ with eigenvalues $\pm{{\mathbf{i}}}$, respectively. We denote the
eigenvectors by

$\begin{split}E^{+}:&=v_{1}+{\mathbf{i}}v_{2}=\left[\begin{array}[]{ccc}0&1&{\mathbf{i}}\\\
1&0&0\\\ {\mathbf{i}}&0&0\end{array}\right],\quad
E^{-}:=v_{1}-{\mathbf{i}}v_{2}=\left[\begin{array}[]{ccc}0&1&-{\mathbf{i}}\\\
1&0&0\\\ -{\mathbf{i}}&0&0\end{array}\right].\end{split}$

###  2.2. Isomorphism between ${\rm PSL}_{2}(\mathbb{R})$ and ${\rm
SO}(2,1)^{\circ}$

We fix an isomorphism between the groups ${\rm PSL}_{2}(\mathbb{R})$ and ${\rm
SO}(2,1)^{\circ}$ once and for all.

Define $\Psi:{\rm SL}_{2}(\mathbb{R})\longrightarrow{\rm SO}(2,1)^{\circ}$ by

$\Psi\left(\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\right)=\left[\begin{array}[]{ccc}\dfrac{1}{2}(a^{2}-b^{2}-c^{2}+d^{2})&ab-
cd&\dfrac{1}{2}(a^{2}+b^{2}-c^{2}-d^{2})\\\ ac-bd&ad+bc&ac+bd\\\
\dfrac{1}{2}(a^{2}-b^{2}+c^{2}-d^{2})&ab+cd&\dfrac{1}{2}(a^{2}+b^{2}+c^{2}+d^{2})\end{array}\right].$

It can be checked that $\Psi$ is a group homomorphism, and Kernel($\Psi$) =
$(\pm I_{2})$. Thus $\Psi$ defines an isomorphism of Lie groups between ${\rm
PSL}_{2}(\mathbb{R})$ and ${\rm SO}(2,1)^{\circ}$. We also note down the
images of some special elements of ${\rm SL}_{2}(\mathbb{R})$ under $\Psi$
here.

$\begin{split}\Psi\left[\begin{array}[]{cc}e^{t}&0\\\
0&e^{-t}\end{array}\right]=a_{2t}=\left[\begin{array}[]{ccc}\cosh 2t&0&\sinh
2t\\\ 0&1&0\\\ \sinh 2t&0&\cosh 2t\end{array}\right]\quad\forall\
t\in\mathbb{R},\\\ \Psi\left[\begin{array}[]{cc}1&u\\\
0&1\end{array}\right]=n_{u}=\left[\begin{array}[]{ccc}1-\dfrac{u^{2}}{2}&u&\dfrac{u^{2}}{2}\\\
-u&1&u\\\
-\dfrac{u^{2}}{2}&u&1+\dfrac{u^{2}}{2}\end{array}\right]\quad\forall\
u\in\mathbb{R},\\\ \Psi\left[\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right]=k_{2\theta}=\left[\begin{array}[]{ccc}\cos
2\theta&-\sin 2\theta&0\\\ \sin 2\theta&\cos 2\theta&0\\\
0&0&1\end{array}\right]\quad\forall\ \theta\in\mathbb{R}.\end{split}$

Thus the Iwasawa decompositions for ${\rm SL}_{2}(\mathbb{R})$ and ${\rm
SO}(2,1)^{\circ}$ are related via the map $\Psi$. We make use of this fact
later in the discussion for $K$-types of irreducible unitary representations
of ${\rm SO}(2,1)^{\circ}$.

##  3\. Representations of ${\rm SO}(2,1)^{\circ}$ and associated functions on
hyperbolic plane

###  3.1. Induced representations of $G$

Fix Haar measures on $G$ and $K$ such that $\int\limits_{K}1~{}dk=1$. We
identify $dk$ with the normalised Lebesgue measure $\dfrac{d\theta}{2\pi}$ on
$[0,2\pi]$. Let $s\in\mathbb{C}$. Define a character of the torus $A$ of
$G={\rm SO}(2,1)^{\circ}$ by

$\mu_{s}:a_{t}\mapsto e^{st}\quad\forall\ t\in\mathbb{R}.$

Inflate $\mu_{s}$ to the Borel subgroup $B:=NA=AN$ of $G$ and then consider
the induced space $V(s)$ consisting of all measurable functions
$f:G\rightarrow\mathbb{C}$ such that $f_{|_{K}}\in L^{2}(K,dk)$ and

$f(bg)=e^{t}\mu_{s}(a_{t})f(g)\quad\forall\ b=a_{t}n_{u}\in B=AN,\ g\in G.$

Define a representation $\rho_{s}$ of $G$ on $V(s)$ by

$\rho_{s}(g)f(y)=f(yg)\quad\forall\ g,y\in G,\ f\in V(s).$

The representation $(\rho_{s},V(s))$ is unitary and irreducible precisely when
$s\in{\mathbf{i}}\mathbb{R}\cup(-1,1)$ (see [2]). Furthermore, the only
possible isomorphisms between $(\rho_{s},V(s))$ and
$(\rho_{s}^{\prime},V(s^{\prime}))$ are when $s=\pm s^{\prime}$. Thus we can
without loss of generality assume $s\in{\mathbf{i}}\mathbb{R}_{\geq
0}\sqcup(0,1)$ while considering the irreducible unitary representation
$V(s)$, and we call $(\rho_{s},V(s))$ by the names principal series  when
$s\in{\mathbf{i}}\mathbb{R}_{\geq 0}$ and complementary series  when
$s\in(0,1)$, respectively.

###  3.2. Associated spherical functions on $\mathcal{H}$

Fix $s\in\mathbb{C}$. Let $\mathcal{H}=\\{z=x+{\mathbf{i}}y:y>0\\}$ be the
hyperbolic $2$-space. Consider the usual action of ${\rm SL}(2,\mathbb{R})$ on
$\mathcal{H}$ by fractional linear transformations and let $g:z\mapsto g\cdot
z:=\psi^{-1}(g)\cdot z$ be the corresponding action of $G={\rm
SO}(2,1)^{\circ}$ on $\mathcal{H}$. We use this to define a function
$\chi_{s}:\mathcal{H}\rightarrow\mathbb{C}$ by

$\chi_{s}(x+{\mathbf{i}}y)=\chi_{s}(n_{x}a_{\ln
y}\cdot{\mathbf{i}})=\mu_{s}(n_{x}a_{\ln y})=y^{s}\quad\forall\
x+{\mathbf{i}}y\in\mathcal{H}.$

Let $\phi_{s}$ be the function defined by

$\phi_{s}(z)=\int\limits_{K}\chi_{s}(k\cdot z)~{}dk\quad\forall\
z\in\mathcal{H},$

where $k\cdot z:=\psi^{-1}(k)\cdot z\ \text{for all}\ z\in\mathcal{H},\ k\in
K$, and $\psi^{-1}(k)\cdot z$ denotes the usual action of ${\rm
SL}(2,\mathbb{R})$ on $\mathcal{H}$ by fractional linear transformations.

A simple calculation (as in [3, p.65-67]) gives that

######  Lemma 3.1.

For every $s\in\mathbb{C}$, the function $\phi_{s+\frac{1}{2}}$ satisfies:

  1. (i)

$\phi_{s+\frac{1}{2}}(k\cdot z)=\phi_{s+\frac{1}{2}}(z)\quad\forall\ k\in K,\
z\in\mathcal{H}$.

  2. (ii)

$\phi_{s+\frac{1}{2}}\in\mathcal{C}^{\infty}(\mathcal{H})$ and
$\Delta\phi_{s+\frac{1}{2}}=\frac{1-s^{2}}{4}\phi_{s+\frac{1}{2}}$ where
$\Delta=-y^{2}(\frac{d^{2}}{dx^{2}}+\frac{d^{2}}{dy^{2}})$ is the hyperbolic
Laplacian operator on $\mathcal{H}$.

###  3.3. Discrete series representations

For every positive integer $m\geq 2$, there are inequivalent unitary discrete
series representations $D^{\pm}(m)$ of the group $G$ that can be realised as
an irreducible sub-representation of the induced space $V(m-1)$. This follows
from the construction of discrete series representations of ${\rm
SL}(2,\mathbb{R})$ (see [2]).

###  3.4. $K$-types of irreducible representations of ${\rm SO}(2,1)^{\circ}$

Let $n\in\mathbb{Z}$. Define $\tau_{n}$ to be a character of the (abelian)
group $K$ by

$\tau_{n}(k_{\theta})=e^{{\mathbf{i}}n\theta}\quad\forall\ k_{\theta}\in K.$

For any unitary irreducible representation $(\pi,V)$ of $G$, we have a Hilbert
direct sum decomposition of $V$ into $K$-isotypes

$V=\widehat{\bigoplus\limits_{n\in\mathbb{Z}}}H_{n},$

where $H_{n}$ is the $K$-isotype $V^{\tau_{n}}$ of $V$ w.r.t. $\tau_{n}$, i.e.

$H_{n}=\\{v\in
V:\pi(k_{\theta})v=\tau_{n}(k_{\theta})v=e^{{\mathbf{i}}n\theta}v\quad\forall\
k_{\theta}\in K\\}.$

It can be shown that for every $n\in\mathbb{Z}$, the complex vector space
$H_{n}$ is either $(0)$ or of dimension $1$ (this is a well-known fact. As
reference, see [2, P. 24]).

We say that the $K$-type $\tau_{n}$ appears in $\pi$ if $H_{n}\neq(0)$ in $V$.
Thus

$V=\widehat{\bigoplus\limits_{n\in\mathbb{Z}}}H_{n}=\widehat{\bigoplus\limits_{\tau_{n}\
\text{appears in}~{}\pi}}H_{n}.$

The irreducible unitary representations $\pi$ of $G$ bijectively correspond to
the irreducible unitary representations $\tilde{\pi}$ of ${\rm
SL}(2,\mathbb{R})$ such that $\tilde{\pi}(-I_{2\times 2})=1$. Thus only ‘half’
of the discrete series representations of ${\rm SL}(2,\mathbb{R})$ appear in
the list for $G$. Using this and the adjoint action of the basis elements of
the Lie algebra $\mathfrak{g}$, we can compute the $K$-types that appear in
$\pi$ for all irreducible unitary representations $\pi$ of $G$. The following
table summarises this information.

$\begin{array}[]{|c|c|}\hline\cr 1&\tau_{0}\\\ \hline\cr D_{m}^{+},\ \ m\geq
2\ \text{ even}&\tau_{n}:\ \ n=\frac{m}{2}+j,\ j\geq 0\\\ \hline\cr
D_{m}^{-},\ \ m\leq-2\ \text{ even}&\tau_{n}:\ \ n=-\frac{m}{2}-j,\ j\geq 0\\\
\hline\cr\rho_{s},\ \ s\in{\mathbf{i}}\mathbb{R}_{\geq
0}\sqcup(0,1)&\tau_{n}:\ \ n\in\mathbb{Z}\\\ \hline\cr\end{array}$

######  Definition 3.2.

An irreducible unitary representation $(\pi,V)$ of $G$ is said to be
$\tau_{n}$-spherical if there exists a nonzero vector $v\in V$ such that

$\pi(k_{\theta})v=\tau_{n}(k_{\theta})v=e^{in\theta}v\quad\forall\
k_{\theta}\in K.$

For a given $n\in\mathbb{Z}$, we can list the $\tau_{n}$-spherical
representations of $G$ as follows:

$\begin{array}[]{|c|c|}\hline\cr
n>0&1,D_{m}:m\in\\{2,4,\ldots,2n\\},\rho_{s}:s\in{\mathbf{i}}\mathbb{R}_{\geq
0}\sqcup(0,1)\\\ \hline\cr n=0&1,\rho_{s}:s\in{\mathbf{i}}\mathbb{R}_{\geq
0}\sqcup(0,1)\\\ \hline\cr
n<0&1,D_{m}:m\in\\{-2,-4,\ldots,-2n\\},\rho_{s}:s\in{\mathbf{i}}\mathbb{R}_{\geq
0}\sqcup(0,1)\\\ \hline\cr\end{array}$

Notation: Let $\widehat{G}_{\tau_{n}}$ denote the $\tau_{n}$-spherical
spectrum of $G$, defined as the set of all equivalence classes of irreducible
unitary representations $\pi$ of $G$ such that $\tau_{n}$ appears in $\pi$.

###  3.5. $\tau_{n}$-spherical representations and $\tau_{n}$-spherical
functions

For $n\in\mathbb{Z}$, we denote by $\mathcal{C}_{c}^{\infty}(G//K,\tau_{n})$,
the space of all smooth functions $f:G\rightarrow\mathbb{C}$ such that

$f(k_{\theta_{1}}xk_{\theta_{2}})=e^{{\mathbf{i}}n(\theta_{1}+\theta_{2})}f(x)\quad\forall\
x\in G,k_{\theta_{1}},k_{\theta_{2}}\in K.$

Let $(\pi,V)$ be a $\tau_{n}$-spherical representation of $G$. Let $e_{\pi}\in
H_{n}=V^{\tau_{n}}\subseteq V$ be a unit vector. Let
$\phi_{\pi}:G\rightarrow\mathbb{C}$ be a function defined by

$\phi_{\pi}(x)=\langle\pi(x)e_{\pi},e_{\pi}\rangle\quad\forall\ x\in G.$

Thus it follows that $\phi_{\pi}\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{n})$.

We state a lemma here which will be used later.

######  Lemma 3.3.

Let $(\pi,V)$ be a $\tau_{n}$-spherical representation of $G$. Then the
character distribution of $\pi$ satisfies

$\chi_{\pi}(f)=\int\limits_{G}f(g)\phi^{\prime}_{\pi}(g)dg\quad\forall\
f\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{n}),$

where $\phi^{\prime}_{\pi}(g)$ is defined by

$\phi^{\prime}_{\pi}(g)=\begin{cases}\langle\pi(x)v_{-n},v_{-n}\rangle&\text{if}\
H_{-n}=\mathbb{C}v_{-n}\neq(0),||v_{-n}||=1\\\ 0&\text{if}\
H_{-n}=(0).\end{cases}$

  * Proof.

Let $k_{\theta}\in K$, $v,w\in V$,
$f\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{n})$. We have

$\begin{split}\langle\pi(k_{\theta})\pi(f)v,w\rangle&=\langle\pi(f)v,\pi(k_{-\theta})w\rangle\\\
&=\int\limits_{G}f(x)~{}\langle\pi(x)v,\pi(k_{-\theta})w\rangle~{}dx\\\
&=\int\limits_{G}f(k_{-\theta}x)~{}\langle\pi(x)v,w\rangle~{}dx\\\
&=e^{-{\mathbf{i}}n\theta}\langle\pi(f)v,w\rangle.\end{split}$

Thus we conclude that

$\pi(f)v\in V^{\tau_{-n}}=H_{-n}\quad\forall\ v\in V,\
f\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{n}).$

Case (1) The isotypic component $V^{\tau_{-n}}=H_{-n}$ in $(\pi,V)$ is non-
zero, say equals $\mathbb{C}v_{-n}$ for some $v_{-n}$ with $||v_{-n}||=1$.
Thus $\chi_{\pi}(f)$ = trace $\pi(f)=\langle\pi(f)v_{-n},v_{-n}\rangle$ since
the components $H_{m}$ are mutually orthogonal and $\pi(f)v\in H_{-n}\
\forall\ v\in V$.

Case (2) The isotypic component $V^{\tau_{-n}}=H_{-n}$ in $(\pi,V)$ is zero.
In this case $\pi(f)=0$ from above calculation.

In both cases, we get the desired conclusion for the character distribution
$\chi_{\pi}$ of $\pi$.

∎

From Lemma 3.3 and its proof, we also conclude that

######  Corollary 3.4.

If $\pi$ is $\tau_{n}$-spherical, and
$f\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{-n})$, then

$\chi_{\pi}(f)=\int\limits_{G}f(x)\phi_{\pi}(x)~{}dx.$

##  4\. Main results

Let $\Gamma_{1}$ and $\Gamma_{2}$ be uniform torsion-free lattices in $G$ so
that the quotient spaces $\Gamma_{1}\backslash G$ and $\Gamma_{2}\backslash G$
are compact. Consider the regular representations of $G$ on
$L^{2}(\Gamma_{i}\backslash G)$ for $i=1,2$. For an irreducible unitary
representation $\pi$ of $G$, let $m(\pi,\Gamma_{i})$ be the multiplicity of
$\pi$ in the decomposition of $L^{2}(\Gamma_{i}\backslash G)$.

We prove the following analogue of the strong multiplicity theorem for
$\tau_{n}$-spherical spectra for $\Gamma_{1}$ and $\Gamma_{2}$.

######  Theorem 4.1.

Let $n\in\mathbb{Z}$. Suppose $\Gamma_{1}$ and $\Gamma_{2}$ are uniform
torsion-free lattices in $G$ such that

$m(\pi,\Gamma_{1})=m(\pi,\Gamma_{2})\quad\text{for all but finitely many
representations}\ \pi\in\widehat{G}_{\tau_{n}}.$ $\text{Then}\quad
m(\pi,\Gamma_{1})=m(\pi,\Gamma_{2})\quad\text{for all representations}\
\pi\in\widehat{G}_{\tau_{n}}.$

Let $\Gamma$ be a uniform torsion-free lattice in $G$. The center
$Z(U_{\mathfrak{g}})$ of the universal enveloping algebra of $G$ acts on the
space of all smooth functions on $\Gamma\backslash G$. Let
$\mathcal{C}^{\infty}(\Gamma\backslash G,\tau_{n})$ be the subspace of
$\mathcal{C}^{\infty}(\Gamma\backslash G)$ consisting of all smooth right
$\tau_{n}$-equivariant $f$ i.e.

$f(gk_{\theta})=e^{{\mathbf{i}}n\theta}f(g)\quad\forall\ g\in G,\
k_{\theta}\in K.$

It can be checked that $\mathcal{C}^{\infty}(\Gamma\backslash G,\tau_{n})$ is
stable under $Z(U_{\mathfrak{g}})$-action, and the eigenspaces
$V(\lambda,\Gamma,\tau_{n})$, w.r.t. characters $\lambda$ of
$Z(U_{\mathfrak{g}})$ are finite dimensional subspaces of
$\mathcal{C}^{\infty}(\Gamma\backslash G,\tau_{n})$. We discuss the following
result that relates the multiplicities $m(\pi,\Gamma_{i})$ for
$\pi\in\widehat{G}_{\tau_{n}}$ with these eigenspaces.

If $(\pi,V_{\pi})$ is a $\tau_{n}$-spherical representation, then we can
associate a character $\lambda_{\pi}$ of $Z(U_{\mathfrak{g}})$ such that
$\phi_{\pi}(x):=\langle\pi(x)e_{\pi},e_{\pi}\rangle$ defines an eigenfunction
of $\lambda_{\pi}$ where $e_{\pi}$ is a vector in $V_{\pi}$ with
$||e_{\pi}||=1$.

######  Lemma 4.2.

If $\pi_{j}:1\leq j\leq r$ are mutually inequivalent $\tau_{n}$-spherical
representations then $\phi_{\pi_{j}}:1\leq j\leq r$ are linearly independent
and analytic functions on $G$.

  * Proof.

The conclusion of this lemma follows at once from the fact that
$\phi_{\pi_{j}}$s are eigenvectors of distinct characters and also of elliptic
essentially self-adjoint differential operators in $Z(U_{\mathfrak{g}})$. ∎

######  Proposition 4.3.

If $\pi$ is a $\tau_{n}$-spherical representation, then

$m(\pi,\Gamma)={\rm\dim}_{\mathbb{C}}V(\lambda_{\pi},\Gamma,\tau_{n}).$

Conversely, if $\lambda$ is a character of $Z(U_{\mathfrak{g}})$ such that
$V(\lambda,\Gamma)\neq 0$, then there exists a unique $\tau_{n}$-spherical
representation $\pi$ of $G$ such that $\lambda=\lambda_{\pi}$ and
$m(\pi,\Gamma)={\rm\dim}_{\mathbb{C}}V(\lambda_{\pi},\Gamma,\tau_{n})$.

  * Proof.

Let $m=m(\pi,\Gamma)$. Let $W_{j}:1\leq j\leq m$ be mutually orthogonal
subspaces in $L^{2}(\Gamma\backslash G)$, each of them isomorphic to $\pi$.
Let $e_{\pi}^{(j)}\in W_{j}$ be a unit vector such that it is a basis of the
one dimensional $\tau_{n}$-isotype $W_{j}^{\tau_{n}}$. Define $h_{j}$ by

$h_{j}(x)=\int\limits_{[0,2\pi]}e^{-{\mathbf{i}}n\theta}e_{\pi}^{(j)}(xk_{\theta})\frac{d\theta}{2\pi}\quad\forall\
x\in G.$

These functions $h_{j}$s are smooth, are in
$\mathcal{C}^{\infty}(\Gamma\backslash G,\tau_{n})$, and are eigenfunctions of
the character $\lambda_{\pi}$ of $Z(U_{\mathfrak{g}})$. Since the vectors
$e_{\pi}^{(j)}\in W_{j}$ and are mutually orthogonal, it follows that
functions $h_{j}$s are also mutually orthogonal and hence we have

$m(\pi,\Gamma)\leq{\rm\dim}_{\mathbb{C}}V(\lambda_{\pi},\Gamma,\tau_{n}).$

Conversely, if $f\in\mathcal{C}^{\infty}(\Gamma\backslash G,\tau_{n})$ is an
eigenfunction of a character $\lambda$ of $Z(U_{\mathfrak{g}})$, then $f\in
L^{2}(\Gamma\backslash G)$. Write

$L^{2}(\Gamma\backslash G)=\widehat{\bigoplus\limits_{j}}V_{j},$

where each $V_{j}$ is an irreducible subspace of $L^{2}(\Gamma\backslash G)$
corresponding to $\pi_{j}\in\widehat{G}$.

Hence $f$ can be written as

$f=\sum\limits_{j}\alpha_{j}v_{j},$

where each $v_{j}$ is a unit vector in $V_{j}$.

Observe that, we in fact get a sum over only those $j$ such that
$V_{j}^{\tau_{n}}\neq 0$. Further, since $f$ is an eigenfunction of $\lambda$,
it follows that $v_{j}$ is also an eigenfunction of $\lambda$ for all $j$ such
that $\alpha_{j}\neq 0$.

Using the linear independence of characters $\lambda_{\pi_{j}}$ for
$\pi_{j}\in\widehat{G}$, the sum is over only those $V_{j}$ for which the
representation is $\pi$ and the associated character is
$\lambda_{\pi}=\lambda$. This implies that

$m(\pi,\Gamma)={\rm\dim}_{\mathbb{C}}V(\lambda_{\pi},\Gamma,\tau_{n}).$

∎

######  Lemma 4.4.

The space $\mathcal{C}_{c}^{\infty}(G//K,\tau_{n})$ separates points on the
orbits of $K\times K$-action on $G$ given by

$(k_{\theta_{1}},k_{\theta_{2}})\cdot g:=k_{\theta_{1}}gk_{\theta_{2}}^{-1}.$

  * Proof.

Let $KxK$ and $KyK$ be disjoint orbits of the above action. Let $f$ be a
smooth and compactly supported function such that $f=0$ on $KyK$ and $f>0$ on
$KxK$. Define $F$ by

$F(g)=\int\limits_{[0,2\pi]}~{}\int\limits_{[0,2\pi]}~{}e^{-{\mathbf{i}}n(\theta_{1}+\theta_{2})}f(k_{\theta_{1}}gk_{\theta_{2}})\frac{d\theta_{1}}{2\pi}\frac{d\theta_{2}}{2\pi}.$

This $F$ is in $\mathcal{C}_{c}^{\infty}(G//K,\tau_{n})$, and separates $KxK$
and $KyK$ because of the assumed properties of the function $f$. ∎

We state two more results about uniform lattices in semisimple Lie groups
here. We refer the reader to [1] for the proofs.

######  Lemma 4.5.

If $\Gamma$ is a torsion-free uniform lattice in $G$, then every nontrivial
conjugacy class $[\gamma]_{G}$ of $\gamma\in\Gamma$ is disjoint from $K$ and
hence $1_{G}\notin K[\gamma]_{G}K$.

######  Lemma 4.6.

Given torsion-free uniform lattices $\Gamma_{1}$ and $\Gamma_{2}$ in $G$,
there exists a non-empty open set $B\subset G$ which is $(K\times K)$-stable
and disjoint from all conjugacy classes $[\gamma]_{G}$ with
$\gamma\in\Gamma_{1}\cup\Gamma_{2}$.

  * Proof of Thm. 4.1.

Let $S$ be a finite subset of $\widehat{G}_{\tau_{n}}$ such that

$m(\pi,\Gamma_{1})=m(\pi,\Gamma_{2})\quad\forall\
\pi\in\widehat{G}_{\tau_{n}}\setminus S.$

Applying the Selberg trace formula for lattices $\Gamma_{1}$ and $\Gamma_{2}$
for a function $f\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{-n})$, we see that

$\sum\limits_{\pi\in
S}\left(m(\pi,\Gamma_{1})-m(\pi,\Gamma_{2})\right)\chi_{\pi}(f)=\sum\limits_{[\gamma]\in[\Gamma_{1}]_{G}\cup[\Gamma_{2}]_{G}}[a(\gamma,\Gamma_{1})-a(\gamma,\Gamma_{2})]O_{\gamma}(f)$

(here the term $O_{\gamma}(f)$ refers to the orbital integral of $f$ w.r.t.
the left invariant measure on the conjugacy class $[\gamma]$ in $G$ (see [1],
[4] for more details).

Let $\phi=\sum\limits_{\pi\in
S}\left(m(\pi,\Gamma_{1})-m(\pi,\Gamma_{2})\right)\phi_{\pi}$. Thus using Lem.
3.3, we have

$\int\limits_{G}f(g)\
\phi(g)d\mu(g)=\sum\limits_{[\gamma]\in[\Gamma_{1}]_{G}\cup[\Gamma_{2}]_{G}}[a(\gamma,\Gamma_{1})-a(\gamma,\Gamma_{2})]O_{\gamma}(f).$

Choose a $K\times K$-stable open set $B$ that avoids all conjugacy classes
appearing on the right hand side of above (as in Lem. 4.6.) For every
$f\in\mathcal{C}_{c}^{\infty}(G//K,\tau_{-n})$ that is supported on $B$, we
have

$\int\limits_{G}f(g)\ \phi(g)d\mu(g)=0,$

showing that $\phi=0$ on $B$ since $\mathcal{C}_{c}^{\infty}(G//K,\tau_{-n})$
separates points on $B$ as seen in Lem. 4.4. Since all $\phi_{\pi}$ with
$\pi\in S$ are analytic and linearly independent, we conclude that

$m(\pi,\Gamma_{1})=m(\pi,\Gamma_{2})\quad\forall\ \pi\in S.$

∎

## References

  * [1] Bhagwat, C., Rajan C. S., On a multiplicity one property for the length spectra of even dimensional compact hyperbolic spaces. J. Number Theory 131 (2011), no. 11, 2239–2244.
  * [2] Lang, Serge, ${\rm SL}_{2}(\mathbb{R})$. Graduate Texts in Mathematics, 105. Springer-Verlag, New York, (1985).
  * [3] Lubotzky, Alexander, Discrete groups, expanding graphs and invariant measures. With an appendix by Jonathan D. Rogawski. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, (2010).
  * [4] Wallach, Nolan R. On the Selberg trace formula in the case of compact quotient. Bull. Amer. Math. Soc. 82 (1976), no. 2, 171-195.


# Probabilistic multi-Stirling numbers of the second kind and probabilistic
multi-Lah numbers

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.

Assume that the moment generating function of the random variable $Y$ exists
in a neighborhood of the origin. We introduce the probabilistic multi-Stirling
numbers of the second kind associated with $Y$ and the probabilistic multi-Lah
numbers associated with $Y$, both of indices $(k_{1},k_{2},\dots,k_{r})$, by
means of the multiple logarithm. Those numbers are respectively probabilistic
extensions of the multi-Stirling numbers of the second kind and the multi-Lah
numbers which, for $(k_{1},k_{2},\dots,k_{r})=(1,1,\dots,1)$, boil down
respectively to the Stirling numbers of the second and the unsigned Lah
numbers. The aim of this paper is to study some properties, related
identities, recurrence relations and explicit expressions of those
probabilistic extension numbers in connection with several other special
numbers.

###### Key words and phrases:

probabilistic multi-Stirling numbers of the second kind associated with $Y$;
probabilistic multi-Lah numbers associated with $Y$

###### 2010 Mathematics Subject Classification:

11B68; 11B73; 11B83

* is corresponding author.

##  1\. Introduction

Assume that $Y$ is a random variable whose moment generating function exists
in a neighborhood of the origin (see (8)). The aim of this paper is to study
the probabilistic multi-Stirling numbers of the second kind ${n\brace
k_{1},k_{2},\dots k_{r}}_{Y}$ associated with $Y$ (see (20)) and the
probabilistic multi-Lah numbers $L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)$
associated with $Y$ (see (28)). We investigate some properties, related
identities, recurrence relations and explicit expressions for those numbers in
connection with the multi-Bernoulli numbers
$B_{n}^{(k_{1},k_{2},\dots,k_{r})}$ (see (5)), the probabilistic Fubini
polynomials of order $r$, $F_{n}^{(r,Y)}(y)$, associated with $Y$ (see (33)),
the probabilistic Stirling numbers of the second kind ${n\brace r}_{Y}$
associated with $Y$ (see (16)), the unsigned multi-Stirling numbers of the
first kind ${n\brack k_{1},k_{2},\dots,k_{r}}$ (see (10)), the unsigned
Stirling numbers of first kind ${n\brack r}$ (see (6)) and the Stirling
numbers of the second kind ${n\brace r}$ (see (7)).

We remark that ${n\brace k_{1},k_{2},\dots k_{r}}_{Y}$ is a probabilistic
extension of the multi-Stirling number of the second kind ${n\brace
k_{1},k_{2},\dots k_{r}}$ (see (12)), which reduces to the Stirling number of
the second kind ${n\brace r}$, for $(k_{1},k_{2},\dots k_{r})=(1,1,\dots,1)$.
We also note that $L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)$ is a probabilistic
extension of the multi-Lah number $L^{(k_{1},\dots,k_{r})}(n,r)$ (see (2)),
which boils down to the unsigned Lah number $L(n,r)$ (see (4)), for
$(k_{1},k_{2},\dots k_{r})=(1,1,\dots,1)$. In addition, ${n\brack
k_{1},k_{2},\cdots,k_{r}}$ and $B_{n}^{(k_{1},k_{2},\dots,k_{r})}$ generalize
respectively ${n\brack r}$ and $\frac{(-1)^{n}}{r!}B_{n}^{(r)}$, for
$(k_{1},k_{2},\dots k_{r})=(1,1,\dots,1)$, where $B_{n}^{(r)}$ are the
Bernoulli numbers of order $r$. The common feature of the six numbers
${n\brace k_{1},k_{2},\dots k_{r}}_{Y}$,
$L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)$, ${n\brace k_{1},k_{2},\dots k_{r}}$,
$L^{(k_{1},k_{2},\dots,k_{r})}(n,r)$, ${n\brack k_{1},k_{2},\cdots,k_{r}}$ and
$B_{n}^{(k_{1},k_{2},\dots,k_{r})}$ is that they are all defined with the help
of the multiple logarithm $\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}(t)$ (see
(1)), which reduces to the polylogarithm $\mathrm{Li}_{k_{1}}(t)$, for $r=1$
and to $\frac{1}{r!}\big{(}-\log(1-t)\big{)}^{r}$, for $(k_{1},k_{2},\dots
k_{r})=(1,1,\dots,1)$.

The outline of this paper is as follows. First, we recall the multiple
logarithms, the multi-Lah numbers and the multi-Bernoulli numbers. We remind
the reader of the unsigned Stirling numbers of the first kind and the Stirling
numbers of the second kind. We recall the probabilistic Lah numbers
$L_{Y}(n,k)$ associated with $Y$. We remind the reader of the unsigned multi-
Stirling numbers of the first kind and the multi-Stirling numbers of the
second kind. Finally, we recall the probabilistic Stirling numbers of the
second. Section 2 contains the main results of this paper. We derive first
that
$\mathrm{Li}_{\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}(t)=\frac{1}{r!}\big{(}-\log(1-t)\big{)}^{r}$.
Then we define the probabilistic multi-Stirling numbers of the second kind
${n\brace k_{1},k_{2},\dots k_{r}}_{Y}$ associated with $Y$, and show
${n\brace\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\frac{}{}_{Y}={n\brace
r}_{Y}$. In Theorem 2.1, we express ${n+1\brace k_{1},k_{2},\dots,k_{r-1},1}$
as a finite sum involving ${m\brace
k_{1},k_{2},\dots,k_{r-1}},\,\,(m=r-1,\dots,n)$, which generalizes a well-
known identity for the Stirling numbers of the second kind to that for the
probabilistic Stirling numbers of the second kind associated with $Y$. In
Theorem 2.2, we express
$\sum_{m=0}^{n}B_{m}^{(k_{1},k_{2},\dots,k_{r})}{n\brace m}_{Y}$ as a finite
sum involving the product ${i\brace j}_{Y}{n-i\brace
k_{1},k_{2},\dots,k_{r}}_{Y}$. In Theorem 2.3, we represent ${n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}$ as a finite sum involving the triple product
${l\brace m}{n\brace l}_{Y}{m\brack k_{1},k_{2},\dots,k_{r}}$. We define the
probabilistic multi-Lah numbers $L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)$
associated with $Y$. In Theorem 2.4, we show that
$L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)=\sum_{k=r}^{n}{n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}{n\brack k}$. In Theorem 2.5, we express ${n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}$ as a finite sum involving the triple product
${l\brace r}{n\brace m+l}_{Y}B_{m}^{(k_{1},k_{2},\dots,k_{r})}$. In Theorem
2.6, we show the identity $\sum_{k=r}^{n}{n\brace
k}_{Y}L^{(k_{1},k_{2},\dots,k_{r})}(k,r)=\sum_{k=r}^{n}\binom{n}{k}{k\brace
k_{1},k_{2},\dots,k_{r}}_{Y}F_{n-k}^{(r,Y)}(1)$, where $F_{n}^{(r,Y)}(y)$ are
the probabilistic Fubini polynomials of order $r$. For the rest of this
section, we recall the facts that are needed throughout this paper.

For integers $k_{i}\ (1\leq i\leq r)$, the multiple logarithm of index
$(k_{1},k_{2},\dots,k_{r})$ is given by

(1)
$\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}(t)=\sum_{0<m_{1}<m_{2}<\cdots<m_{r}}\frac{t^{m_{r}}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{r}^{k_{r}}},\quad(|t|<1).\quad(\mathrm{see}\ [19,24]).$

In particular, for $r=1$, we have

$\mathrm{Li}_{k_{1}}(t)=\sum_{m=1}^{\infty}\frac{t^{m}}{m^{k_{1}}},\quad(\mathrm{see}\
[19,24,28]),$

where $\mathrm{Li}_{k_{1}}(t)$ is called the polylogarithm.

Recently, the multi-Lah number of index $(k_{1},k_{2},\dots,k_{r})$ is defined
by

(2)
$\frac{\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}(1-e^{-t})}{(1-t)^{r}}=\sum_{n=r}^{\infty}L^{(k_{1},\dots,k_{r})}(n,r)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[19,24]).$

Note that

(3)
$\sum_{n=r}^{\infty}L^{\overbrace{(1,1,\dots,1)}^{r-times}}(n,r)\frac{t^{n}}{n!}=\frac{\mathrm{Li}_{1,1,\dots,1}(1-e^{-t})}{(1-t)^{r}}=\frac{1}{r!}\bigg{(}\frac{t}{1-t}\bigg{)}^{r}=\sum_{n=r}^{\infty}L(n,r)\frac{t^{n}}{n!}.$

Here $L(n,r)$ are the unsigned Lah numbers given by

(4) $\langle x\rangle_{n}=\sum_{k=0}^{n}L(n,k)(x)_{k},\quad(n\geq
0),\quad(\mathrm{see}\ [12,17,18,20-25,27]),$

where $\langle x\rangle_{0}=1,\ \langle
x\rangle_{n}=x(x+1)\cdots(x+n-1),\quad(n\geq 1)$, and
$(-1)^{n}\langle-x\rangle_{n}=(x)_{n}$.

In [19], Kim-Kim introduced the multi-Bernoulli number of index
$(k_{1},k_{2},\dots,k_{r})$ defined by

(5)
$\frac{1}{(1-e^{-t})^{r}}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}(1-e^{-t})=\sum_{n=0}^{\infty}B_{n}^{(k_{1},k_{2},\dots,k_{r})}\frac{t^{n}}{n!}.$

Note that
$B_{n}^{\overbrace{(1,1,\dots,1)}^{r-times}}=\frac{(-1)^{n}}{r!}B_{n}^{(r)}$,
where $B_{n}^{(r)}$ are the Bernoulli numbers of order $r$, given by

$\bigg{(}\frac{t}{e^{t}-1}\bigg{)}^{r}=\sum_{n=0}^{\infty}B_{n}^{(r)}\frac{t^{n}}{n!},\quad(\mathrm{see}\
[1-32]).$

The unsigned Stirling numbers of the first kind are defined by

(6) $\langle x\rangle_{n}=\sum_{k=0}^{n}{n\brack k}x^{k},\quad(n,k\geq
0),\quad(\mathrm{see}\ [17,18,20]).$

The Stirling numbers of the second kind are defined by

(7) $x^{n}=\sum_{k=0}^{n}{n\brace k}(x)_{k},\quad(n\geq 0),\quad(\mathrm{see}\
[8,12,29]).$

Throughout this paper, we assume that $Y$ is random variable such that the
moment generating function of $Y$,

(8)
$E\big{[}e^{tY}\big{]}=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}E\big{[}Y^{n}\big{]},\quad(|t|<r),\quad\textrm{exists
for some $r>0$.}$

Let $(Y_{j})_{j\geq 1}$ be a sequence of mutually independent copies of random
variable $Y$, and let

$S_{k}=Y_{1}+Y_{2}+\cdots+Y_{k},\ (k\geq 1),\quad\mathrm{with}\quad S_{0}=0.$

The probabilistic Lah numbers associated with $Y$ are given by

(9)
$\frac{1}{k!}\bigg{(}E\bigg{[}\bigg{(}\frac{1}{1-t}\bigg{)}^{Y}\bigg{]}-1\bigg{)}^{k}=\sum_{n=k}^{\infty}L_{Y}(n,k)\frac{t^{n}}{n!},\quad(k\geq
0),\quad(\mathrm{see}\ [2,11,20,21]).$

When $Y=1$, $L_{Y}(n,k)=L(n,k),\ (n\geq k\geq 0)$.

From (9), we note that

$L_{Y}(n,k)=\frac{1}{k!}\sum_{l=0}^{k}\binom{k}{l}(-1)^{k-l}E\big{[}\langle
S_{l}\rangle_{n}\big{]},\quad(\mathrm{see}\ [27]).$

In [24], the unsigned multi-Stirling numbers of the first kind are defined by

(10) $\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}(t)=\sum_{n=r}^{\infty}{n\brack
k_{1},k_{2},\dots,k_{r}}\frac{t^{n}}{n!}.$

From (10), we note that

(11)
$\sum_{n=r}^{\infty}{n\brack\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\frac{t^{n}}{n!}=\mathrm{Li}_{\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}(t)=\frac{1}{r!}\big{(}-\log(1-t)\big{)}^{r}=\sum_{n=r}^{\infty}{n\brack
r}\frac{t^{n}}{n!}.$

Thus, by (11), we get

${n\brack\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}={n\brack r},\quad(n\geq
r\geq 0).$

As the inverse relation of (11), the multi-Stirling numbers of the second kind
are given by

(12)
$\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-e^{t}}\Big{)}=\sum_{n=r}^{\infty}{n\brace
k_{1},k_{2},\dots,k_{r}}\frac{t^{n}}{n!},\quad(\mathrm{see}\ [24]).$

From (12), we note that

(13)
$\sum_{n=r}^{\infty}{n\brace\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\frac{t^{n}}{n!}=\mathrm{Li}_{\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\Big{(}1-e^{1-e^{t}}\Big{)}=\frac{1}{r!}\big{(}e^{t}-1\big{)}^{r}=\sum_{n=r}^{\infty}{n\brace
r}\frac{t^{n}}{n!}.$

Thus, by (13), we get

(14) ${n\brace\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}={n\brace
r},\quad(n\geq r\geq 1).$

By (11) and (12), we get for $n\geq 0$ and $r\in\mathbb{N}$, we have

(15) ${n+1\brace
k_{1},k_{2},\dots,k_{r-1},1}=\sum_{m=r-1}^{n}\binom{n}{m}{m\brace
k_{1},k_{2},\dots,k_{r-1}},\quad(\mathrm{see}\ [24]).$

The probabilistic Stirling numbers of the second kind associated with $Y$ are
defined by

(16) ${n\brace
k}_{Y}=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}E\big{[}S_{j}^{n}\big{]},\quad(n\geq
k\geq 0),\quad(\mathrm{see}\ [20-22,25,32]).$

When $Y=1$,

${n\brace
k}_{Y}=\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}(-1)^{k-j}j^{n}={n\brace k}.$

##  2\. Probabilistic multi-Stirling numbers of the second kind and
probabilistic multi-Lah numbers

For $k_{i}\geq 1$, $(i=1,2,\dots,r)$, by (1), we get

(17) $\displaystyle\frac{d}{dt}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}(t)$
$\displaystyle=\frac{d}{dt}\sum_{0<m_{1}<m_{2}<\cdots<m_{r}}\frac{t^{m_{r}}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{r}^{k_{r}}}$
$\displaystyle=\sum_{0<m_{1}<\cdots<m_{r}}\frac{t^{m_{r}-1}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{r}^{k_{r}-1}}=\frac{1}{t}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1},k_{r}-1}(t).$

Letting $k_{r}=1$ in (17), we have

(18) $\displaystyle\frac{d}{dt}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1},1}(t)$
$\displaystyle=\sum_{0<m_{1}<m_{2}<\cdots<m_{r-1}}\frac{1}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{r-1}^{k_{r-1}}}\sum_{m_{r}=m_{r-1}+1}^{\infty}t^{m_{r}-1}$
$\displaystyle=\frac{1}{1-t}\sum_{0<m_{1}<\cdots<m_{r-1}}\frac{t^{m_{r-1}}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{r-1}^{k_{r-1}}}=\frac{1}{1-t}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1}}(t).$

Thus, by (18), we get

$\mathrm{Li}_{1,1}(t)=\frac{1}{2!}\big{(}-\log(1-t)\big{)}^{2},\quad\mathrm{Li}_{1,1,1}(t)=\frac{1}{3!}\big{(}-\log(1-t)\big{)}^{3}.$

Continuing this process, we have

(19)
$\mathrm{Li}_{\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}(t)=\frac{1}{r!}\big{(}-\log(1-t)\big{)}^{r},\quad(r\geq
1).$

Now, we define the probabilistic multi-Stirling numbers of the second kind
associated with $Y$ as

(20)
$\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}=\sum_{n=r}^{\infty}{n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{t^{n}}{n!}.$

When $Y=1$, we have

${n\brace k_{1},k_{2},\dots,k_{r}}_{Y}={n\brace k_{1},k_{2},\dots,k_{r}}.$

Note that

(21)
$\displaystyle\sum_{n=r}^{\infty}{n\brace\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\frac{}{}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\mathrm{Li}_{\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\frac{1}{r!}\Big{(}E\big{[}e^{Yt}\big{]}-1\Big{)}^{r}=\frac{1}{r!}\sum_{j=0}^{r}\binom{r}{j}(-1)^{r-j}E\big{[}e^{S_{j}t}\big{]}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{r!}\sum_{j=0}^{r}\binom{r}{j}(-1)^{r-j}E\big{[}S_{j}^{n}\big{]}\frac{t^{n}}{n!}.$

Thus, by (16) and (21), we get

(22)
${n\brace\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\frac{}{}_{Y}={n\brace
r}_{Y},\quad(n\geq r\geq 1).$

From (1), we note that

(23) $\displaystyle\frac{d}{dt}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1},1}$
$\displaystyle\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}=\frac{d}{dt}\sum_{0<m_{1}<m_{2}<\cdots<m_{r}}\frac{(1-e^{1-E[e^{Yt}]})^{m_{r}}}{m_{1}^{k_{1}}m_{2}^{k_{2}}\cdots
m_{r-1}^{k_{r-1}}m_{r}}$
$\displaystyle=\sum_{0<m_{1}<\cdots<m_{r-1}}\frac{E[Ye^{Yt}]e^{1-E[e^{Yt}]}}{m_{1}^{k_{1}}\cdots
m_{r-1}^{k_{r-1}}}\sum_{n_{r}=m_{r-1}+1}^{\infty}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}^{m_{r}-1}$
$\displaystyle=\sum_{0<m_{1}<\cdots<m_{r-1}}\frac{E[Ye^{Yt}]e^{1-E[e^{Yt}]}}{m_{1}^{k_{1}}\cdots
m_{r-1}^{k_{r-1}}}\frac{(1-e^{1-E[e^{Yt}]})^{m_{r-1}}}{e^{1-E[e^{Yt}]}}$
$\displaystyle=E\big{[}Ye^{Yt}\big{]}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\sum_{l=0}^{\infty}E\big{[}Y^{l+1}\big{]}\frac{t^{l}}{l!}\sum_{m=r-1}^{\infty}{m\brace
k_{1},\dots,k_{r-1}}_{Y}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{n=r-1}^{\infty}\sum_{m=r-1}^{n}\binom{n}{m}{m\brace
k_{1},k_{2},\dots,k_{r-1}}_{Y}E\big{[}Y^{n-m+1}\big{]}\frac{t^{n}}{n!}.$

On the other hand, by (20), we get

(24)
$\displaystyle\frac{d}{dt}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1},1}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\frac{d}{dt}\sum_{n=r}^{\infty}{n\brace
k_{1},k_{2},\dots,k_{r-1},1}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=r}^{\infty}{n\brace
k_{1},k_{2},\dots,k_{r-1},1}_{Y}\frac{t^{n-1}}{(n-1)!}$
$\displaystyle=\sum_{n=r-1}^{\infty}{n+1\brace
k_{1},k_{2},\dots,k_{r-1},1}_{Y}\frac{t^{n}}{n!}.$

Therefore, by (22), (23) and (24), we obtain the following theorem,

######  Theorem 2.1.

For $n,r\in\mathbb{N}$ with $n\geq r-1$, we have

$\sum_{m=r-1}^{n}\binom{n}{m}E\big{[}Y^{n-m+1}\big{]}{m\brace
k_{1},k_{2},\dots,k_{r-1}}_{Y}={n+1\brace k_{1},k_{2},\dots,k_{r-1},1}_{Y}.$

In particular, for $n\geq r$, we have

$\sum_{m=r-1}^{n-1}\binom{n-1}{m}{m\brace r-1}_{Y}E[Y^{n-m}]={n\brace
r}_{Y},\quad\sum_{m=r-1}^{n-1}\binom{n-1}{m}{m\brace r-1}={n\brace r}.$

Replacing $t$ by $E[e^{Yt}]-1$ in (5), on the one hand, we get

(25)
$\displaystyle\frac{1}{(1-e^{1-E[e^{Yt}]})^{r}}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1},k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\sum_{m=0}^{\infty}B_{m}^{(k_{1},\dots,k_{r})}\frac{1}{m!}\Big{(}E[e^{Yt}]-1\Big{)}^{m}$
$\displaystyle=\sum_{m=0}^{\infty}B_{m}^{(k_{1},k_{2},\dots,k_{r})}\sum_{n=m}^{\infty}{n\brace
m}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{\infty}\bigg{(}\sum_{m=0}^{n}B_{m}^{(k_{1},k_{2},\dots,k_{r})}{n\brace
m}_{Y}\bigg{)}\frac{t^{n}}{n!}.$

On the other hand, by (20), we get

(26)
$\displaystyle\frac{1}{(1-e^{1-E[e^{Yt}]})^{r}}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r-1},k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\sum_{l=0}^{\infty}\binom{r+l-1}{l}e^{l(1-E[e^{Yt}])}\sum_{m=r}^{\infty}{m\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{l=0}^{\infty}\binom{r+l-1}{l}\sum_{j=0}^{\infty}l^{j}(-1)^{j}\frac{1}{j!}\Big{(}E[e^{Yt}]-1\Big{)}^{j}\sum_{m=r}^{\infty}{m\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{l=0}^{\infty}\binom{r+l-1}{l}\sum_{j=0}^{\infty}l^{j}(-1)^{j}\sum_{i=j}^{\infty}{i\brace
j}_{Y}\frac{t^{i}}{i!}\sum_{m=r}^{\infty}{m\brace
k_{1},k_{2},\dots,k_{r}}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{l=0}^{\infty}\binom{r+l-1}{l}\sum_{i=0}^{\infty}\bigg{(}\sum_{j=0}^{i}l^{j}(-1)^{j}{i\brace
j}_{Y}\frac{t^{i}}{i!}\bigg{)}\sum_{m=r}^{\infty}{m\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{t^{m}}{m!}$
$\displaystyle=\sum_{n=r}^{\infty}\bigg{(}\sum_{l=0}^{\infty}\sum_{i=0}^{n-r}\sum_{j=0}^{i}\binom{n}{i}\binom{r+l-1}{l}l^{j}(-1)^{j}{i\brace
j}_{Y}{n-i\brace k_{1},k_{2},\dots,k_{r}}_{Y}\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (25) and (26), we obtain the following theorem.

######  Theorem 2.2.

For $n,r\in\mathbb{N}$ with $n\geq r$, we have

$\sum_{m=0}^{n}B_{m}^{(k_{1},k_{2},\dots,k_{r})}{n\brace
m}_{Y}=\sum_{l=0}^{\infty}\sum_{i=0}^{n-r}\sum_{j=0}^{i}\binom{n}{i}\binom{r+l-1}{l}l^{j}(-1)^{j}{i\brace
j}_{Y}{n-i\brace k_{1},k_{2},\dots,k_{r}}_{Y}.$

Replacing $t$ by $1-e^{1-E[e^{Yt}]}$ in (10), we get

(27)
$\displaystyle\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\sum_{m=r}^{\infty}{m\brack
k_{1},k_{2},\dots,k_{r}}\frac{1}{m!}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}^{m}$
$\displaystyle=\sum_{m=r}^{\infty}{m\brack
k_{1},k_{2},\dots,k_{r}}(-1)^{m}\sum_{l=m}^{\infty}{l\brace
m}\frac{1}{l!}\Big{(}1-E\big{[}e^{Yt}\big{]}\Big{)}^{l}$
$\displaystyle=\sum_{l=r}^{\infty}\sum_{m=r}^{l}{m\brack
k_{1},k_{2},\dots,k_{r}}(-1)^{m-l}{l\brace m}\sum_{n=l}^{\infty}{n\brace
l}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=r}^{\infty}\sum_{m=r}^{l}{m\brack
k_{1},k_{2},\dots,k_{r}}(-1)^{m-l}{l\brace m}\sum_{l=r}^{n}{n\brace
l}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=r}^{\infty}\bigg{(}\sum_{l=r}^{n}\sum_{m=r}^{l}{m\brack
k_{1},k_{2},\dots,k_{r}}(-1)^{m-l}{l\brace m}{n\brace
l}_{Y}\bigg{)}\frac{t^{n}}{n!}.$

Therefore ,by (20) and (27), we obtain the following theorem.

######  Theorem 2.3.

For $n,r\in\mathbb{N}$ with $n\geq r$, we have

${n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}=\sum_{l=r}^{n}\sum_{m=r}^{l}(-1)^{m-l}{l\brace
m}{n\brace l}_{Y}{m\brack k_{1},k_{2},\dots,k_{r}}.$

Now, we consider the probabilistic multi-Lah numbers associated with $Y$ which
are given by

(28)
$\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\bigg{(}1-e^{1-E\big{[}\big{(}\frac{1}{1-t}\big{)}^{Y}\big{)}\big{]}}\bigg{)}=\sum_{n=r}^{\infty}L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)\frac{t^{n}}{n!}.$

By (19), we note that

(29)
$\displaystyle\sum_{n=r}^{\infty}L_{Y}^{\overbrace{(1,1,\dots,1)}^{r-\mathrm{times}}}(n,r)\frac{t^{n}}{n!}$
$\displaystyle=\mathrm{Li}_{\underbrace{1,1,\dots,1}_{r-\mathrm{times}}}\bigg{(}1-e^{1-E\big{[}\big{(}\frac{1}{1-t}\big{)}^{Y}\big{]}}\bigg{)}$
$\displaystyle=\frac{1}{r!}\bigg{(}E\bigg{[}\bigg{(}\frac{1}{1-t}\bigg{)}^{Y}\bigg{]}-1\bigg{)}^{r}$
$\displaystyle=\sum_{n=r}^{\infty}L_{Y}(n,r)\frac{t^{n}}{n!}.$

Thus, by (29), we get

(30)
$L_{Y}^{\overbrace{(1,1,\dots,1)}^{r-\mathrm{times}}}(n,r)=L_{Y}(n,r),\quad(n\geq
r>0).$

Using (28) and (20) with $t$ replaced by$-\log(1-t)$, we get

(31)
$\displaystyle\sum_{n=r}^{\infty}L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)\frac{t^{n}}{n!}$
$\displaystyle=\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-E\big{[}\big{(}\frac{1}{1-t}\big{)}^{Y}\big{]}}\Big{)}$
$\displaystyle=\sum_{k=r}^{\infty}{k\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{1}{k!}\big{(}-\log(1-t)\big{)}^{k}$
$\displaystyle=\sum_{k=r}^{\infty}{n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\sum_{n=k}^{\infty}{n\brack k}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=r}^{\infty}\sum_{k=r}^{n}{n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}{n\brack k}\frac{t^{n}}{n!}.$

Therefore, by comparing the coefficients on both sides of (31), we obtain the
following theorem.

######  Theorem 2.4.

For $n,r\in\mathbb{N}$ with $n\geq r$, we have

$L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)=\sum_{k=r}^{n}{n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}{n\brack k}.$

Now, from (5) and (20), we observe that

(32) $\displaystyle\sum_{n=r}^{\infty}$ $\displaystyle{n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{t^{n}}{n!}=\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\frac{\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\big{(}1-e^{1-E[e^{Yt}]}\big{)}}{\big{(}1-e^{1-E[e^{Yt}]}\big{)}^{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}^{r}$
$\displaystyle=\sum_{m=0}^{\infty}B_{m}^{(k_{1},k_{2},\dots,k_{r})}\frac{1}{m!}\Big{(}E[e^{Yt}]-1\Big{)}^{m}(-1)^{r}r!\frac{1}{r!}\Big{(}e^{1-E[e^{Yt}]}-1\Big{)}^{r}$
$\displaystyle=\sum_{m=0}^{\infty}B_{m}^{(k_{1},k_{2},\dots,k_{r})}\frac{1}{m!}\Big{(}E[e^{Yt}]-1\Big{)}^{m}(-1)^{r}r!\sum_{l=r}^{\infty}{l\brace
r}\frac{1}{l!}\Big{(}1-E[e^{Yt}]\Big{)}^{l}$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{l=r}^{\infty}B_{m}^{(k_{1},k_{2},\dots,k_{r})}r!(-1)^{l-r}{l\brace
r}\frac{(m+l)!}{m!l!}\frac{1}{(m+l)!}\Big{(}E\big{[}e^{Yt}\big{]}-1\Big{)}^{m+l}$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{l=r}^{\infty}B_{m}^{(k_{1},k_{2},\dots,k_{r})}r!(-1)^{l-r}{l\brace
r}\binom{m+l}{m}\sum_{n=m+l}^{\infty}{n\brace m+l}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=r}^{\infty}\bigg{(}\sum_{m=0}^{n-r}\sum_{l=r}^{n-m}B_{m}^{(k_{1},k_{2},\dots,k_{r})}r!(-1)^{l-r}\binom{m+l}{m}{l\brace
r}{n\brace m+l}_{Y}\bigg{)}\frac{t^{n}}{n!}.$

By comparing the coefficients on both sides of (32), we obtain the following
theorem.

######  Theorem 2.5.

Let $n,r$ be positive integers with $n\geq r$. Then we have

${n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}=\sum_{m=0}^{n-r}\sum_{l=r}^{n-m}r!(-1)^{l-r}\binom{m+l}{m}{l\brace
r}{n\brace m+l}_{Y}B_{m}^{(k_{1},k_{2},\dots,k_{r})}.$

In particular, we have

${n\brace
r}_{Y}=\sum_{m=0}^{n-r}\sum_{l=r}^{n-m}(-1)^{m+l-r}\binom{m+l}{m}{l\brace
r}{n\brace m+l}_{Y}B_{m}^{(r)}.$

For $r\in\mathbb{N}$, the probabilistic Fubini polynomials of order $r$
associated with $Y$ are defined by

(33)
$\frac{1}{\big{(}1-y(E[e^{Yt}]-1)\big{)}^{r}}=\sum_{n=0}^{\infty}F_{n}^{(r,Y)}(y)\frac{t^{n}}{n!},\quad(\mathrm{see}\
[32]).$

Replacing $t$ by $E[e^{Yt}]-1$ in (2), on the one hand, we have

(34)
$\displaystyle\frac{1}{(2-E[e^{Yt}])^{r}}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\sum_{k=r}^{\infty}L^{(k_{1},k_{2},\dots,k_{r})}(k,r)\frac{1}{k!}\Big{(}E\big{[}e^{Yt}\big{]}-1\Big{)}^{k}$
$\displaystyle=\sum_{k=r}^{\infty}L^{(k_{1},k_{2},\dots,k_{r})}(k,r)\sum_{n=k}^{\infty}{n\brace
k}_{Y}\frac{t^{n}}{n!}$
$\displaystyle=\sum_{n=r}^{\infty}\bigg{(}\sum_{k=r}^{n}{n\brace
k}_{Y}L^{(k_{1},k_{2},\dots,k_{r})}(k,r)\bigg{)}\frac{t^{n}}{n!}.$

On the other hand, by (20) and (33), we get

(35)
$\displaystyle\frac{1}{(2-E[e^{Yt}])^{r}}\mathrm{Li}_{k_{1},k_{2},\dots,k_{r}}\Big{(}1-e^{1-E[e^{Yt}]}\Big{)}$
$\displaystyle=\sum_{k=r}^{\infty}{k\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\frac{t^{k}}{k!}\sum_{l=0}^{\infty}F_{l}^{(r,Y)}(1)\frac{t^{l}}{l!}$
$\displaystyle=\sum_{n=r}^{\infty}\bigg{(}\sum_{k=r}^{n}{k\brace
k_{1},k_{2},\dots,k_{r}}_{Y}\binom{n}{k}F_{n-k}^{(r,Y)}(1)\bigg{)}\frac{t^{n}}{n!}.$

Therefore, by (34) and (35), we obtain the following theorem.

######  Theorem 2.6.

For $n,r\in\mathbb{N}$ with $n\geq r$, we have

$\sum_{k=r}^{n}{n\brace
k}_{Y}L^{(k_{1},k_{2},\dots,k_{r})}(k,r)=\sum_{k=r}^{n}\binom{n}{k}{k\brace
k_{1},k_{2},\dots,k_{r}}_{Y}F_{n-k}^{(r,Y)}(1).$

##  3\. Conclusion

Let $Y$ be a random variable whose moment generating function exists in a
neighborhood of the origin. The Stirling numbers of the second ${n\brace r}$
are generalized to the multi-Stirling numbers of the second kind ${n\brace
k_{1},k_{2},\dots,k_{r}}$ and further to the probabilistic multi-Stirling
numbers of the second kind associated with $Y$, ${n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}$. Likewise, the unsigned Lah numbers $L(n,r)$ are
generalized to the multi-Lah numbers $L^{(k_{1},k_{2},\dots,k_{r})}(n,r)$ and
further to the probabilistic multi-Lah numbers associated with
$L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)$. In connection with several other
special numbers, we investigated some properties, related identities,
recurrence relations and explicit expressions of the numbers ${n\brace
k_{1},k_{2},\dots,k_{r}}_{Y}$ and $L_{Y}^{(k_{1},k_{2},\dots,k_{r})}(n,r)$.

In recent years, degenerate versions, $\lambda$-analogues and probabilistic
extensions of many special numbers and polynomials have been studied and many
remarkable results have been obtained (see [2,3,10,11,16-18,20,21,23,25-28,32]
and the references therein). It is one of our future research projects to
continue to study degenerate versions, $\lambda$-analogues and probabilistic
extensions of various special numbers and polynomials and to find their
applications to physics, science and engineering as well as to mathematics.

Declarations

Ethics approval and consent to participate The submitted article is an
original research paper and has not been published anywhere else.

Consent for publication All authors consent for publication.

Data Availability Statement There is no data used in this article.

Competing interests The authors declare no competing interests.

Funding This research has been conducted by the Research Grant of Kwangwoon
University in 2024.

Authors’ contributions All authors contributed equally in the preparation of
this manuscript.

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