# Generalized $q$-Stirling numbers and normal ordering

Roberto B. Corcino  Mathematics and ICT Department, Cebu Normal University,
6000 Cebu City, Philippines<EMAIL_ADDRESS>,  Richell O. Celeste  and
Ken Joffaniel M. Gonzales  Institute of Mathematics, University of the
Philippines Diliman, 1101 Quezon City, Philippines<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The normal ordering coefficients of strings consisting of $V,U$ which satisfy
$UV=qVU+hV^{s}$ ($s\in\mathbb{N}$) are considered. These coefficients are
studied in two contexts: first, as a multiple of a sequence satisfying a
generalized recurrence, and second, as $q$-analogues of rook numbers under the
row creation rule introduced by Goldman and Haglund. A number of properties
are derived, including recurrences, expressions involving other $q$-analogues
and explicit formulas. We also give a Dobinsky-type formula for the associated
Bell numbers and the corresponding extension of Spivey’s Bell number formula.
The coefficients, viewed as rook numbers, are extended to the case
$s\in\mathbb{R}$ via a modified rook model.

###### Key words and phrases:

Generalized Stirling numbers, normal ordering, rook theory, Bell numbers

###### 2010 Mathematics Subject Classification:

Primary 05A15; Secondary, 11B65, 11B73

##  1\. Introduction

Let $V$ and $U$ be operators (or variables) that satisfy the commutation
relation $[U,V]=UV-VU=1$. Given a string $w$ consisting of $U$’s and $V$’s,
the _normally ordered_ form of $w$ is an equivalent operator expressed as
$\sum c_{i,j}V^{i}U^{j}$. The normally ordered form may be computed using the
commutation relation alone, i.e., by replacing all occurrences of $UV$ with
$VU+1$, a task which can be cumbersome especially for long strings. It has
been shown, however, that the normal ordering coefficients are numbers studied
in combinatorics. In quantum physics, for instance, one encounters the boson
operators $a$ and $a^{\dagger}$ which satisfy
$[a,a^{\dagger}]=aa^{\dagger}-a^{\dagger}a=1$. It is known that
$(a^{\dagger}a)^{n}=\sum_{k=0}^{n}S(n,k){a^{\dagger}}^{k}a^{k}$, where the
$S(n,k)$ are the Stirling numbers of the second kind. Combinatorially, the
number $S(n,k)$ counts the number of partitions of $\\{1,2,\ldots,n\\}$ into
$k$ disjoint, non-empty subsets. Since only the commutation relation is
considered, it follows that any pair of operators that satisfies the same
commutation relation will have the same coefficients. One such pair of
operators is given by $X$ and $D$ which act on the monomial $x^{n}$ by
$Vx^{n}=x^{n+1}$ and $Ux^{n}=nx^{n-1}$. Since $[U,V]=1$, then
$(VU)^{n}=\sum_{k=0}^{n}S(n,k)V^{k}U^{k}$.

The connection between normal ordering and rook theory was demonstrated by
Navon [15] who proved that the normal ordering coefficients are given by rook
numbers on a Ferrers board. Varvak [19] derived explicit formulas for these
coefficients using rook factorization. Blasiak [1], and El-Desouky Cakić and
Mansour [3] also computed explicit formulas using other methods. These
coefficients may be considered as some form of generalized Stirling numbers.

Generalizations of the classical commutation relation have also been
introduced and studied. Katriel and Kibler [7] considered the $q$-deformed
commutation relation $[a,a^{\dagger}]_{q}=aa^{\dagger}-qa^{\dagger}a=1$ and
showed that the coefficients of $(a^{\dagger}a)^{n}$ are a $q$-analogue of the
Stirling numbers of the second kind, i.e.,
$(a^{\dagger}a)^{n}=\sum_{k=0}^{n}S_{q}[n,k]{a^{\dagger}}^{k}a^{k}$. By
letting $q=1$, we recover the non-deformed commutation relation and the
Stirling numbers of the second kind. Mansour and Schork [10] extensively
studied normal ordering relations of the form $xy=qyx+hf(y)$. For the case
where $f(y)=y^{s}$, they derived the normal ordering of the expression
$(x+y)^{n}$, thereby extending Newton’s binomial formula. In [11], they
focused on the coefficients of $(VU)^{n}$, where $V,U$ satisfy $UV-VU=hV^{s}$
and obtained properties of the corresponding generalization of Stirling
numbers and Bell numbers. They continued the study of these numbers in [12]
where they obtained more properties and introduced a $q$-analogue through the
relation $UV-qVU=hV^{s}$. (We also mention that the papers [10, 11, 12] also
contain an excellent account of some literature on the subject.)

In this paper, we continue the study of normal ordering coefficients given the
commutation relation $UV-qVU=hV^{s}$ where $s\in\mathbb{N}$. The outline of
the paper is as follows. In Section 2, we study the properties of the
coefficients of $(VU)^{n}$ in the context of a sequence satisfying a
generalized recurrence relation. In Section 3, we obtain explicit formulas for
the normal ordering coefficients of arbitrary strings
$H_{\textnormal{{r}},\textnormal{{s}}}=V^{r_{n}}U^{s_{n}}\cdots
V^{r_{2}}U^{s_{2}}V^{r_{1}}U^{s_{1}}$ using an interpretation of normal
ordering in terms of rook placements. In Section 4, we resume to the
coefficients of $(VU)^{n}$ and further exploit the rook model to obtain
identities for the associated Bell numbers. Finally, in Section 5, we extend
the coefficients to the case $s\in\mathbb{R}$ using a modified rook model.

As we shall be dealing with many forms of generalized Stirling numbers, we
summarize some notation in the table below for the convenience of the reader.
The presence of the parameters $\textbf{r},\textbf{s},h,s$ and $q$ means that
they are assumed to be arbitrary. Square brackets are used to refer to
$q$-analogues. A similar convention is applied to generalized Bell numbers.

String | Parameters | Coefficients | Original Notation
---|---|---|---
$(VU)^{n}$ |  $q=1$, $h=1,s=0$ | $S(n,k)$ |
$(VU)^{n}$ |  $q$ arbitrary, $h=1,s=0$ | $S_{q}[n,k]$ |
$(VU)^{n}$ |  $q=1,h\neq 0,s$ arbitrary | $S_{s,h}(n,k)$ |  $\mathfrak{S}_{s;h}(n,k)$ [11, 12]
$(VU)^{n}$ |  $q,h\neq 0,s$ arbitrary | $S_{s,h,q}[n,k]$ |  $\mathfrak{S}_{s;h}(n,k|q)$ [12]
$H_{\textbf{r},\textbf{s}}$ | $q=1,h=1,s=0$ | $S^{\textbf{r},\textbf{s}}(k)$ |  $S_{\textbf{r},\textbf{s}}(k)$ [1]
$H_{\textbf{r},\textbf{s}}$ |  $q,h\neq 0,s$ arbitrary | $S^{\textbf{r},\textbf{s}}_{s,h,q}[k]$ |

##  2\. The numbers $S_{s,h,q}[n,k]$

For $x\in\mathbb{R}$, we define the $q$-analogue of $x$ by
$[x]_{q}=\frac{q^{x}-1}{q-1}$ for $x\neq 0$ and $[0]_{q}=0$. As $q\rightarrow
1^{-}$, $[x]_{q}\rightarrow x$. If $x\in\mathbb{N}$, then
$[x]_{q}=1+q+\cdots+q^{x-1}$ so that the replacement $q=1$ is sufficient to
recover $x$.

Let $h\in\mathbb{C}\backslash\\{0\\}$ and $s\in\mathbb{N}$. For $U,V$ that
satisfy the commutation relation $UV-qVU=hV^{s}$. Mansour, Schork and Shattuck
[11] showed that the coefficients $S_{s,h,q}[n,k]$ in

$(VU)^{n}=\sum_{k=0}^{n}S_{s,h,q}[n,k]V^{sn-(s-1)k}U^{k}$

satisfy the recurrence relation

$S_{s,h,q}[n,k]=q^{s(n-1)-(s-1)(k-1)}S_{s,h,q}[n-1,k-1]+h[s(n-1)-(s-1)k]_{q}S_{s,h,q}[n-1,k]\,,$
(1)

with initial conditions $S_{s,h,q}[n,0]=S_{s,h,q}[0,n]=\delta_{0,n}$. If
$q=1$, we obtain the commutation relation $UV-VU=hV^{s}$ and the numbers
$S_{s,h}(n,k)$ which were studied in [11]. Mansour, Schork and Shattuck [12]
also showed that the number $S_{s,h,q}[n,k]$ can be expressed in terms of
$b_{n,k}=q^{(s-1)\binom{k}{2}-s\binom{n}{2}}S_{s,h,q}[n,k]$ where $b_{n,k}$
satisfies

$b_{n,k}=b_{n-1,k-1}+(h/q)([s(n-1)]_{1/q}-[(s-1)k]_{1/q})b_{n-1,k}\,.$ (2)

The recurrence (2) is useful in deriving some basic properties. However, we
will find it more convenient to work on a more general setting. Specifically,
let $\textbf{v}=(v_{0},v_{1},\ldots)$ and $\textbf{w}=(w_{0},w_{1},\ldots)$ be
sequences from a ring $K$ and define the numbers
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$ by the recurrence relation

$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}=A^{\textnormal{{v}},\textnormal{{w}}}_{n-1,k-1}+(v_{n-1}+w_{k})A^{\textnormal{{v}},\textnormal{{w}}}_{n-1,k}\,,$
(3)

with initial conditions
$A^{\textnormal{{v}},\textnormal{{w}}}_{0,n}=\delta_{0,n}$ and
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,0}=(v_{n-1}+w_{0})(v_{n-2}+w_{0})\cdots(v_{0}+w_{0})$.
We also call v and w weight functions. When only the value of a weight
function at $i=0,1,\ldots$ is specified (for instance, $v_{i}$), it is
understood that the corresponding weight function is the same letter in
boldface without the subscripts. If $v_{i}=[si]_{1/q},w_{i}=-[(s-1)i]_{1/q}$,
then by (2),
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}=(h/q)^{-(n-k)}q^{(s-1)\binom{k}{2}-s\binom{n}{2}}S_{s,h,q}[n,k]$.

We mention that for linear weight functions, the recurrence relation (3) has
been considered by Xu [20], Hsu and Shuie [5] and El-Desouky and Cakić [4].
Some $q$\- and $p,q$-analogues have also been studied by Corcino, Hsu and Tan
[2], and Remmel and Wachs [16] (although we note that the type-II
$p,q$-analogue in [16] does not seem to fall as
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$). An explicit formula was
obtained in [9] and [20], which is given by

$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}=\sum_{j=0}^{k}\frac{\prod_{i=0}^{n-1}(w_{j}+v_{i})}{\prod_{\begin{subarray}{c}i=0,\neq
j\end{subarray}}^{k}(w_{j}-w_{i})}\,,$

provided that the $w_{i}$’s are distinct. For $s\neq 1$, the formula above
gives us (see [12, Theorem 45])

$S_{s,h,q}[n,k]=h^{n-k}q^{s\binom{n}{2}-(s-1)\binom{k}{2}-(n-k)}\sum_{j=0}^{k}\frac{\prod_{i=0}^{n-1}([si]_{1/q}-[(s-1)j]_{1/q})}{\prod_{\begin{subarray}{c}i=0,\neq
j\end{subarray}}^{k}([(s-1)i]-[(s-1)j]_{1/q})}\,.$

The theorem that follows gives different formulations for the numbers
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$ in terms of expressions that are
analogous to elementary and complete symmetric functions.

######  Theorem 1.

The following identities hold:

$\displaystyle A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{0\leq i_{1}<i_{2}<...<i_{n-k}\leq
n-1}~{}\prod_{j=1}^{n-k}(v_{i_{j}}+w_{i_{j}-j+1})$ (4) $\displaystyle
A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$ $\displaystyle=\sum_{0\leq
i_{1}\leq i_{2}\leq...\leq i_{n-k}\leq
k}~{}\prod_{j=1}^{n-k}(v_{i_{j}+j-1}+w_{i_{j}})$ (5) $\displaystyle
A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{i_{0}+i_{1}+i_{2}+...+i_{k}=n-k}~{}\prod_{j=0}^{k}~{}\prod_{l=0}^{i_{j}-1}(v_{j+l+i_{0}+i_{1}+i_{2}+...+i_{j-1}}+w_{j})\,.$
(6)

###### Proof.

It can be shown by partitioning the set of indices that the $RHS$ of (4) and
(5) satisfy the recursion (3). Identity (6) is a restatement of (5). ∎

Let $H$ be a set. The $m$-th _elementary symmetric function_ $e$ and $m$-th
_complete symmetric function_ $h$ are defined as follows: $e_{m}(H)$ (resp.
$h_{m}(H)$) is the sum of all products of $m$ elements from $H$ taken without
(resp. with) replacement. Denote by 0 the function that is identically 0.
Observe that by (4),
$A^{\textbf{v},\textbf{0}}_{n,k}=e_{n-k}(\\{v_{0},v_{1},\ldots,v_{n-1}\\})$
and by (5) or (6),
$A^{\textbf{0},\textbf{v}}_{n,k}=h_{n-k}(\\{v_{0},v_{1},\ldots,v_{k}\\})$.

In the next theorem, we denote the matrix whose $(n,k)$-th entry is
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$ by
$\left[A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}\right]$.

######  Theorem 2.

The following identities hold:

  1. (1)

A matrix factorization

$\displaystyle\left[A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}\right]=\left[A^{\textnormal{{v}},\textnormal{{0}}}_{n,k}\right]\left[A^{\textnormal{{0}},\textnormal{{w}}}_{n,k}\right]\mbox{,
or equivalently, }~{}~{}A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{j=k}^{n}A^{\textnormal{{v}},\textnormal{{0}}}_{n,j}A^{\textnormal{{0}},\textnormal{{w}}}_{j,k}$
(7)

  2. (2)

Orthogonality relation

$\displaystyle\sum_{k=m}^{n}A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}A^{\textnormal{{-w}},\textnormal{{-v}}}_{k,m}$
$\displaystyle=\delta_{n,m}$ (8)

  3. (3)

A pair of inverse relations

$\displaystyle(x+v_{0})(x+v_{1})\cdots(x+v_{n-1})$
$\displaystyle=\sum_{k=0}^{n}A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}(x-w_{0})(x-w_{1})\cdots(x-w_{k-1})$
(9) $\displaystyle(x-w_{0})(x-w_{1})\cdots(x-w_{n-1})$
$\displaystyle=\sum_{k=0}^{n}A^{\textnormal{{-w}},\textnormal{{-v}}}_{n,k}(x+v_{0})(x+v_{1})\cdots(x+v_{k-1})$
(10)

  4. (4)

Convolution formula

$\displaystyle A^{\textnormal{{v}},\textnormal{{w}}}_{l+m,n}$
$\displaystyle=\sum_{k=0}^{n}A^{\textnormal{{v}},\textnormal{{w}}}_{l,k}A^{\textnormal{{v}}_{+l},\textnormal{{w}}_{+k}}_{m,n-k}\,,\mbox{~{}where
for any weight function $\textnormal{{f}}$, we define
$\textnormal{{f}}_{+m}=(f_{m},f_{m+1},\ldots)$}\,.$ (11)

###### Proof.

We prove the second equation in (7) by showing that the quantity on the $RHS$
satisfies the recurrence relation (3). Let $k\leq j\leq n$. By (3),

$\displaystyle A^{\textbf{v},\textbf{0}}_{n,j}A^{\textbf{0},\textbf{w}}_{j,k}$
$\displaystyle=(A^{\textbf{v},\textbf{0}}_{n-1,j-1}+v_{n-1}A^{\textbf{v},\textbf{0}}_{n-1,j})(A^{\textbf{0},\textbf{w}}_{j-1,k-1}+w_{k}A^{\textbf{0},\textbf{w}}_{j-1,k})$
$\displaystyle=A^{\textbf{v},\textbf{0}}_{n-1,j-1}A^{\textbf{0},\textbf{w}}_{j-1,k-1}+v_{n-1}A^{\textbf{v},\textbf{0}}_{n-1,j}A^{\textbf{0},\textbf{w}}_{j,k}+w_{k}A^{\textbf{v},\textbf{0}}_{n-1,j-1}A^{\textbf{0},\textbf{w}}_{j-1,k}$

Substituting this into (7) and collecting the necessary terms proves the
identity.

For the orthogonality relation (8), we use
$\left[A^{\textbf{0},\textbf{v}}_{n,k}\right]^{-1}=\left[(-1)^{n-k}A^{\textbf{v},\textbf{0}}_{n,k}\right]=\left[A^{\textbf{-v},\textbf{0}}_{n,k}\right]$
(see [14, Identity (2.10)]). By (7),
$\left[A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}\right]^{-1}=\left[A^{\textbf{0},\textbf{w}}_{n,k}\right]^{-1}\left[A^{\textbf{v},\textbf{0}}_{n,k}\right]^{-1}=\left[A^{\textbf{-w},\textbf{0}}_{n,k}\right]\left[A^{\textbf{0},\textbf{-v}}_{n,k}\right]=\left[A^{\textbf{-w},\textbf{-v}}_{n,k}\right]$.

An inductive argument and the recursion (3) proves (9). Identity (8) then
establishes the equivalence of (9) with (10). For (11), we use (4) and break
each $(l+m-n)$-tuple of indices into two, namely, into those that satisfy
$0\leq i_{1}<\ldots<i_{l-k}\leq l-1$ and $l\leq i_{l-k+1}<\ldots<i_{l+m-n}\leq
m+l-1$, for some $k$. This $k$ is unique since each coordinate in every
$(l+m-n)$-tuple is unique. The first set of indices will produce
$A^{\textnormal{{v}},\textnormal{{w}}}_{l,k}$. On the other hand, the second
set will produce

$\displaystyle\sum_{l\leq i_{l-k+1}<\ldots<i_{l+m-n}\leq
m+l-1}~{}\prod_{j=l-k+1}^{l+m-n}(v_{i_{j}}+w_{i_{j}-(j-1)})$
$\displaystyle=\sum_{0\leq i_{1}<\ldots<i_{m-(n-k)}\leq
m-1}\prod_{j=1}^{m-(n-k)}(v_{i_{j}+l}+w_{i_{j}-(j-1)+k})$
$\displaystyle=A^{\textbf{v}_{+l},\textbf{w}_{+k}}_{m,n-k}\,.$

All identities have now been proved. ∎

By Identity (7), we can write $S_{s,h,q}[n,k]$ in terms of other $q$-Stirling
numbers. Note that the second kind of this analogue is different from the
$q$-analogue $S_{0,1,q}[n,k]=S_{q}[n,k]$. By (1), the numbers $S_{q}[n,k]$
satisfy the recurrence

$S_{q}[n,k]=q^{k-1}S_{q}[n-1,k-1]+[k]_{q}S_{q}[n-1,k]\,.$

On the other hand, the other $q$-analogues satisfy

$\displaystyle\hat{S}_{q}[n,k]$
$\displaystyle=\hat{S}_{q}[n-1,k-1]+[k]_{q}\hat{S}_{q}[n-1,k]$
$\displaystyle\hat{c}_{q}[n,k]$
$\displaystyle=\hat{c}_{q}[n-1,k-1]+[n-1]_{q}\hat{c}_{q}[n-1,k]\,.$

with
$\hat{S}_{q}[n,0]=\hat{S}_{q}[0,n]=\hat{c}_{q}[n,0]=\hat{c}_{q}[0,n]=\delta_{n,0}$.
Using these recurrence relations, we see that when $v_{i}=[i]_{q}$,
$A^{\textbf{v},\textbf{0}}_{n,k}=\hat{c}_{q}[n,k]$ and
$A^{\textbf{0},\textbf{v}}_{n,k}=\hat{S}_{q}[n,k]$. These analogues have been
studied extensively by Médicis and Leroux [13].

######  Corollary 3.

The numbers $S_{s,h,q}[n,k]$ satisfy

$S_{s,h,q}[n,k]=h^{(n-k)}q^{s\binom{n}{2}-(s-1)\binom{k}{2}-n+k}\sum_{j=k}^{n}[s]_{1/q}^{n-j}(-[s-1]_{1/q})^{j-k}\hat{c}_{1/{q^{s}}}[n,j]\hat{S}_{1/{q^{s-1}}}[j,k]\,.$
(12)

Consequently, the numbers $S_{s,h}(n,k)$ are given by

$S_{s,h}(n,k)=h^{n-k}\sum_{j=k}^{n}s^{n-j}(1-s)^{j-k}c(n,j)S(j,k)\,.$

where $c(n,k)$ and $S(n,k)$ are the (classical) Stirling numbers of the first
kind and second kind, respectively.

###### Proof.

The result follows from (7) and the property
$[si]_{1/q}=[s]_{1/q}[i]_{1/q^{s}}$,
$[(s-1)i]_{1/q}=[s-1]_{1/q}[i]_{1/q^{s-1}}$. ∎

Define $c_{s,h,q}[n,k]$ by the recursion

$c_{s,h,q}[n,k]=q^{(s-1)(n-1)-s(k-1)}c_{s,h,q}[n-1,k-1]+h[(s-1)(n-1)-sk]_{q}c_{s,h,q}[n-1,k]\,,$

with initial conditions $c_{s,h,q}[n,0]=c_{s,h,q}[0,n]=\delta_{0,n}$. The
orthogonality relation for $S_{s,h,q}[n,k]$ involves $c_{s,h,q}[n,k]$ and is
given in the next corollary.

######  Corollary 4.

For $m\leq n$, we have the following orthogonality relations

$\displaystyle\sum_{k=m}^{n}S_{s,h,q}[n,k]c_{s,h,q}[k,m]$
$\displaystyle=\delta_{m,n}$
$\displaystyle\sum_{k=m}^{n}c_{s,h,q}[n,k]S_{s,h,q}[k,m]$
$\displaystyle=\delta_{m,n}\,.$

###### Proof.

Let $v_{i}=[si]_{1/q},w_{i}=-[(s-1)i]_{1/q}$. One can verify that
$A^{\textbf{-w},\textbf{-v}}_{n,k}=(h/q)^{-(n-k)}q^{s\binom{k}{2}-(s-1)\binom{n}{2}}c_{s,h,q}[n,k]$.
The orthogonality relation (8) implies that

$\sum_{k=m}^{n}(h/q)^{-(n-k)}q^{(s-1)\binom{k}{2}-s\binom{n}{2}}S_{s,h,q}[n,k](h/q)^{-(k-m)}q^{s\binom{m}{2}-(s-1)\binom{k}{2}}c_{s,h,q}[k,m]=\delta_{m,n}\,.$

After clearing the powers of $h$ and $q$ we obtain the desired result. ∎

The next theorem gives some recurrence relations for
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$. Identities (13) and (15) reduce
into $q$-analogues of [11, Theorems 5.5 and 5.4], respectively, which we state
in the following corollary.

######  Theorem 5.

The following recurrence relations for
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$ hold

$\displaystyle A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{j=k}^{n}A^{\textnormal{{v}},\textnormal{{w}}}_{j-1,k-1}\prod_{i=j}^{n-1}(v_{i}+w_{k})$
(13) $\displaystyle A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{j=k}^{n}(-1)^{j-k}A^{\textnormal{{v}},\textnormal{{w}}}_{n+1,j+1}\prod_{i=k+1}^{j}(v_{n}+w_{i})$
(14) $\displaystyle A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{j=0}^{k}(v_{n-j-1}+w_{k-j})A^{\textnormal{{v}},\textnormal{{w}}}_{n-j-1,k-j}\,.$
(15)

###### Proof.

The proof of the three identities uses (4).

For (13), we use the fact that for every $(n-k)$-tuple of indices
$(i_{1},\ldots,i_{n-k})$, there exists a unique $j$ satisfying $k\leq j\leq n$
such that $(i_{j-k+1},i_{j-k+2}\ldots,i_{n-k})=(j,j+1,\ldots,n-1)$ and
$(i_{1},\ldots,i_{j-k})$ satisfies $0\leq i_{1}<\ldots<i_{j-k}\leq j-2$. A
similar approach proves (15).

To prove identity (14), let $I_{j}$ be the set of $(n-j)$-tuples of indices
satisfying $0\leq i_{1}<\ldots<i_{n-j}\leq n$. Let
$I^{\prime}_{j}=I_{j}\times(\underbrace{n,\ldots,n}_{j-k})$ and $\mathcal{I}$
be the multiset consisting of the union of all $I^{\prime}_{j}$’s, $k\leq
j\leq n$. Assign the weight $(-1)^{j-k}$ to each element of $\mathcal{I}$
belonging to $I^{\prime}_{j}$. Then each $(n-k)$-tuple in $\mathcal{I}$ which
does not satisfy $0\leq i_{1}<\ldots<i_{n-k}\leq n-1$ occurs twice but with
weights that are negatives of each other. ∎

######  Corollary 6.

The following recurrence relations for $S_{s,h,q}[n,k]$ hold

$\displaystyle S_{s,h,q}[n,k]$
$\displaystyle=\sum_{j=k}^{n}h^{n-j}q^{s(j-1)-(s-1)(k-1)}S_{s,h,q}[j-1,k-1]\prod_{i=j}^{n-1}[si-(s-1)k]_{q}$
(16) $\displaystyle S_{s,h,q}[n,k]$
$\displaystyle=\sum_{j=k}^{n}(-h)^{j-k}q^{sn(k-j-1)+(s-1)\binom{j+1}{2}-(s-1)\binom{k}{2}}S_{s,h,q}[n+1,j+1]\prod_{i=k+1}^{j}[sn-(s-1)i]_{q}$
(17) $\displaystyle S_{s,h,q}[n,k]$
$\displaystyle=\sum_{j=0}^{k}hq^{j(sn-(s-1)k)-\binom{j+1}{2}}[s(n-j-1)-(s-1)(k-j)]_{q}S_{s,h,q}[n-j-1,k-j]\,.$
(18)

###### Proof.

We use Theorem 5 with $v_{i}=[si]_{1/q}$ and $w_{i}=-[(s-1)i]_{1/q}$, and the
observation that for any integers $b,c$, we have
$[b-c]_{q}=q^{b-1}([b]_{1/q}-[c]_{1/q})$. ∎

Another identity of interest is the one by Carlitz (see [13, Identities (4.5)
and (4.7)]) which gives the following equivalent relations between
$q$-binomial coefficients and $q$-Stirling numbers of the second kind

$\displaystyle\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}$
$\displaystyle=\sum_{j=k}^{n}\binom{n}{j}(q-1)^{j-k}\hat{S}_{q}[j,k]\,.$ (19)
$\displaystyle(q-1)^{n-k}\hat{S}_{q}[n,k]$
$\displaystyle=\sum_{j=k}^{n}(-1)^{n-j}\binom{n}{j}\genfrac{[}{]}{0.0pt}{}{j}{k}_{q}\,.$

Here, $\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}$ denotes the $q$-binomial
coefficients which is defined by
$\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}$, where
for $n\in\mathbb{N}$, $[n]_{q}!=[n]_{q}[n-1]_{q}\cdots[2]_{q}[1]_{q}$. For
generalizations of Carlitz’s identity, see [14, Theorem 2.2] where they are
proved by distributing weights on certain tableaux.

Identity (19) can be directly derived from
$A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$ using (7). Let $f_{i}=q^{i}$ and
$v_{i}=1$ and $w_{i}=q^{i}-1$. Then,
$A^{\textnormal{{0}},\textnormal{{f}}}_{n,k}=A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}=\sum_{j=k}^{n}A^{\textnormal{{1}},\textnormal{{0}}}_{n,j}A^{\textnormal{{0}},\textnormal{{w}}}_{j,k}$.
It suffices to note that
$A^{\textnormal{{0}},\textnormal{{f}}}_{n,k}=\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}$,
$A^{\textnormal{{1}},\textnormal{{0}}}_{n,j}=\binom{n}{j}$ and (using
$q^{i}-1=(q-1)[i]_{q}$),
$A^{\textnormal{{0}},\textnormal{{w}}}_{j,k}=(q-1)^{j-k}\hat{S}_{q}[j,k]$ by
the symmetric functions forms of these numbers.

The next theorem gives a generalization of Carlitz’s identity. Equivalent
identities may be obtained by manipulating the matrices involved using the
matrix formulation of identities (7) and (8).

######  Theorem 7.

Let $c$ and $d$ be constants, and
$\textnormal{{v}}^{*},\textnormal{{w}}^{*},\textnormal{{v}},\textnormal{{w}}$
be weight functions such that $v_{i}=c+v^{*}_{i},w_{i}=d+w^{*}_{i}$. Then,

$\displaystyle A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}$
$\displaystyle=\sum_{k\leq t_{1}\leq t_{2}\leq t_{3}\leq
n}\binom{t_{3}}{t_{2}}\binom{t_{2}}{t_{1}}c^{t_{3}-t_{2}}d^{t_{2}-t_{1}}A^{\textnormal{{v}}^{*},\textnormal{{0}}}_{n,t_{3}}A^{\textnormal{{0}},\textnormal{{w}}^{*}}_{t_{1},k}$
(20) $\displaystyle A^{\textnormal{{v}}^{*},\textnormal{ {w}}^{*}}_{n,k}$
$\displaystyle=\sum_{k\leq t_{1}\leq t_{2}\leq t_{3}\leq
n}\binom{t_{3}}{t_{2}}\binom{t_{2}}{t_{1}}(-c)^{t_{3}-t_{2}}(-d)^{t_{2}-t_{1}}A^{\textnormal{{v}},\textnormal{{0}}}_{n,t_{3}}A^{\textnormal{{0}},\textnormal{{w}}}_{t_{1},k}\,.$
(21)

###### Proof.

Let c and d denote the constant functions equal to $c$ and $d$, respectively.
By repeated application of (7),
$\left[A^{\textnormal{{v}},\textnormal{{w}}}_{n,k}\right]=\left[A^{\textnormal{{v}},\textnormal{{0}}}_{n,k}\right]\left[A^{\textnormal{{0}},\textnormal{{w}}}_{n,k}\right]=\left[A^{\textnormal{{v}}^{*},\textnormal{{c}}}_{n,k}\right]\left[A^{\textnormal{{d}},\textnormal{{w}}^{*}}_{n,k}\right]=\left[A^{\textnormal{{v}}^{*},\textnormal{{0}}}_{n,k}\right]\left[c^{n-k}\binom{n}{k}\right]\left[d^{n-k}\binom{n}{k}\right]\left[A^{\textnormal{{0}},\textnormal{{w}}^{*}}_{n,k}\right]$,
which proves (20). Identity (21) is similarly proved. ∎

As a corollary, we obtain the following identity which expresses
$S_{s,h,q}[n,k]$ in terms of $q$-binomial coefficients.

######  Corollary 8.

Let $q\neq 1$. The numbers $S_{s,h,q}[n,k]$ may be written as

$S_{s,h,q}[n,k]=h^{n-k}(1-q)^{k-n}\sum_{\begin{subarray}{c}k\leq t_{1}\leq
t_{3}\leq n\end{subarray}}(-1)^{t_{3}-t_{2}+t_{1}-k}\\\
q^{(s-1)\binom{t_{1}-k}{2}-(s-1)\binom{t_{1}}{2}+s\binom{t_{3}}{2}}\binom{t_{3}}{t_{2}}\binom{t_{2}}{t_{1}}\genfrac{[}{]}{0.0pt}{}{n}{t_{3}}_{q^{s}}\genfrac{[}{]}{0.0pt}{}{t_{1}}{k}_{q^{s-1}}\,.$

###### Proof.

Let $Q=1/q$, $v^{*}_{i}=Q^{si}-1$, $w^{*}_{i}=-(Q^{(s-1)i}-1)$, $c=1,d=-1$,
$v_{i}=Q^{si},w_{i}=-Q^{(s-1)i}$. Observe that $v^{*}_{i}=(Q-1)[si]_{Q}$ and
$w^{*}_{i}=(Q-1)[(s-1)i]_{Q}$. Hence, by (21),

$(Q-1)^{n-k}A^{\textnormal{{v}}^{*},\textnormal{ {w}}^{*}}_{n,k}=\sum_{k\leq
t_{1}\leq t_{2}\leq t_{3}\leq
n}\binom{t_{3}}{t_{2}}\binom{t_{2}}{t_{1}}(-1)^{t_{3}-t_{2}+t_{1}-k}Q^{s\binom{n-t3}{2}}\genfrac{[}{]}{0.0pt}{}{n}{t_{3}}_{Q^{s}}\genfrac{[}{]}{0.0pt}{}{t_{1}}{k}_{Q^{s-1}}\,.$

Finally, we use the property
$\genfrac{[}{]}{0.0pt}{}{n}{k}_{1/p}=\genfrac{[}{]}{0.0pt}{}{n}{k}_{p}p^{-\binom{n}{2}+\binom{k}{2}+\binom{n-k}{2}}$.
∎

##  3\. Explicit Formulas

Let
$\textbf{r}=(r_{1},r_{2},\ldots,r_{n}),\textbf{s}=(s_{1},s_{2},\ldots,s_{n})$.
We will use $|\cdot|$ to denote the sum of the elements of a finite sequence.
We now consider the normal ordering of strings of the form
$H_{\textnormal{{r}},\textnormal{{s}}}=V^{r_{n}}U^{s_{n}}\cdots
V^{r_{2}}U^{s_{2}}V^{r_{1}}U^{s_{1}}$ which can be written in the form

$H_{\textnormal{{r}},\textnormal{{s}}}=\sum_{k=s_{1}}^{|\textbf{s}|}S^{\textbf{r},\textbf{s}}_{s,h,q}[k]V^{|\textbf{r}|-(|\textbf{s}|-k)(1-s)}U^{k}\,.$
(22)

In this section, we obtain explicit formulas for the numbers
$S^{\textbf{r},\textbf{s}}_{s,h,q}[k]$.

Our first identity is a $q$-analogue of the following explicit formula for
$S^{\textnormal{{r}},\textnormal{{s}}}_{0,1,1}(k)=S^{\textbf{r},\textbf{s}}(k)$
which was computed by El-Desouky, Cakić and Mansour [3] by repeated
application of the Leibniz formula

$S^{\textbf{r},\textbf{s}}(k)=\sum_{j_{1}+\cdots+j_{n-1}=s_{1}+\cdots+s_{n}-k}\prod_{i=1}^{n-1}\binom{s_{i+1}}{j_{i}}\left(r_{1}+\cdots+r_{i}-(j_{1}+\cdots+j_{i-1})\right)^{(\underline{j_{i}})}$
(23)

where $x^{(\underline{j})}=x(x-1)\cdots(x-(j-1))$ denotes the falling
factorial. Our proof uses a rook theoretic interpretation of normal ordering.
This interpretation was pointed out by Varvak [19, Section 7] and is similar
to the row creation rule introduced by Goldman and Haglund [6].

The process of computing the normal ordering of a string $w$ can be considered
as forming the collection of finite sequences of two operations, namely the
conversion $UV$ to $qVU$ and $UV$ to $hV^{s}$, starting from the rightmost
$UV$, such that successive applications of each element in the sequence
produces a different string, until a string in normally ordered form is
obtained. Let us denote the two operations by $\alpha$ and $\beta$,
respectively. Applying a sequence containing $k$ $\beta$’s results to an
expression $cV^{|\textbf{r}|-k(1-s)}U^{|\textbf{s}|-k}$. The sum of all such
coefficients $c$ is then the coefficient of
$V^{|\textbf{r}|-k(1-s)}U^{|\textbf{s}|-k}$ in the normal ordering of
$H_{\textnormal{{r}},\textnormal{{s}}}$. The process we just described can be
translated in terms of rook placements on a Ferrers board (or simply, board).

Let $U$ correspond to a horizontal step and $V$ a vertical step. Then $w$
outlines a board which we denote by $B(w)$. The conversion $\alpha$
corresponds to leaving a cell empty while the conversion $\beta$ corresponds
to placing a rook on a cell such that the rook cancels all cells on top of it
and divides the row to its left into $s$ rows. Note that if $s=0$, then a
placement of a rook converts a row into zero rows, or equivalently, it cancels
all the cells to its left. We denote a rook by marking a cell with “$\bullet$”
and a canceled cell with “$\times$”. Since the $\beta$’s convert the rightmost
$UV$, the rooks are placed in some chosen columns from right to left. We can
think of a cell lying in a divided row as containing _subcells_ , with each
cell containing 1 subcell by default. A canceled cell is a assigned the weight
1 while a cell containing a rook is assigned the weight $h$. All other cells
are assigned the weight $q^{a}$, where $a$ is the number of subcells in a
cell. The weight of the rook placement is the product of the weight of the
cells. Finally, the normally ordered string resulting from a particular rook
placement is $\omega V^{i}U^{j}$, where $\omega$ is the weight of the rook
placement, $i$ is the number of rows in the leftmost column plus $s-1$ if the
leftmost column contains a rook, and $j$ is the number of columns not
containing rooks. Note that the original model by Goldman and Haglund [6]
involves creating $s$ new rows to the left of a cell containing a rook and
then canceling the original row. The model we described involves dividing the
row into $s$ rows, ie., creating $s-1$ new rows without canceling the original
row. It is apparent that these models are equivalent, the only other notable
difference being the orientation of the boards. Alternatively, we also say the
placement of a rook in a cell adds $s-1$ subcells to every cell lying to its
left. Let us call the rook placement rule we just described as the _row
creation rule_.

Figure 1 shows a rook placement on $B(w)$, where $w=V^{2}U^{3}V^{3}U^{2},s=2$.
This placement corresponds to applying the sequence
$(\alpha,\alpha,\alpha,\alpha,\alpha,\beta,\alpha,\alpha,\beta,\alpha)$ from
right to left. The string $h^{2}q^{8}V^{7}U^{3}$ is produced.

| $\times$ | $\times$ |  |
---|---|---|---|---
| $\bullet$ | $\bullet$ |  |
|  |
|  |  |
Figure 1.

Let $B$ be a board. Denote by $C_{k}(B;s)$ the collection of all placements of
$k$ rooks on $B$ under the row creation rule. For a rook placement $\phi\in
C_{k}(B,s)$, denote the weight of $\phi$ by $\omega(\phi)$. We define the
_rook number_ $R_{s,h,q}[B,k]$ by

$R_{s,h,q}[B,k]=\sum_{\phi\in C_{k}(B;s)}\omega(\phi)\,.$ (24)

One sees that the number of rooks completely determines the exponents of $V$
and $U$ in the word resulting from the rook placement. In particular, if $k$
rooks are placed, then $k$ columns are cancelled and $k(s-1)$ rows are added.
Hence, we can write $H_{\textnormal{{r}},\textnormal{{s}}}$ as

$H_{\textnormal{{r}},\textnormal{{s}}}=\sum_{k=0}^{|\textbf{s}|-s_{1}}R_{s,h,q}[B(H_{\textnormal{{r}},\textnormal{{s}}}),k]V^{|\textbf{r}|-k(1-s)}U^{|\textbf{s}|-k}\,.$
(25)

Comparing this with (22) gives
$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]=R_{s,h,q}[B(H_{\textnormal{{r}},\textnormal{{s}}}),|\textbf{s}|-k]$.
It is important to note that
$R_{s,h,q}[B(H_{\textnormal{{r}},\textnormal{{s}}}),|\textbf{s}|-k]=0$ when
$k>|\textbf{s}|-s_{1}$ since the number of rooks cannot exceed the number of
rows of positive length. This is also reflected in the fact that the exponent
of $U$ in the normal ordering of $H_{\textnormal{{r}},\textnormal{{s}}}$ is
between $s_{1}$ and $|\textbf{s}|$, or equivalently, that
$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]=0$ when $k<s_{1}$ or
$k>|\textbf{s}|$.

The following lemma will be used in deriving an analogue of (23). An
equivalent formula was derived in [10] using a different method.

######  Lemma 9.

We have

$U^{s^{\prime}}V^{r^{\prime}}=\sum_{j=0}^{s^{\prime}}\left(h^{j}q^{r^{\prime}(s^{\prime}-j)}\genfrac{[}{]}{0.0pt}{}{s^{\prime}}{j}_{q^{s-1}}\prod_{i=0}^{j-1}[r^{\prime}+i(s-1)]_{q}\right)V^{r^{\prime}+j(s-1)}U^{s^{\prime}-j}\,.$

###### Proof.

Suppose that $j$ columns have been chosen where rooks will be placed. If the
first rook is placed on the cell in the $i$th row, $1\leq i\leq r^{\prime}$,
then the cells below the rook will contribute a weight of $q^{i-1}$. As $i$
varies, a total weight of $h(1+q+\cdots+q^{r^{\prime}-1})=h[r^{\prime}]_{q}$
will be contributed by all possible placements of the first rook. Since the
placement of the first rook adds $s-1$ subcells to every cell to its left, the
total weight contributed by all possible placement of the second rook is
$[r^{\prime}+(s-1)]_{q}$. Continuing this process with the other columns, we
see that the weight contributed by all possible placements of $j$ rooks in the
chosen columns is $h^{j}\prod_{i=0}^{j-1}[r+i(s-1)]_{q}$, and that this weight
is the same for any choice of $j$ columns.

We now consider the weight contributed by the other columns in which no rooks
are placed. For such a column, the weight is completely determined by the
number of columns to its right that contains a rook, i.e., if there are $t$
columns to its right containing a rook, then the column will assume a weight
of $q^{r^{\prime}+t(s-1)}$. Note that $t$ varies from $0$ to $j$ and that for
a given placement of $j$ rooks, the weight contributed by all the columns
containing no rooks is
$q^{r^{\prime}t_{0}}q^{(r^{\prime}+(s-1))t_{1}}q^{(r^{\prime}+2(s-1))t_{2}}\cdots
q^{(r^{\prime}+j(s-1))t_{j}}$ for some
$t_{0}+t_{1}+\cdots+t_{j}=s^{\prime}-j$. Summing this up on all such possible
collections $\\{t_{0},t_{1},\ldots,t_{i}\\}$, we have

$\displaystyle\sum_{t_{0}+t_{1}+\cdots+t_{j}=s^{\prime}-j}$ $\displaystyle
q^{r^{\prime}t_{0}}q^{(r^{\prime}+(s-1))t_{1}}q^{(r^{\prime}+2(s-1))t_{2}}\cdots
q^{(r^{\prime}+j(s-1))t_{j}}$
$\displaystyle=q^{r^{\prime}(s^{\prime}-j)}\sum_{t_{0}+t_{1}+\cdots+t_{j}=s^{\prime}-j}q^{0(s-1)t_{0}}q^{1(s-1)t_{1}}q^{2(s-1)t_{2}}\cdots
q^{j(s-1)t_{j}}$
$\displaystyle=q^{r^{\prime}(s^{\prime}-j)}\genfrac{[}{]}{0.0pt}{}{s^{\prime}}{j}_{q^{s-1}}\,,$

by (6) with $v_{i}=0$ and $w_{i}=q^{i(s-1)}$. This proves the lemma. ∎

######  Theorem 10.

The string $H_{\textnormal{{r}},\textnormal{{s}}}$ may be written as

$H_{\textnormal{{r}},\textnormal{{s}}}=\sum_{j_{1}=0}^{s_{2}}\sum_{j_{2}=0}^{s_{3}}\cdots\sum_{j_{n-1}=0}^{s_{n}}\prod_{i=1}^{n-1}h^{j_{1}+\cdots+j_{n-1}}\Gamma_{q,s}[j_{i},r_{1}+\cdots+r_{i}+(j_{1}+\cdots+j_{i-1})(s-1),s_{i+1}]\\\
V^{r_{1}+\cdots+r_{n}+(j_{1}+\cdots+j_{n-1})(s-1)}U^{s_{1}+\cdots+s_{n}-(j_{1}+\cdots+j_{n-1})}\,.$
(26)

where

$\Gamma_{q,s}[j,r^{\prime},s^{\prime}]=q^{r^{\prime}(s^{\prime}-j)}\genfrac{[}{]}{0.0pt}{}{s^{\prime}}{j}_{q^{s-1}}\prod_{i=0}^{j-1}[r^{\prime}+i(s-1)]_{q}\,.$

Hence, the numbers $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$ are
given by

$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]=h^{|\textnormal{{s}}|-k}\sum_{j_{1}+\cdots+j_{n-1}=s_{1}+\cdots+s_{n}-k}~{}\prod_{i=1}^{n-1}\Gamma_{q,s}[j_{i},r_{1}+\cdots+r_{i}+(j_{1}+\cdots+j_{i-1})(s-1),s_{i+1}]\,.$
(27)

###### Proof.

Identity (26) is proved by repeated application of (9) beginning from
$U^{s_{2}}V^{r_{1}}$. Identity (27) follows by comparing the coefficient of
$U^{k}$ in (26) and (22). ∎

######  Corollary 11.

The following explicit formula for $S_{s,h,q}[n,k]$ holds

$S_{s,h,q}[n,k]=h^{n-k}\sum_{j_{1}+\cdots+j_{n-1}=n-k}~{}\prod_{i=1}^{n-1}q^{(i+(j_{1}+\cdots+j_{i-1})(s-1))(1-j_{i})}\genfrac{[}{]}{0.0pt}{}{i+(j_{1}+\cdots+j_{i-1})(s-1)}{j_{i}}_{q}\,.$
(28)

Varvak’s [19] use of rook factorization to obtain an explicit formula adapts
readily in the case of $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$
after some modification. We will need the following analogues of the falling
factorial and factorial: for $r\in\mathbb{R},j\in\mathbb{N}$, define
$[r]^{(\underline{j})}_{q,1-s}=[r(1-s)]_{q}[(r-1)(1-s)]_{q}\cdots[(r-j+1)(1-s)]_{q}$
and for $n\in\mathbb{N}$, define $[n]_{q,1-s}!=[n]^{(\underline{n})}_{q,1-s}$.

######  Theorem 12.

Let $s\neq 1$. The coefficients
$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$ satisfy the explicit
formula

$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]=\frac{h^{|\textnormal{{s}}|-k}}{[k]_{q,1-s}!}\sum_{j=0}^{k}(-1)^{k-j}q^{\binom{k-j}{2}(1-s)}\genfrac{[}{]}{0.0pt}{}{k}{j}_{q^{1-s}}\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]$
(29)

where

$\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]=\prod_{t=1}^{n}[j-(s_{1}+s_{2}+\cdots+s_{t-1})+(r_{1}+r_{2}+\cdots+r_{t-1})/(1-s)]_{q,1-s}^{(\underline{s_{t}})}\,.$

###### Proof.

We use a representation of $V,U$ as linear operators whose action on the
monomial $t^{j}$ is given by $Vt^{j}=t^{j+1}$ and $Ut^{j}=h[n]_{q}t^{j+s-1}$.
One can verify that these operators satisfy $VU-qVU=hV^{s}$ and that
$U^{k}t^{n(1-s)}=h^{k}[n]_{q,1-s}^{(\underline{k})}t^{(n-k)(1-s)}$. We then
apply both sides of (22) to $t^{x(1-s)}$. After letting $t=1$ to the resulting
equation and using
$[x]^{(\underline{k})}_{q,1-s}=[1-s]_{q}^{k}[x]_{q^{1-s}}[x-1]_{q^{1-s}}\cdots[x-k+1]_{q^{1-s}}$,
we obtain

$h^{|\textbf{s}|}\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[x]=\sum_{k=s_{1}}^{|\textbf{s}|}h^{k}S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k][1-s]_{q}^{k}[x]_{q^{1-s}}[x-1]_{q^{1-s}}\cdots[x-k+1]_{q^{1-s}}\,.$
(30)

Let $E$ denote the shift operator $EP(x)=P(x+1)$ and $\Delta_{Q}^{k}$ the
$k$-th $Q$-difference operator defined by
$\Delta_{Q}^{k}=(E-1)(E-Q)\cdots(E-Q^{k-1})$. If
$P(x)=\sum_{k}p_{k}[x]_{Q}[x-1]_{Q}\cdots[x-k+1]_{Q}$, then
$p_{k}=\frac{1}{[k]_{Q}!}\Delta_{Q}^{k}P(x)|_{x=0}$. By the $q$-binomial
theorem,
$\Delta_{Q}^{k}=\sum_{j=0}^{k}(-1)^{j}Q^{\binom{j}{2}}\genfrac{[}{]}{0.0pt}{}{k}{j}_{Q}E^{k-j}$.
The result then follows by letting $Q=q^{1-s}$,
$p_{k}=h^{k}S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k][1-s]_{q}^{k}$ and
$P(x)=h^{|\textbf{s}|}\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[x]$. ∎

######  Corollary 13.

For $s\neq 1$ or $s\neq 0$, the numbers $S_{s,h,q}[n,k]$ has the following
explicit formula

$S_{s,h,q}[n,k]=\frac{h^{n-k}[s]_{q}^{n}}{[k]_{q^{1-s}}![1-s]_{q}^{k}}\sum_{j=0}^{k}(-1)^{k-j}q^{\binom{k-j}{2}(1-s)}\genfrac{[}{]}{0.0pt}{}{k}{j}_{q^{1-s}}\prod_{t=1}^{n}[(j/s)+t-j-1]_{q^{s}}\,.$

When $s=0$,

$S_{0,h,q}[n,k]=\frac{h^{n-k}}{[k]_{q}!}\sum_{j=0}^{k}(-1)^{k-j}q^{\binom{k-j}{2}}\genfrac{[}{]}{0.0pt}{}{k}{j}_{q}[j]_{q}^{n}\,.$

##  4\. $q$-Bell numbers

The Bell polynomial $B_{n}(x)$ is defined as the sum
$\sum_{k=0}^{n}S(n,k)x^{k}$ while the Bell number $B_{n}$ is given by
$B_{n}(1)$. Analogously, we define the generalized $q$-Bell polynomials
$B^{\textbf{r},\textbf{s}}_{s,h,q}[x]$ and generalized $q$-Bell numbers
$B^{\textbf{r},\textbf{s}}_{s,h,q}$ as

$B^{\textbf{r},\textbf{s}}_{s,h,q}[x]=\sum_{k=s_{1}}^{|\textbf{s}|}S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]x^{k},\hskip
18.06749ptB^{\textbf{r},\textbf{s}}_{s,h,q}=B^{\textbf{r},\textbf{s}}_{s,h,q}[1]\,.$

If $\textbf{r}=(1,1,\ldots,1),\textbf{s}=(1,1,\ldots,1)$, we define

$B_{s,h,q}[n;x]=B^{\textbf{r},\textbf{s}}_{s,h,q}[x],\hskip
18.06749ptB_{s,h,q}[n]=B_{s,h,q}[n;1]\,.$

For all the other particular cases, we apply the same notational convention in
Table 1. The numbers $B_{s,h,q}[n;x]$ and $B_{s,h,q}[n]$ reduce to the usual
Bell polynomial $B(n;x)$ and Bell number $B(n)$, respectively, when
$q=1,h=1,s=0$. The Bell polynomial $B(n;x)$ have the following expression as
an infinite series known as the _Dobinsky formula_ (see [20])

$B(n;x)=\frac{1}{e^{x}}\sum_{j=0}^{\infty}j^{n}\frac{x^{j}}{j!}\,.$

The corollary that follows gives the Dobinsky-type formula for
$B^{\textbf{r},\textbf{s}}_{s,h,q}[x]$.

######  Corollary 14.

Let $s\neq 1$ and $\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]$ be as
in Theorem 12. Then,

$B^{\textbf{r},\textbf{s}}_{s,h,q}[x]=\left(\sum_{j=0}^{\infty}h^{|\textbf{s}|-j}(-1)^{j}q^{\binom{j}{2}(1-s)}\frac{x^{j}}{[j]_{q,1-s}!}\right)\left(\sum_{j=0}^{\infty}\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]\frac{x^{j}}{h^{j}[j]_{q,1-s}!}\right)\,.$
(31)

###### Proof.

By the property $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]=0$ when
$|\textbf{s}|<k<s_{1}$ and by Theorem 12 ,

$\displaystyle B^{\textbf{r},\textbf{s}}_{s,h,q}[hx]$
$\displaystyle=\sum_{k=s_{1}}^{|\textbf{s}|}S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]h^{k}x^{k}$
$\displaystyle=\sum_{k=0}^{\infty}S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]h^{k}x^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{j=0}^{k}\frac{h^{|\textnormal{{s}}|}}{[k]_{q,1-s}!}(-1)^{k-j}q^{\binom{k-j}{2}(1-s)}\genfrac{[}{]}{0.0pt}{}{k}{j}_{q^{1-s}}\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]x^{k}\,.$

Using
$\genfrac{[}{]}{0.0pt}{}{k}{j}_{q^{1-s}}=\frac{[k]_{q^{1-s}}!}{[j]_{q^{1-s}}![k-j]_{q^{1-s}}!}=\frac{[k]_{q,1-s}!}{[j]_{q,1-s}![k-j]_{q,1-s}!}$
and the Cauchy product rule,

$\displaystyle B^{\textbf{r},\textbf{s}}_{s,h,q}[hx]$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{j=0}^{k}\left(\frac{h^{|\textnormal{{s}}|}(-1)^{k-j}q^{\binom{k-j}{2}(1-s)}}{[k-j]_{q,1-s}!}\right)\left(\frac{\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]}{[j]_{q,1-s}!}\right)x^{k}$
$\displaystyle=\left(\sum_{j=0}^{\infty}h^{|\textbf{s}|}(-1)^{j}q^{\binom{j}{2}(1-s)}\frac{x^{j}}{[j]_{q,1-s}!}\right)\left(\sum_{j=0}^{\infty}\Omega^{\textnormal{{r}},\textnormal{{s}}}_{s,q}[j]\frac{x^{j}}{[j]_{q,1-s}!}\right)\,.$

∎

Our goal in the remainder of this section is to obtain $q$-analogues of the
following Bell number identities derived by Mansour, Schork and Shattuck [11,
Theorems 4.4 and 5.3]:

$\displaystyle B_{s,h}(n)$
$\displaystyle=\sum_{r=0}^{n-1}h^{n-r-1}\binom{n-1}{r}B_{s,h}(r)\prod_{i=0}^{n-r-2}(1+si)$
(32) $\displaystyle B_{s,h}(n+m)$
$\displaystyle=\sum_{r=0}^{n}\sum_{j=0}^{m}h^{n-r}\binom{n}{r}S(m,j)B_{s,h}(r)\prod_{i=0}^{n-r-1}(j(1-s)+sm+si)\,.$
(33)

One verifies that (33) reduces to (32) when $n$ is replaced with $n-1$ and $m$
with $1$. The corresponding identity for the classical Bell numbers was first
derived by Spivey [18] and is given by

$B(n+m)=\sum_{r=0}^{n}\sum_{j=0}^{m}\binom{n}{r}S(m,j)B(r)j^{n-r}\,.$ (34)

In addition to (33), generalizations of the identity (34) have been proved
using different methods (see [21] and the references therein). The approach we
present here uses the rook model in Section 3.

Recall that $C_{k}(B;s)$ is the collection of all placements of $k$ rooks in
the board $B$. Denote by $J_{n}$ the board outlined by the string $(VU)^{n}$.
In the proofs that follow, we call a cell a _bottom cell_ if it is the
bottommost cell in a column.

######  Lemma 15.

Let $s\in\mathbb{N}$ and $\phi\in C_{k}(J_{n};s)$. Then, there exists a unique
(possibly empty) collection $\mathcal{C}$ of columns in $\phi$ such that if
$|\mathcal{C}|=m+1$, then (a) each of these columns has a rook in the bottom
$1,1+s,1+2s,\ldots,1+ms$ subcells and (b) every column not in $\mathcal{C}$
contains at least $1+st$ uncanceled subcells not containing a rook, where $t$
is the number of columns in $\mathcal{C}$ to the right of that column.

###### Proof.

Let $\phi\in C_{k}(J_{n};s)$. The set $\mathcal{C}$ may be obtained as
follows. Let $c_{1}$ be the first column of $\phi$ from the right containing a
rook in the bottom cell. If $c_{1}$ does not exist, then
$\mathcal{C}=\varnothing$ and $m=-1$. If $c_{1}$ exists, then all columns to
the right of $c_{1}$ have at least 1 uncanceled cell not containing a rook.
Let $c_{2}$ be the first column to the left of $c_{1}$ containing a rook in
the bottom $1+s$ subcells. If $c_{2}$ does not exist, then all columns to the
left of $c_{1}$ contain at least $1+s$ uncanceled subcells not containing a
rook. Hence, $\mathcal{C}=\\{c_{1}\\}$ satisfies (a) and (b) and $m=0$.
Otherwise, if such a column $c_{2}$ exists, then all columns to the right of
$c_{2}$ and to the left of $c_{1}$ contain $1+s$ uncanceled subcells not
containing a rook. Let $c_{3}$ be the first column to the left of $c_{2}$
containing a rook in the bottom $1+2s$ subcells. If $c_{3}$, does not exist,
then $\mathcal{C}=\\{c_{1},c_{2}\\}$ satisfies (a) and (b) and $m=1$. We
repeat the process with the succeeding columns as long as needed until all the
elements of $\mathcal{C}$ are determined. This process shows both the
existence and uniqueness of $\mathcal{C}$, which proves the lemma. ∎

######  Theorem 16.

Let $n,k\in\mathbb{N}$. Then

$S_{s,h,q}[n,k]=\sum_{r=k-1}^{n-1}h^{n-r-1}q^{r}\genfrac{[}{]}{0.0pt}{}{n-1}{r}_{q^{s}}S_{s,h,q}[r,k-1]\prod_{i=0}^{n-r-2}[1+si]_{q}\,.$
(35)

Furthermore, the numbers $B_{s,h,q}[n]$ are given by

$B_{s,h,q}[n]=\sum_{r=0}^{n-1}h^{n-r-1}q^{r}\genfrac{[}{]}{0.0pt}{}{n-1}{r}_{q^{s}}B_{s,h,q}[r]\prod_{i=0}^{n-r-2}[1+si]_{q}\,.$
(36)

###### Proof.

We first prove (35). The number $S_{s,h,q}[n,k]$ equals the sum of the weights
of all rook placements in $C_{n-k}(J_{n};s)$. Let $R_{n-k}(J_{n};s;n-r-2)$ be
the subset of $C_{n-k}(J_{n};s)$ consisting of rook placements that satisfy
conditions (a) and (b) in Lemma 15 with $m=n-r-2$. Then, the same lemma
implies that the collection $\\{R_{n-k}(J_{n};s;n-r-2)\,|\,k-1\leq r<n-1\\}$
forms a partition of $C_{n-k}(J_{n};s)$. We want to show that

$\sum_{\phi\in
R_{n-k}(J_{n};s;n-r-2)}\omega(\phi)=h^{n-r-1}q^{r}\genfrac{[}{]}{0.0pt}{}{n-1}{r}_{q^{s}}S_{s,h,q}[r,k-1]\prod_{i=0}^{n-r-2}[1+si]_{q}\,,$

from which (35) follows.

For $\phi\in R_{n-k}(J_{n};s;n-r-2)$, denote its set of columns satisfying the
conditions in Lemma 15 by $\mathcal{C}_{\phi}$. Let
$\lambda_{\mathcal{C}_{\phi}}$ be the cells of $\phi$ consisting of its cells
of the columns $\mathcal{C}_{\phi}$ and the bottom $st+1$ cells of the other
columns, where $t$ is the number of columns from $\mathcal{C}_{\phi}$ that lie
to the right of one such column. Also, let $\phi-\lambda_{\mathcal{C}_{\phi}}$
be the cells of $\phi$ not in $\lambda_{\mathcal{C}_{\phi}}$. One sees that
the cells in $\phi-\lambda_{\mathcal{C}_{\phi}}$ form a rook placement in
$C_{r-k+1}(J_{r};s)$. We can therefore write every rook placement $\phi\in
R_{n-k}(J_{n};s;n-r-2)$ uniquely as a pair $(\lambda_{\mathcal{C}},\rho)$, for
some set of columns $\mathcal{C}$ satisfying Lemma 15 with $m=n-r-2$, and some
rook placement $\rho$ in $C_{r-k+1}(J_{r};s)$. The sum of the weights of rook
placements in $C_{r-k+1}(J_{r};s)$ is $S_{s,h,q}[r,k-1]$. We now compute the
sum of the weights of the cells in $\lambda_{\mathcal{C}}$ over all such
possible set of columns $\mathcal{C}$, which we denote by $L_{n-r-2}$.

Clearly, the cells in $\lambda_{\mathcal{C}}$ which contain rooks contribute a
weight of $h^{n-r-1}\prod_{i=0}^{n-r-2}[1+si]_{q}$. In addition,
$\lambda_{\mathcal{C}}$ contains $r$ bottom cells, which collectively
contribute a weight of $q^{r}$. The weight contributed by the remaining cells
depends on the location of the rooks. To get a better picture of how the
contribution by the remaining cells varies, let us distribute the cells of
$\lambda_{\mathcal{C}}$ so that the $s$ subcells lie to the left of each rook.
We illustrate this in Figure 2 for $n=9$ and $n-r-2=3$. Here, the second
column of $\lambda_{\mathcal{C}}$ from the left lies to the left of three
columns of $\lambda_{\mathcal{C}}$ containing rooks. Hence, this column
contains $1+3s$ bottom cells. We moved each of the three $s$ subcells
(indicated by marking a cell with an $s$) so that they lie to the left of each
of the three rooks. Going back to the general case, let
$t_{1},t_{2},\ldots,t_{n-r-1}$ be the number of cells marked $s$ in the rows
containing the rooks in $\lambda_{\mathcal{C}}$ starting from bottommost row.
Then the $t_{i}$’s satisfy $0\leq t_{1}\leq t_{2}\leq\ldots\leq t_{n-r-1}\leq
r$. Hence, if $\omega(\lambda_{\mathcal{C}})$ denotes the product of the
weights of the cells in $\lambda_{\mathcal{C}}$, we have

$\displaystyle\sum_{\mathcal{C}\in L_{n-r-2}}\omega(\lambda_{\mathcal{C}})$
$\displaystyle=h^{n-r-1}q^{r}\prod_{i=0}^{n-r-2}[1+si]_{q}\sum_{0\leq
t_{1}\leq t_{2}\leq\ldots\leq t_{n-r-1}\leq r}q^{st_{1}}q^{st_{2}}\cdots
q^{st_{n-r-1}}$
$\displaystyle=h^{n-r-1}q^{r}\prod_{i=0}^{n-r-2}[1+si]_{q}\genfrac{[}{]}{0.0pt}{}{n-1}{r}_{q^{s}}\,.$

where the second equality follows from (5) with $v_{i}=0,w_{i}=q^{is}$. This
proves (35).

To prove (36), we take the sum of both sides of (35) over all $0\leq k\leq n$.
∎

$\times$ | $s$ | $s$ | $s$ | $\times$ | $s$ | $\times$ | $\bullet$ |
---|---|---|---|---|---|---|---|---
$\times$ | $s$ | $s$ | $s$ | $\times$ | $s$ | $\bullet$ |  |
$\times$ |  |  |  | $\times$ |  |  |  |
$\times$ | $s$ | $s$ | $s$ | $\bullet$ |  |  |  |
$\times$ |  |  |  |  |  |  |  |
$\times$ |  |  |  |  |  |  |  |
$\times$ |  |  |  |  |  |  |  |
$\bullet$ |  |  |  |  |  |  |  |
Figure 2.

######  Lemma 17.

Denote by $J_{n\oplus\alpha}$ the board outlined by $(VU)^{n}V^{\alpha}$. Let
$s\in\mathbb{N}$ and $\phi\in C_{k}(J_{n\oplus\alpha};s)$. Then, there exists
a unique (possibly empty) collection $\mathcal{C}$ of columns in $\phi$ such
that if $|\mathcal{C}|=m+1$, then (a) each of these columns has a rook in the
bottom $\alpha,\alpha+s,\ldots,\alpha+sm$ subcells and (b) every column not in
$\mathcal{C}$ contains at least $\alpha+st$ uncanceled subcells not containing
a rook, where $t$ is the number of columns in $\mathcal{C}$ to the right of
that column.

###### Proof.

The proof is similar to that of Lemma 15. ∎

######  Theorem 18.

Let $n,m,k\in\mathbb{N}$. We have

$S_{s,h,q}[n+m,k]=\sum_{r=0}^{n}\sum_{j=0}^{m}h^{n-r}q^{r(j(1-s)+sm)}\genfrac{[}{]}{0.0pt}{}{n}{r}_{q^{s}}S_{s,h,q}[m,j]S_{s,h,q}[r,k-j]\prod_{i=0}^{n-r-1}[j(1-s)+sm+si]_{q}\,.$
(37)

Moreover, the numbers $B_{s,h,q}[n+m]$ are given by

$B_{s,h,q}[n+m]=\sum_{r=0}^{n}\sum_{j=0}^{m}h^{n-r}q^{r(j(1-s)+sm)}\genfrac{[}{]}{0.0pt}{}{n}{r}_{q^{s}}S_{s,h,q}[m,j]B_{s,h,q}[r]\prod_{i=0}^{n-r-1}[j(1-s)+sm+si]_{q}\,.$
(38)

###### Proof.

We first prove $\eqref{nmk}$. The number $S_{s,h,q}[n+m,k]$ equals the total
weight of all rooks placements in $C_{n+m-k}(J_{n+m};s)$. The rooks may be
placed as follows: Number the columns from right to left. For some $j$ such
that $0\leq j\leq m$, place $m-j$ rooks in columns $2,\ldots,m$ and the
remaining $n+j-k$ rooks in columns $m+1,\ldots,n$. The total weight of all
placements of $m-j$ rooks in columns $2,\ldots,m$ is $S_{s,h,q}[m,j]$. As a
consequence of placing $m-j$ rooks, each of columns $m+1,\ldots,n$ have
${s(m-j)+j=j(1-s)+sm}$ subcells in their first $m$ cells from the top. Hence,
these columns form the board $J_{n\oplus\alpha}$, with $\alpha=j(1-s)+sm$.
Using Lemma 17, we can form a suitable partition of
$C_{k}(J_{n\oplus\alpha};s)$ and proceed as in the proof Identity (35) in
Theorem 16. We leave the details to the reader.

To obtain (38), take the sum of both sides of (37) over all $0\leq k\leq n+m$.
∎

######  Corollary 19.

The recurrence relation for the generalized $q$-Bell polynomials is given by

$B_{q}[n+m;x]=\sum_{r=0}^{n}\sum_{j=0}^{m}h^{n-r}q^{r(j(1-s)+sm)}\genfrac{[}{]}{0.0pt}{}{n}{r}_{q^{s}}S_{s,h,q}[m,j]B_{q}[r;x]x^{j}\prod_{i=0}^{n-r-1}[j(1-s)+sm+si]_{q}\,.$

In particular,

$B_{s,h,q}[n;x]=\sum_{r=0}^{n-1}h^{n-r-1}q^{r}\genfrac{[}{]}{0.0pt}{}{n-1}{r}_{q^{s}}B_{s,h,q}[r;x]x\prod_{i=0}^{n-r-2}[1+si]_{q}\,.$

Moreover, a $q$-analogue of Spivey’s identity (34) is

$B_{q}[n+m]=\sum_{r=0}^{n}\sum_{j=0}^{m}q^{rj}\binom{n}{r}S_{s,h,q}[m,j]B_{q}[r][j]_{q}^{n-r}$

where we take $0^{0}=1$.

##  5\. The numbers $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$ and
$S_{s,h,q}[n,k]$ for $s\in\mathbb{R}$

Goldman and Haglund introduced, in the latter part of their paper [6], a rook
placement rule which is defined for $s\in\mathbb{R}$, such that the
corresponding rook numbers agree with the rook numbers under the row creation
rule when $s\in\mathbb{N}$. We give an presentation of this rule using the
concept of _pre-weights_. As before, we place the rooks in some chosen columns
from right to left such that at most one rook occupies each column. We say
that every cell has a default pre-weight of 1 and a placement of a rook adds a
pre-weight of $s-1$ to every cell to its left. Every cell lying above a rook
is assigned the weight 1. If a cell does not lie above a rook and has pre-
weight $p$, then it is assigned the weight $h[p]_{q}$ if it contains a rook,
and $q^{p}$ if otherwise. Let us call the rook placement rule we just
described as the _pre-weight rule_. We denote the weight of a rook placement
$\psi$ by $\omega(\psi)$, which is defined as the product of the weights of
the cells. Given a board $B$, we denote by $C^{*}_{k}(B;s)$ the collection of
all placement of $k$ rooks on $B$ under the pre-weight rule. When
$s\in\mathbb{N}$, the pre-weight of a cell is exactly the number of its
subcells. In this case, we have

$R_{s,h,q}[B(w),k]=\sum_{\phi\in C_{k}(B;s)}\omega(\phi)=\sum_{\psi\in
C^{*}_{k}(B;s)}\omega(\psi)\,.$

We can therefore define $R[B(w),k]$ as

$R_{s,h,q}[B(w),k]=\sum_{\psi\in C^{*}_{k}(B;s)}\omega(\psi)\,.$

when $s\in\mathbb{R}$ without ambiguity. Using this definition, we can extend
numbers the $S^{\textbf{r},\textbf{s}}_{s,h,q}[k]$ for $s\in\mathbb{R}$ by
defining them as
$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]=R_{s,h,q}[B(H_{\textnormal{{r}},\textnormal{{s}}}),|\textbf{s}|-k]$.
Analogously, we define $S_{s,h,q}[n,k]=R_{s,h,q}[B((VU)^{n}),|\textbf{s}|-k]$.
It can be shown that under this definition, $S_{s,h,q}[n,k]$ satisfies the
same recursion(1). Since the identities for $S_{s,h,q}[n,k]$ in Section 2 were
derived using this recursion, all of them also hold when $s\in\mathbb{R}$.
Note that the numbers $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[n,k]$
lose their interpretation as normal ordering coefficients when $s$ is not a
nonnegative integer.

The proof of Lemma 9, and hence, of Theorem 10 (explicit formula for
$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$) and Corollary 11 (explicit
formula for $S_{s,h,q}[n,k]$) extends readily under the pre-weight rule. We
leave the details to the reader. On the other hand, the proof of Theorem 12
(explicit formula for $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$), and
hence, of Corollary 13 (explicit formula for $S_{s,h,q}[n,k]$) and Corollary
14 (Dobinsky-type formula), uses the rook factorization (30), which was
computed using the normal ordering interpretation. The result [6, Theorem 7.1]
furnishes the needed rook factorization for $h=1$, which will allow us to
extend the said theorem and its corollaries. This result can be extended to
arbitrary $h$ without much difficulty.

To prove Theorem 16 (recursion for $S_{s,h,q}[n,k]$ and $B_{s,h,q}[n]$),
Theorem 18 (recursion for $S_{s,h,q}[n+m,k]$ and $B_{s,h,q}[n+m]$) and
Corollary 19 for the case $s\in\mathbb{R}$, we impose a modification in the
assignment of pre-weights when the board is $J_{n}$. First, as usual, choose
columns where we will place rooks. The first rook adds a pre-weight of $s-1$
to each cell that lies above the bottom cell in every column to its left, the
second rook adds a pre-weight of $s-1$ to the cell that lies above the cell in
every column to its left which has been added a pre-weight of $s-1$ by the
first rook, etc. We denote by $C^{\\#}_{k}(J_{n};s)$ the set of all rook
placements on $J_{n}$ under this rule, which we call the _modified pre-weight
rule_. This modification preserves the sum of the weights of all rook
placements given a choice of columns where rooks are to be placed. Hence,

$\sum_{\psi\in C^{*}_{k}(J_{n};s)}\omega(\psi)=\sum_{\rho\in
C^{\\#}_{k}(J_{n};s)}\omega(\rho)\,.$ (39)

###### Proof of Theorems 16 and 18 for $s\in\mathbb{R}$.

We only describe some portions of the necessary changes in the proofs of
Theorems 16 and 18 for $s\in\mathbb{R}$. We leave the rest of the details to
the reader.

For Theorem 16, we will need the following modification of Lemma 15: _Let
$\phi\in C^{\\#}_{k}(J_{n};s)$. Then, there exists a unique (possibly empty)
collection $\mathcal{C}$ of columns in $\phi$ such that if
$|\mathcal{C}|=m+1$, then (a) each of these columns has a rook in the bottom
$1,2,3,\ldots,1+m$ cells and (b) every column not in $\mathcal{C}$ contains at
least $1+t$ uncanceled cells not containing a rook, where $t$ is the number of
columns in $\mathcal{C}$ to the right of that column._

For $1\leq j\leq 1+m$, the bottom $j$ cells referred to in (a) have a combined
pre-weight of $1+s(j-1)$, and hence, the sum of the weights of all possible
placement of rooks on these $j$ cells is $h[1+s(j-1)]_{q}$. On the other hand,
the $1+t$ uncanceled cells in (b) have a combined pre-weight of $1+st$ and
hence, contributes a weight of $q^{1+st}$.

We now turn to Theorem 18. As in the proof of the case $s\in\mathbb{N}$, we
place $m-j$ rooks in columns $2,\ldots,m$ of $J_{n+m}$. As a consequence, the
first $m$ cells from the top of columns $m+1,\ldots,n$ have a total pre-weight
of $j(1-s)+sm$. Let $B$ be the board consisting of columns $m+1,\ldots,n$ of
$J_{n+m}$ after $m-j$ rooks were placed.

It can be verified that the relation (39) also holds when some of the cells of
$J_{n}$ have default pre-weights other than 1. Let $J^{\prime}_{n,\alpha}$ be
the board $(VU)^{n}V$ such that the bottom cells have a default pre-weight of
$\alpha$. One sees that sum of the pre-weights of the $t$-th column of $B$ and
the sum of the pre-weights of the $t$-th column of $J^{\prime}_{n,\alpha}$
with $\alpha=j(1-s)+sm$ are equal. This implies that the sum of the weights of
the rook placements in $C^{\\#}_{k}(B;s)$ equals that of
$C^{\\#}_{k}(J^{\prime}_{n,\alpha};s)$. The rest of the proof uses a
generalization of Lemma 17, which is the same statement as the modification of
Lemma 15, except that we replace $C^{\\#}_{k}(J_{n};s)$ with
$C^{\\#}_{k}(J^{\prime}_{n,\alpha};s)$. ∎

##  6\. Conclusion

We have obtained a number of identities for $S_{s,h,q}[n,k]$, including an
orthogonality relation, recurrence formulas and expressions involving other
$q$-analogues. In particular, it was shown that $S_{s,h}(n,k)$ can be written
in terms of the classical Stirling numbers and $S_{s,h,q}[n,k]$ in terms of
$q$-Stirling numbers. Explicit formulas for
$S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$, and consequently,
$S_{s,h,q}[n,k]$, were also given. Recurrence relations for $S_{s,h,q}[n,k]$
and $B_{s,h,q}[n]$ have also been proved using rook placements. As a
consequence, we have also provided alternative proofs of the original
identities in [11] which were proved using weighted Laguerre configurations.
It is possible that the identities for $S_{s,h,q}[n,k]$ in Section 2 also have
rook theoretic proofs. When viewed as normal ordering coefficients, the
numbers $S^{\textnormal{{r}},\textnormal{{s}}}_{s,h,q}[k]$ are defined only
for $s\in\mathbb{N}$. These numbers, as well as the identities they possess
which were derived in the earlier sections, were extended to the case
$s\in\mathbb{R}$ using a modified rook model.

We have not considered the corresponding boson operators and their action on
coherent states that arise as a consequence of the generalized commutation
relation. We note that the $q$-deformed case where $h=1,s=0$ was studied by
Schork [17] while Blasiak [1] studied the undeformed case and obtained the
normal ordering of a more general class of expressions involving the boson
operators.

## Acknowledgment

K.J.M. Gonzales would like to thank the Department of Science and Technology
(DOST) through the ASTHRD Program for financial support during his stay at the
University of the Philippines - Diliman.

## References

  * [1] P. Blasiak, Combinatorics of boson normal ordering and some applications, Ph.D. Thesis, University of Paris VI and Polish Academy of Sciences (2005), arXiv:quant-ph/0507206.
  * [2] R.B. Corcino, L.C. Hsu and E.L. Tan, A $q$-analogue of generalized Stirling numbers, Fibonacci Quart. 44 (2006), 154-166.
  * [3] B.S. El-Desouky, N. Cakić and T. Mansour, Modified approach to generalized Stirling numbers via differential operators, App. Math. Lett. 23 (2010), 115-120.
  * [4] B.S. El-Desouky and N. Cakić, Generalized higher ordering Stirling numbers, Math. Comp. Modelling 54 (2011), 2848-2857.
  * [5] L. Hsu and P.J. Shiue, A unified approach to generalized Stirling numbers, Adv. App. Math. 20 (1998), 366-384.
  * [6] J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory Ser. A 91 (2000), 509-530.
  * [7] J. Katriel and M. Kibler, Normal ordering for deformed boson operators and operator valued deformed Stirling numbers, J. Phys. A: Math. Gen. 25 (1992), 2683-2691.
  * [8] W. Lang, On generalizations of the Stirling number triangles, J. Integer Seq. 3 (2000), Article 00.2.4.
  * [9] T. Mansour, S. Mulay and M. Shattuck, A general two-term recurrence and its solution, European J. Combin. 33 (2012), 20-26.
  * [10] T. Mansour and M. Schork, The commutation relation $xy=qyx+hf(y)$, Ramanujan J. 25 (2011), 405-445.
  * [11] T. Mansour, M. Schork and M. Shattuck, On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011), #P77.
  * [12] T. Mansour, M. Schork and M. Shattuck, The generalized Stirling and Bell numbers revisited, J. Integer Seq. 15 (2012), Article 12.8.3.
  * [13] A. Médicis and P. Leroux, A unified combinatorial approach for $q$\- ($p,q$-) Stirling numbers, J. Stat. Planning and Inference 34 (1993), 89-105.
  * [14] A. Médicis and P. Leroux, Generalized Stirling numbers, convolution formulae and $p,q$-analogues, Canad. J. Math 47 (1995), 474- 499.
  * [15] A. Navon, Combinatorics and fermion algebra, Nuovo Cimento 16 (1973) 324-330.
  * [16] J.B. Remmel and M. Wachs, Rook theory, generalized Stirling numbers and $(p,q)$-analogues, Electron. J. Combin. 11 (2004), #P84.
  * [17] M. Schork, Normal ordering $q$-bosons and combinatorics, J. Phys. Lett. A 355 (2006), 293-297.
  * [18] M. Spivey, A generalized recurrence for Bell numbers, J. Integer Seq. 11 (2008), Article 08.2.5.
  * [19] A. Varvak, Rook numbers and the normal ordering problem, J. Combin. Theory Ser. A 112 (2005), 292-307.
  * [20] A. Xu, A Newton interpolation approach to generalized Stirling numbers, J. App. Math. (2012), Article ID 351935.
  * [21] A. Xu, Extensions of Spivey’s Bell number formula, Electron. J. Combin. 19(2) (2012), #P6.