\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage{bm} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\pq}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} \begin{center} \vskip 1cm{\Large\bf Two Approaches to Normal Order Coefficients} \vskip 1cm \large Richell O. Celeste\\ Institute of Mathematics and Natural Sciences Research Institute\\ University of the Philippines --- Diliman\\ Quezon City, Philippines 1101\\ \href{mailto:ching@math.upd.edu.ph}{\tt ching@math.upd.edu.ph} \\ \ \\ Roberto B. Corcino\\ Mathematics Department\\ Cebu Normal University\\ Cebu City, Philippines 6000\\ \href{mailto:rcorcino@yahoo.com}{\tt rcorcino@yahoo.com} \\ \ \\ Ken Joffaniel M. Gonzales\\ Department of Physical Sciences and Mathematics\\ University of the Philippines --- Manila\\ Manila, Philippines 1004\\ \href{mailto:kmgonzales@upd.edu.ph}{\tt kmgonzales@upd.edu.ph} \\ \ \\ \end{center} \vskip .2 in \baselineskip14pt \begin{abstract} We consider the normal ordering coefficients of strings consisting of the symbols $V,U$ which satisfy the commutation rule $UV-qVU=hV^s$. These coefficients are studied using two approaches. First, we continue the study by Varvak, where the coefficients were interpreted as $q$-rook numbers under the row creation rook model introduced by Goldman and Haglund. Second, we express the coefficients in terms of a kind of generalization of some symmetric functions. We derive identities involving the coefficients including some explicit formulas. \end{abstract} \section{Introduction} Let $V,U$ be operators (or variables) that satisfy the commutation rule $UV-qVU=hV^s$, where $s\in\mathbb N$ and $h,q\in\mathbb R$. For example, if $s=0, h=1, q=1$, then $V,U$, respectively, can be represented by the creation operator and annihilation operator in quantum physics \cite{Bla}, or by the operators $X,\partial_x$ whose action on a monomial $x^n$ are given by $Xx^n=x^{n+1}$ and $\partial_xx^n=nx^{n-1}$. Given a string $w$ consisting of $V$'s and $U$'s, the \emph{normally ordered} form of $w$ is an equivalent operator expressed as a sum $\sum c_{i,j} V^i U^j$. The normally ordered form can be computed using the commutation rule alone, i.e., by replacing all occurrences of $UV$ with $qVU+hV^s$, but this task can be cumbersome especially for long strings. It turns out that the coefficients $c_{i,j}$, called \emph{normal ordering coefficients}, can be computed more efficiently using combinatorial techniques. In the classical case $s=0, h=1, q=1$, Navon \cite{Nav} showed that the normal ordering coefficients of an arbitrary string are given by rook numbers on a Ferrers board. Varvak \cite{Var} generalized Navon's result for arbitrary $q$ and derived explicit formulas for these coefficients using rook factorization. Blasiak \cite{Bla}, El-Desouky et al.~\cite{DesCakMan}, and Mansour et al.~\cite{ManSchSha1,ManSchSha2} also computed explicit formulas using other methods. In this paper, we study normal ordering coefficients using two approaches. In Section \ref{first}, we study the coefficients as $q$-rook numbers under the row creation rule introduced by Goldman and Haglund \cite{GolHag}. In Section \ref{second}, we study the coefficients by expressing them in terms of some generalization of elementary and complete homogeneous symmetric functions. Lastly, some special cases are given in Section \ref{special}. \section{First approach: rook numbers}\label{first} Let $\textbf v=(v_1,v_2,\ldots,v_n), \textbf u=(u_1,u_2,\ldots,u_n)$, and $H_{{\textbf{v}},{\textbf{u}}}=V^{v_n}U^{u_n}\cdots V^{v_2}U^{u_2}V^{v_1}U^{u_1}$. In this section, we obtain explicit formulas for the normal ordering coefficients of $H_{{\textbf{v}},{\textbf{u}}}$ which uses the known rook theoretic interpretation directly. We also give a representation of $V,U$ in terms of linear operators and use it to find another explicit formula which generalizes Varvak's \cite[Corollary 4.2]{Var}. Following Blasiak \cite{Bla}, we write the string $H_{{\textbf{v}},{\textbf{u}}}$ in the form \begin{equation}\label {wnorm} H_{{\textbf{v}},{\textbf{u}}} = \sum_ {k=u_1}^{|\textbf u|} S^{\textbf v, \textbf u}_{s,h;q}[k]\, V^{|\textbf v| - (|\textbf u|-k)(1-s)} U^k\,, \end{equation} where $|\textbf u|=u_1+u_2+\cdots+u_n$ and $|\textbf v|=v_1+v_2+\cdots+v_n$. Varvak \cite{Var} showed that for $h=1,q=1$, the coefficients $S^{\textbf v,\textbf u}_{s,h;q}[k]$ also occur as rook numbers under the rook model introduced by Goldman and Haglund \cite{GolHag} which we now describe. An $s$-\emph{rook placement} on a Ferrers board $B$ is obtained as follows. First, choose the columns where rooks will be placed. The rooks are then placed one by one from the right such that every time a rook is placed in a cell the entire row to its left is divided into $s$ rows. When $s=0$, ``division'' into $s$ rows can be interpreted as cancellation of the entire row lying to the left of a rook. Denote by $\mathcal R_s(B,k)$ the set of all placements of $k$ rooks on $B$. An example of a rook placement where $s=2$ is shown in Figure \ref{rooktab}. The $k$-th $s$-rook number of a board $B$ is then defined as $R_{s}(B(w),k)=|\mathcal R_s(B,k)|$. \begin{figure}[htbp] \begin{center} \epsfxsize=1.5in \leavevmode\epsffile{imw.eps} \caption{A placement of 3 rooks.} \label{rooktab} \end{center} \end{figure} Varvak \cite[Theorem 7.1]{Var} showed, in our notation (and after correcting for the typo pointed out by Mansour et al.~\cite{ManSchSha1}), that \begin{equation}\label{rooknorm} H_{{\textbf{v}},{\textbf{u}}} = \sum_ {k=0}^{|\textbf u|-u_1} R_{s,1;1}[B(H_{{\textbf{v}},{\textbf{u}}}),k]\, V^{|\textbf v|-k(1-s)} U^{|\textbf u|-k}. \end{equation} Comparing \eqref{rooknorm} with \eqref{wnorm} gives $S^{{\textbf{v}}, {\textbf{u}}}_{s,1;1}[k] = R_{s,1;1}[B(H_{{\textbf{v}},{\textbf{u}}}),|\textbf u|-k]$. The connection between rook numbers and normal ordering lies on the fact that a string $w$ determines a unique Ferrers board (where a column or row is allowed to have length zero) and that the placement or non-placement of a rook corresponds to the choice of replacement for each occurrence of $UV$. Specifically, a string $w$ outlines a Ferrers board $B(w)$ whose underside border is the lattice path obtained from $w$ by replacing $V$ with a unit step up and $U$ with a unit step to the right. For example, the Ferrers board in Figure \ref{rooktab} is outlined by $VUVUUVVUVVU$. Placing a rook on the northeast inner corner of $B(w)$ corresponds to replacing the rightmost $UV$ with $hV^s$ while leaving a cell empty corresponds to replacing $UV$ with $qVU$. Under this correspondence, it is now clear how Varvak's result can be extended to arbitrary $q$. Given a rook placement $\phi$, assign the weight $w(\phi)=h^{t'}q^t$ where $t'$ is the number of rooks, and $t$ is the number of cells not containing a rook and not lying above a rook if $s\neq 0$ and $t$ is the number of cells not containing a rook and not lying above or to the left of a rook if $s=0$. For example, the rook placement in Figure \ref{rooktab} has weight $h^3q^{11}$. We define the generalized rook numbers by \[ R_{s,h;q}[B,k] = \sum_{\phi\in \mathcal R_s(B,k)} w(\phi). \] When all three parameters $s,h,q$ are arbitrary, the normally ordered form of the string $H_{\textbf v,\textbf u}$ is then given by \begin{equation} H_{{\textbf{v}},{\textbf{u}}} = \sum_ {k=0}^{|\textbf u|-u_1} R_{s,h;q}[B(H_{{\textbf{v}},{\textbf{u}}}),k]\, V^{|\textbf v|-k(1-s)} U^{|\textbf u|-k}.\label{neweq} \end{equation} Again, comparing \eqref{neweq} with \eqref{wnorm} gives $S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k] = R_{s,h;q}[B(H_{{\textbf{v}},{\textbf{u}}}),|\textbf u|-k]$. Our first result gives the normally ordered form of strings of the form $U^{m}V^{l}$. The following notation will be used. For $n\in\mathbb N$, let $[n]_q=1+q+q^2+\cdots+q^{n-1}$ if $n>0$ and $[0]_q=0$. Let $[n]_q!=[n]_q[n-1]_q\cdots[2]_q[1]_q$ for $n>0$ and $[0]_q!=1$. The $q$-binomial coefficient is defined by $\pq{n}{k}_q=\frac{[n]_q!}{[k]_q![n-k]_q!}$. They satisfy the property (see [6,\,Table 1 and Identity 2.2]) \begin{align} \pq{n}{k}_q = \sum_{0 \leq i_1 \leq i_2 \leq \ldots \leq i_{n-k} \leq k} q^{i_1+i_2+\cdots+i_{n-k}}\,. \label{qbinomp} \end{align} \begin{theorem}\label{lemrep} For $s\in\mathbb N$, the normally ordered form of $U^{m}V^{l}$ is given by \begin{equation} U^{m}V^{l} = \sum_{j=0}^{m} \left( h^jq^{l(m-j)} \pq{m}{j}_{q^{s-1}} \prod_{i=0}^{j-1} [l+i(s-1)]_q \right)V^{l+j(s-1)} U^{m-j}.\label{repx} \end{equation} \begin{proof} The string $U^mV^l$ outlines a rectangular Ferrers board with $m$ columns and $l$ rows. The total weight of all placements of $j$ rooks on this board equals the coefficient of $V^{l+j(s-1)}U^{m-j}$ in the normally ordered form of $U^mV^l$. We now compute the total weight of such rook placements as follows. Choose $j$ columns where rooks will be placed. If the first rook is placed on the cell in the $i$th row, $1\leq i\leq l$, then the cells below the rook will contribute a weight of $q^{i-1}$. As $i$ varies, a total weight of $h(1+q+\cdots+q^{l-1})=h[l]_q$ 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 $[l+(s-1)]_q$. 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 $h^j\prod_{i=0}^{j-1} [l+i(s-1)]_q$, 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 $q^{l+t(s-1)}$. 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 $q^{lt_0} q^{(l+(s-1))t_1} q^{(l+2(s-1))t_2} \cdots q^{(l+j(s-1))t_j}$ for some $t_0+t_1+\cdots+t_j=m-j$. Summing this up over all such possible collections $\{t_0,t_1,\ldots,t_i\}$, we have \begin{align*} \sum_{t_0+t_1+\cdots+t_j=m-j} &q^{(l+0(s-1))t_0} q^{(l+1(s-1))t_1} q^{(l+2(s-1))t_2} \cdots q^{(l+j(s-1))t_j} \\ &= q^{l(m-j)} \sum_{t_0+t_1+\cdots+t_j=m-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}\\ &= q^{l(m-j)} \sum_{0 \leq i_1 \leq i_2 \leq \ldots \leq i_{m-j} \leq j} q^{(s-1)(i_1+i_2+\cdots+i_{m-j})}\\ &= q^{l(m-j)} \pq{m}{j}_{q^{s-1}}\,, \end{align*} where the last equality follows from \eqref{qbinomp}. This proves the theorem. \end{proof}\end{theorem} \begin{corollary}\label{theoremrep} Let $s\in\mathbb N$. The string $H_{{\textbf{v}},{\textbf{u}}}$ may be written as \begin{multline}\label{longsum} H_{{\textbf{v}},{\textbf{u}}} = \sum_{j_1=0}^{u_2} \sum_{j_2=0}^{u_3} \cdots \sum_{j_{n-1}=0}^{u_n} \prod_{i=1}^{n-1} h^{j_1+\cdots+j_{n-1}} \Gamma_{q,s} [j_i,v_1+\cdots+v_i+(j_1+\cdots+j_{i-1})(s-1),u_{i+1}]_q \\ V^{v_1+\cdots+v_n + (j_1+\cdots+j_{n-1})(s-1)} U^{u_1+\cdots+u_n-(j_1+\cdots+j_{n-1})}. \end{multline} where \begin{equation*} \Gamma_{q,s}[j,l,m] = q^{l(m-j)} \pq{m}{j}_{q^{s-1}} \prod_{i=0}^{j-1} [l+i(s-1)]_q. \end{equation*} Hence, the numbers $S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k]$ are given by \begin{equation}\label{exp22} S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k] = h^{|{\textbf{u}}|-k}\sum_{j_1+\cdots+j_{n-1}=u_1+\cdots+u_n-k}~\prod_{i=1}^{n-1 } \Gamma_{q,s} [j_i,v_1+\cdots+v_i+(j_1+\cdots+j_{i-1})(s-1),u_{i+1}]. \end{equation} \begin{proof} Identity \eqref{longsum} is proved by repeated application of \eqref{repx} beginning from $U^{u_2}V^{v_1}$. Identity \eqref{exp22} follows by comparing the coefficient of $U^k$ in \eqref{longsum} and \eqref{wnorm}. \end{proof} \end{corollary} We note that the case $s=0, h=1, q=1$ of Corollary \ref{theoremrep} was derived by El Desouky et al.~\cite{DesCakMan} using the Leibniz formula. \begin{corollary}\label{correp} Let $s\in\mathbb N$ and set $\textbf v=\textbf u=(\underbrace{1,1,\ldots,1}_{n})$. Then the following explicit formula for $S^{\textbf v,\textbf u}_{s,h;q}[k]$ holds \begin{equation*} S^{\textbf v,\textbf u}_{s,h;q}[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)} \pq{i+(j_1+\cdots+j_{i-1})(s-1)}{j_i}_q. \end{equation*} \end{corollary} Varvak's \cite{Var} use of rook factorization to obtain an explicit formula adapts readily in the case of $S^{{\textbf{v}},{\textbf{u}}}_{s,h;q}[k]$ 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 $[n]_{q,1-s}!=[n]^{(\underline{n})}_{q,1-s}$. \begin{theorem}\label{expexp} Let $s\neq 1$. The coefficients $S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k]$ satisfy the explicit formula \begin{equation*} S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k] = \frac{h^{|{\textbf{u}}|-k}}{[k]_{q,1-s}!} \sum_{j=0}^k (-1)^{k-j} q^{\binom{k-j}{2}(1-s)} \pq{k}{j}_{q^{1-s}} \Omega^{{\textbf v},{\textbf u}}_{s;q}[j], \end{equation*} where \begin{equation*} \Omega^{{\textbf v},{\textbf u}}_{s;q}[j]=\prod_{t=1}^{n} [j-(u_1+u_2+\cdots+u_{t-1})+(v_1+v_2+\cdots+v_{t-1})/(1-s)]_{q,1-s}^{(\underline{u_t})}. \end{equation*} \begin{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 $Ut^j=h[j]_qt^{j+s-1}$. 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 \eqref{wnorm} to $t^{x(1-s)}$. After letting $t=1$ to the resulting equation and using the property $[x]^{(\underline{k})}_{q,1-s}=[1-s]_q^k [x]_{q^{1-s}}[x-1]_{q^{1-s}} \cdots[x-k+1]_{q^{1-s}}$, we obtain \begin{equation*} h^{|\textbf u|} \Omega^{{\textbf v},{\textbf u}}_{s;q}[x]= \sum_{k=u_1}^{|\textbf u|} h^{k} S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k][1-s]_q^k [x]_{q^{1-s}}[x-1]_{q^{1-s}} \cdots[x-k+1]_{q^{1-s}}. \end{equation*} 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 $p_k=\frac{1}{[k]_Q!} \Delta_Q^k P(x) \vert_{x=0}$. By the $q$-binomial theorem, $\Delta_{Q}^k = \sum_{j=0}^k (-1)^{j} Q^{\binom{j}{2}} \pq{k}{j}_{Q} E^{k-j}$. The result then follows by letting $Q=q^{1-s}$, $p_k=h^{k} S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k][1-s]_q^k$ and $P(x)=h^{|\textbf u|} \Omega^{{\textbf v},{\textbf u}}_{s;q}[x]$. \end{proof} \end{theorem} \begin{corollary}\label{secexp} Let $s\in\mathbb N \backslash \{0,1\}$. If $\textbf v=\textbf u=(\underbrace{1,1,\ldots,1}_{n})$, then the numbers $S^{\textbf v,\textbf u}_{s,h;q}[k]$ have the following explicit formula \begin{equation*} S^{\textbf v,\textbf u}_{s,h;q}[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)} \pq{k}{j}_{q^{1-s}} \prod_{t=1}^n [(j/s)+t-j-1]_{q^s}. \end{equation*} When $s=0$, \begin{equation*} S^{\textbf v,\textbf u}_{0,h;q}[k] = \frac{h^{n-k}}{[k]_q!} \sum_{j=0}^{k} (-1)^{k-j} q^{\binom{k-j}{2}} \pq{k}{j}_{q} [j]_q^n. \end{equation*} \end{corollary} Theorem \ref{expexp} can be used to derive a Dobinsky formula for the Bell numbers corresponding to $S^{\textbf v,\textbf u}_{s,h;q}[k]$. Define the generalized $q$-Bell polynomials $B^{\textbf v,\textbf u}_{s,h;q}[x]$ and generalized $q$-Bell numbers $B^{\textbf v,\textbf u}_{s,h;q}$ by \begin{equation*} B^{\textbf v,\textbf u}_{s,h;q}[x] = \sum_{k=u_1}^{|\textbf u|} S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k] x^{k}, \hspace{0.25in} B^{\textbf v,\textbf u}_{s,h;q}=B^{\textbf v,\textbf u}_{s,h;q}[1]. \end{equation*} When $s=0,h=1,q=1$ and the associated string $H_{\textbf v,\textbf u}$ outlines a staircase board, i.e., when $\textbf v=\textbf u=(\underbrace{1,1,\ldots,1}_{n})$, the numbers $B^{\textbf v,\textbf u}_{s,h;q}[x]$ and $B^{\textbf v,\textbf u}_{s,h;q}$ reduce to the usual Bell polynomial $B(n;x)$ and Bell number $B(n)$, respectively. The classical Dobinsky formula is given by \begin{equation*} B(n;x) = \frac{1}{e^x} \sum_{j=0}^{\infty} j^n \frac{x^j}{j!}. \end{equation*} The Dobinsky formula corresponding to $B^{\textbf v,\textbf u}_{s,h;q}[x]$ is as follows. \begin{corollary}\label{dobcor} Let $s\in\mathbb N \backslash \{1\}$ and $\Omega^{{\textbf v},{\textbf u}}_{s;q}[j]$ be as in Theorem \ref{expexp}. Then, \begin{equation*} B^{\textbf v,\textbf u}_{s,h;q}[x]=\left(\sum_{j=0}^{\infty} h^{|\textbf u|-j} (-1)^j q^{\binom{j}{2}(1-s)} \frac{x^{j}}{[j]_{q,1-s}!} \right)\left(\sum_{j=0}^{\infty} \Omega^{{\textbf v},{\textbf u}}_{s;q}[j] \frac{x^{j}}{h^j[j]_{q,1-s}!}\right). \end{equation*} \begin{proof} Using Theorem \ref{expexp}, the property $S^{{\textbf{v}}, {\textbf{u}}}_{s,h;q}[k]=0$ when $|\textbf u|