% Latex file % The $q$-boson operator algebra and $q$-Hermite polynomials % J. Van der Jeugt % Lett. Math. Phys. 24 (1992), 267-274. % \documentclass[12pt]{article} \usepackage{latexsym,amssymb} % % definitions concerning automatic numbering of theorems, lemmas, etc. % \newtheorem{theo}{Theorem}[section] \newtheorem{defi}[theo]{Definition} \newtheorem{prop}[theo]{Proposition} % % \setlength{\topmargin}{0cm} \setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm} \setlength{\textheight}{21.5cm} \setlength{\textwidth}{15cm} % % The Greek symbols defined by the first two letters of their name % \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\ep{\epsilon} \def\vep{\varepsilon} \def\ze{\zeta} \def\et{\eta} \def\th{\theta} \def\io{\iota} \def\ka{\kappa} \def\la{\lambda} \def\rh{\rho} \def\si{\sigma} \def\ta{\tau} \def\ph{\phi} \def\ch{\raise 2pt\hbox{$\chi$}} % raise this a bit \def\ps{\psi} \def\om{\omega} \def\Ga{\Gamma} \def\De{\Delta} \def\Th{\Theta} \def\La{\Lambda} \def\Si{\Sigma} \def\Ph{\Phi} \def\Ps{\Psi} \def\Om{\Omega} % \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\beas{\begin{eqnarray*}} \def\eeas{\end{eqnarray*}} \def\nn{\nonumber} % % the end-of-proof box \def\mybox{\hfill\llap{$\Box$}} % \def\Nat{\mathbb{N}} \def\Zah{\mathbb{Z}} \def\Real{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} % % % special definitions for the paper % \def\ad{a^\dagger} \def\bd{b^\dagger} \def\ds{\displaystyle} \def\D{{\cal D}} \def\q{\tilde{q}} \def\nn{\nonumber} % % automatic numbering of equations: % \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} % \begin{document} \addtolength{\baselineskip}{2mm} \begin{center} {\Large \bf The $q$-boson operator algebra and $q$-Hermite polynomials}\\[2cm] J.~Van der Jeugt\footnote{Research Associate N.F.W.O. (National Fund for Scientific Research of Belgium)} \\[8mm] Laboratorium voor Numerieke Wiskunde en Informatica, Universiteit Gent,\\ Krijgslaan 281--S9, B9000 Gent, Belgium \end{center} \vskip 3cm \begin{abstract} The $q$-boson algebra is defined as an associative algebra with generators and relations. Some examples are given, and then the $q$-boson algebra is extended such that roots of the ``diagonal generators'' are also defined. It is shown that a family of transformations exist mapping one set of standard generators of the $q$-boson algebra to another set of standard generators. Using such a transformation, one obtains expressions for $q$-bosons for which the $k$th $q$-boson state is expressed in terms of a $q$-Hermite polynomial $p_k(x;q)$ which reduces to the ordinary Hermite polynomial of degree $k$ when $q=1$. \end{abstract} \newpage \section{Introduction and definition} The $q$-generalisation of boson creation and annihilation operators was introduced by Biedenharn~[1] and Macfarlane~[2], who showed that Schwinger's boson realisation of $su(2)$ could be modified in order to obtain a realisation of the quantum enveloping algebra $su_q(2)$. In this Letter, the $q$-boson operator algebra is defined as an associative algebra with generators and relations, some examples of the algebra and of representations are given, and a transformation preserving the $q$-boson structure is presented. For a particular transformation one obtains an interesting $q$-boson algebra in which the $q$-boson creation and annihilation operators $\bd$ and $b$ are expressed in terms of a coordinate $x$ and the $q$-derivative $D_{q,x}$. In this case, the $k$th excited $q$-boson state $|k\rangle$ can be written by means of a $q$-polynomial $p_k(x;q)$. These polynomials reduce to the ordinary Hermite polynomials when $q=1$, and they are related to a variety of definitions for the $q$-Hermite polynomials and also to the continuous $q$-ultraspherical polynomials. In papers on quantum groups and $q$-bosons, it is common to use ${q^n-q^{-n} \over q-q^{-1}}$ as the definition for a $q$-number. In the literature on $q$-special functions~[3], however, it is common to use the quantity ${1-q^{n} \over 1-q}$. Wishing to keep our $q$-boson operators equivalent with those defined elsewhere~[1,2], we have chosen for $[n]={q^n-q^{-n} \over q-q^{-1}}$. Surprisingly, it turns out that the $q$-special functions related to these $q$-boson operators, appearing in Section~4, are nevertheless the classical ones~[3, and references therein] in which the quantity ${1-\q^{n} \over 1-\q}$ with $\q=q^{-2}$ plays a role. Let $q$ be a fixed complex number, $q\not=\pm 1$. \begin{defi} The $q$-boson algebra $B_q$ is the associative algebra over $\C$ with standard generators $a, \ad, t, t^{-1}$ and relations \beq \begin{array}{ll} \rm (a) & tt^{-1}=t^{-1}t=1 , \\ \rm (b) & ta-q^{-1}at=0 , \\ \rm (c) & t\ad - q \ad t =0 , \\ \rm (d) & a\ad - q^{-1} \ad a = t , \\ \rm (e) & a\ad - q \ad a = t^{-1} . \end{array} \label{def} \eeq \end{defi} Note that relations (d) and (e) are equivalent to \beq \begin{array}{ll} \rm (d') & a\ad = (qt-q^{-1}t^{-1})/(q-q^{-1}) , \\ \rm (e') & \ad a = (t-t^{-1})/(q-q^{-1}) . \end{array} \label{altdef} \eeq In Macfarlane~[2] and other papers~[4], $t$ is written as $q^N$ with $N$ a ``number operator''; then some of the relations in (\ref{def}) can be appropriately rewritten in terms of $N$. \begin{defi} The Fock space $F_q$ is a $B_q$ module with basis vectors $v_n$ ($n\in\Nat$), and action given by \beq \begin{array}{l} t^{\pm1} v_n = q^{\pm n} v_n ,\\ \ad v_n = \sqrt{[n+1]} v_{n+1} , \\ a v_n = \sqrt{[n]} v_{n-1} . \end{array} \eeq \end{defi} Herein, $[r]$ stands for the $q$-number $(q^r-q^{-r})/(q-q^{-1})$. The Fock space $F_q$ becomes an inner product space by \beq \langle v_m , v_n \rangle = \de_{m,n} . \eeq In fact, it is a Hilbert space, and the representatives of $a$ and $\ad$ are each others adjoint, whereas those of $t$ and $t^{-1}$ are self-adjoint operators. \begin{defi} A lowest weight module $V_\la$ is a cyclic $B_q$ module generated by a vector $v$ satisfying $av=0$ and $tv=q^\la v$. \end{defi} The vector $v$ is called the lowest weight vector, or the vacuum vector. From the relations (\ref{def}) one can easily deduce that a basis for $V_\la$ is given by $\{ (\ad)^n v | n\in\Nat\}$, and that \beq \begin{array}{l} t (\ad)^n v = q^{\la+n} (\ad)^n v . \\ a (\ad)^n v = [n+\la] (\ad)^{n-1} v . \end{array} \label{low} \eeq Thus if $q$ is not a root of unity (which shall be assumed to be the case throughout this Letter), then the lowest weight modules $V_\la$ are irreducible $B_q$ modules if $\la\not\in\{-1,-2,-3,\ldots\}$. \setcounter{equation}{0} \section{Examples and extensions} The two-dimensional quantum plane $A_q$ is defined as follows~[5]. \begin{defi} $A_q$ is the associative algebra over $\C$ with generators $x,y,x^{-1},y^{-1}$ and relations \beq \begin{array}{ll} \rm (a) & x x^{-1} = x^{-1}x = y y^{-1} = y^{-1} y = 1 , \\ \rm (b) & yx = q xy . \end{array} \eeq \end{defi} From these relations, one easily obtains the other commutation relations among the generators, such as $y^{-1}x=q^{-1}xy^{-1}$. \begin{prop} Let $x,x^{-1},y,y^{-1}$ be the generators of the quantum plane. Then $\ad = x$, $a=x^{-1}(y-y^{-1})/(q-q^{-1})$, $t=y$ and $t^{-1}=y^{-1}$ satisfy the defining relations (\ref{def}) of the $q$-boson operator algebra. \end{prop} It is a straightforward exercise to verify that with the given expressions all relations are indeed satisfied. Another example of the $q$-boson operator algebra, which is well-known~[6], is the following. Let $N=x\partial_x$; $x$ and $N$ are supposed to be acting on differentiable functions of $x$. Then $x, x^{-1}, y=q^N, y^{-1}=q^{-N}$ satisfy the defining relations of the quantum plane generators. Hence $\ad=x$, $a=x^{-1}(q^N-q^{-N})/(q-q^{-1})=D_{q,x}$, $t=q^N$ and $t^{-1}=q^{-N}$ satisfy the defining relations of the $q$-boson algebra. Herein, $D_{q,x}$ is the $q$-differential operator with respect to $x$. Note that the Fock space is $\C[x]$ with basis $v_n=x^n/\sqrt{[n]!}$, where $[n]!=[n][n-1]\cdots[1]$. In the literature, $q$-boson operator algebras are usually defined together with an action on representations in which $t$ and $t^{-1}$ are diagonal operators (such as the Fock space, or lowest weight modules). When acting on such representations, operators such as $t^{1/2}$ or $(t-t^{-1})^{-1}$ are well-defined, although strictly speaking they do not belong to the associative algebra generated by $a,\ad,t, t^{-1}$. Therefore, it is sometimes useful to work in an extended $q$-boson algebra where such expressions are defined. For this purpose, let $\C(t)$ be the field of rational expressions in $t$ (by definition, $t^{-1}=1/t$ and $tt^{-1}=t^{-1}t=1$ is automatically satisfied). Let $\C_t$ be the algebraic extension of $\C(t)$ (generally speaking, its elements are roots of rational expressions in $t$). \begin{defi} The extended $q$-boson operator algebra $B'_q$ is the associative algebra over $\C$ generated by $a,\ad$ and the elements $p(t)$ of $\C_t$ subject to the relations relations \beq \begin{array}{ll} \rm (a) & p(t)p'(t)=p(t)\cdot p'(t) \quad\hbox{(the rhs denotes the product in $\C_t$),} \\ \rm (b) & p(t)a-ap(q^{-1}t)=0 , \\ \rm (c) & p(t)\ad - \ad p(qt) =0 , \qquad \forall p(t),p'(t)\in \C_t\\ \rm (d) & a\ad - q^{-1} \ad a = t , \\ \rm (e) & a\ad - q \ad a = t^{-1} . \end{array} \eeq \end{defi} The $q$-boson operator algebra $B_q$ is naturally embedded in $B'_q$, and every $(t,t^{-1})$-diagonal $B_q$ module $V$ can also be regarded as a $B'_q$ module; note however, that the action of elements of $\C_t$ which lead to ``division by zero'' is not defined. \setcounter{equation}{0} \section{Transformations for $q$-bosons} It is the purpose of this section to find certain transformations of the standard generators $a,\ad,t,t^{-1}$ that satisfy again the relations of a $q$-boson operator algebra. The transformations we intend to study are ``linear'' in $a$ and $\ad$, in the following sense~: \beq \begin{array}{l} b = \al(t)a+\be(t)\ad , \\ \bd= \ga(t)a + \de(t) \ad , \end{array} \label{defb} \eeq where $\al(t), \be(t), \ga(t)$ and $\de(t)$ are elements of $\C_t$ (roots of rational forms in $t$). With (\ref{defb}), we define $s=b\bd-q^{-1}\bd b$ and $s^{-1}=b\bd - q \bd b$. The conditions to be satisfied are then~: $ss^{-1}=1$, $s^{-1}s=1$, $sb=q^{-1}bs$ and $s\bd=q\bd s$. A careful analysis of these conditions shows that $\al, \be, \ga$ and $\de$ have to satisfy~: \beq \begin{array}{l} \al(t)/\al(qt) = q^{-1} \ga(t)/\ga(qt) , \\ \be(t)/\be(qt) = q^{-1} \de(t)/\de(qt) , \\ \al(t)\de(t) = -1/(qt-q^{-1}t^{-1}) , \\ \be(t)\ga(t) = 1/(qt-q^{-1}t^{-1}) . \end{array} \eeq This implies the following result~: \begin{prop} The most general transformation of the $q$-boson of the form (\ref{defb}) is given by \beq \begin{array}{l} \ds b=\ep t \ga(t) a + {\ep'\over t-t^{-1}}\ad {1\over\ga(t)} , \\[5mm] \ds \bd = \ep'\ga(t) a - {\ep\over t-t^{-1}}\ad {t^{-1}\over\ga(t)} , \end{array} \label{sol} \eeq where $\ep^2=\ep'^2=1$ and $\ga(t)\in\C_t$. The elements $(b,\bd,s,s^{-1})$, where $s=b\bd-q^{-1}\bd b$ and $s^{-1}=b\bd - q \bd b$, satisfy the defining relations of the $q$-boson operator algebra (\ref{def} a--e). \end{prop} Note that (\ref{sol}) has no ``classical'' counterpart in the limit $q\rightarrow 1$. The realisation of (\ref{sol}) has some interesting consequences. For instance, having one $q$-boson with generators $a,\ad,t,t^{-1}$ and a corresponding Fock space, one can study the Fock space of the transformed $q$-boson with generators $b,\bd,s,s^{-1}$. This second Fock space is usually very different from the first one. For example, let $(a,\ad,t,t^{-1}) = ( D_{q,x}, x, q^N, q^{-N})$. Choosing $\ga(q^N)=q^{-N/2}$ and $\ep=\ep'=1$, one has \beq \begin{array}{l} \ds b=q^{N/2}D_{q,x} + {1\over q^N-q^{-N}} x q^{N/2} ,\\[5mm] \ds \bd=q^{-N/2}D_{q,x} - {1\over q^N-q^{-N}} x q^{-N/2} , \end{array} \eeq Let the vacuum state $v_0$ for the new Fock space have coefficients $a_k$ in the old Fock space, i.e. \beq v_0=\sum_{k=0}^\infty a_k x^k . \eeq The condition $bv_0=0$ leads to $a_{2k+1}=0$ and $a_{2k} = (-1)^k a_0 / (\sqrt{q}r)^k [2k]! $, with $r= q-q^{-1}$. Thus one finds \begin{eqnarray} v_0 &=& a_0 \sum_{k=0}^\infty {(-x^2/\sqrt{q}r)^k \over [2k]!} \nn\\ &=&a_0 \cos_q ({q^{-1/4}x \over \sqrt{q-q^{-1}}}) . \end{eqnarray} Herein, $\cos_q$ is the $q$-generalised cosine function, studied by Jackson~[7,8]. \setcounter{equation}{0} \section{$q$-Hermite polynomials as boson states} Consider again the case $(a,\ad,t,t^{-1})=(D_{q,x}\equiv D,x, q^{N},q^{-N})$ and the transformation~(\ref{sol}). Choosing $\ep=-1$ and $\ep'=1$, and \beq \ga(q^N)= {c q^{-N/2} \over q^{N+1}-q^{-N-1} } , \eeq where $c$ is to be determined later, one finds the following expressions for $b$ and $\bd$~: \begin{eqnarray} b&=& {x\over c} q^{N/2} - {q^{N/2}\over q^{N+1}-q^{-N-1}} c D , \label{b}\\ \bd&=& {x\over c} q^{-N/2} + {q^{-N/2}\over q^{N+1}-q^{-N-1}} c D .\label{bd} \end{eqnarray} The vacuum state $v_0(x)=\sum_k a_k x^k$ is found by requiring $b v_0(x)=0$. This leads to $a_{2k+1}=0$ and $a_{2k}=a_0(r/c^2q)^j$, with again $r=q-q^{-1}$. Thus $v_0(x)=a_0 \sum_{k=0}^\infty (rx^2/qc^2)^j$, which can formally be rewritten as \beq v_0(x) = {a_0 \over 1 - rx^2/qc^2 } . \eeq It can be verified that $s^{-1}v_0(x)=v_0(x)$, thus also $sv_0=v_0$. One can then define $|k\rangle = (\bd)^k v_0(x)$. Using~(\ref{low}), the following relations are obtained~: \begin{eqnarray} b|k\rangle & = &[k] |k-1\rangle ,\label{bk}\\ \bd|k\rangle &=& |k+1\rangle ,\label{bdk}\\ s|k\rangle &=& q^k |k\rangle .\label{sk} \end{eqnarray} The following proposition shows that the $q$-boson states $|k\rangle$ can be written in terms of a polynomial of degree $k$. \begin{prop} The states $|k\rangle$ have the form \beq |k\rangle = q^{-k(N+1)/2+k^2/4} p_k(x;q) v_0(x) , \label{state} \eeq where $p_k(x;q)$ is a polynomial of degree $k$ in $x$ with coefficients in $\C[q]$. \end{prop} \noindent {\sl Proof.} The statement is true when $k=0$. Assume that it is valid for all values up to $k$; we shall prove that is then also valid for $k+1$. Using~(\ref{bk}) and~(\ref{b}), one finds the following expression~: \beq {q^{N/2}\over q^{N+1}-q^{-N-1}}cD|k\rangle = {x\over c}q^{N/2} |k\rangle - [k] |k-1\rangle . \eeq This can now be used in the calculation of $\bd|k\rangle$, and we find~: \begin{eqnarray} |k+1\rangle & = & \bd |k\rangle \nn \\ &=& {x\over c} q^{-N/2} |k\rangle + q^{-N} {q^{N/2}\over q^{N+1}-q^{-N-1}} cD|k\rangle \nn\\ &=& ({x\over c}q^{-N/2}+q^{-N}{x\over c}q^{N/2}) |k\rangle -q^{-N}[k] |k-1\rangle . \label{rhs} \end{eqnarray} Using the induction argument, i.e.~expression~(\ref{state}), for $|k\rangle$ and $|k-1\rangle$, the right hand side of~(\ref{rhs}) reduces to~: \beq q^{-(k+1)(N+1)/2+(k+1)^2/4} \left( {q^{-1/4}(1+q)\over c} x p_k(x;q) - q^{-k+1}[k]p_{k-1}(x;q)\right) v_0(x). \eeq Thus $|k+1\rangle = q^{-(k+1)(N+1)/2+(k+1)^2/4} p_{k+1}(x;q) v_0(x)$, where \beq p_{k+1}(x;q) = {1+q \over cq^{1/4}} x p_k(x;q) - {1-q^{-2k}\over 1-q^{-2}} p_{k-1}(x;q) . \label{recursion} \eeq From this last equation the statement follows. \mybox Equation~(\ref{recursion}), together with $p_{-1}(x;q)=0$ and $p_0(x;q)=1$, can be used as a recursion relation to calculate the $p_k(x;q)$. Choosing $c=q^{-1/4}(1+q)$, this relation becomes $p_{k+1}=xp_k-kp_{k-1}$ in the limit $q\rightarrow 1$, which is the recursion relation defining Hermite polynomials. So, in the present case, the $q$-boson states are written in terms of $q$-generalised Hermite polynomials satisfying~(\ref{recursion}). Hermite polynomials have been $q$-generalised in different ways, and there exists a huge amount of literature on the subject of $q$-special functions~[3]. The polynomials~(\ref{recursion}) can be related to many of the known orthogonal $q$-polynomials. Choosing for example $c=q^{-1/4}(1+q)(1-q^{-2})^{1/2}$, one obtains \beq p_{k+1}(x;q)=(1-q^{-2})^{-1/2} x p_k(x;q) - {1-q^{-2k} \over 1-q^{-2}} p_{k-1}(x;q) , \eeq or, with $\q=q^{-2}$ and $P_k(x;\q)=p_k(x;q)$~: \beq P_{k+1}(x;\q)=(1-\q)^{-1/2} x P_k(x;\q) - {1-\q^k \over 1-\q} P_{k-1} (x;\q). \eeq From now onwards, we shall continue to work with $\q$; however, for sake of notation we shall denote $\q$ by $q$ in what follows. The polynomials $P_k(x;q)$ are related to the $q$-Hermite polynomials $R_k(x;q)$ of Allaway~[8] by \beq R_k(x;q)=(1-q)^{k/2} P_k(x;q) . \eeq These polynomials satisfy the recursion relation \beq R_{k+1}(x;q)=xR_k(x;q)-(1-q^k)R_{k-1}(x;q), \label{R} \eeq with $R_{-1}=0$ and $R_0=1$, and have been studied in detail~[8,9]. They form a special case of the polynomials $R_k(x;q;a,b,c)$ introduced by Al-Salam and Chihara~[9], and studied in a rather general context by Dehesa~[10]. The $R_k(x)\equiv R_k(x;q;a,b,c)$ are defined by \beq R_{k+1}(x)=(x-aq^k)R_k(x)-(c-bq^{k-1})(1-q^k)R_{k-1}(x) , \eeq with $R_{-1}(x)=0$ and $R_0(x)=1$. This coincides with~(\ref{R}) for $(a,b,c)=(0,0,1)$. Note that the so-called discrete $q$-Hermite polynomials $H_k(x;q)$, as defined in~[3, p.~193], also fall in this class since $H_k(x;q)=R_k(x;q;0,-1,0)$. There is also a relation with the continuous $q$-ultraspherical polynomials~[3, p.~26], defined by \beq C_k(x;\be|q)=\sum_{j=0}^k {(\be;q)_j (\be;q)_{k-j} \over (q;q)_j (q;q)_{k-j} } e^{i(k-2j)\th} , \eeq where $x=\cos\th$ and \beq (a;q)_n = \cases {1 , & if $n=0$;\cr (1-a)(1-aq)\cdots(1-aq^{n-1}),& if $n=1,2,\ldots$\cr} \eeq The little $c_k$, \beq c_k(x;\be|q)={(q;q)_k\over (\be^2;q)_k} C_k(x;\be|q) , \eeq introduced in~[3, p.~188], satisfy \beq (1-\be^2q^k)c_{k+1}(x;\be|q)=(1-\be q^k)2xc_k(x;\be|q) -(1-q^k)c_{k-1}(x;\be|q), \eeq with $c_{-1}=0$ and $c_0=1$. Thus, the polynomials appearing here can also be identified as follows~: \beq P_k(x;q)=(1-q)^{-k/2} c_k(x/2;0|q). \eeq Finally, it should be noted that in a recent publication~[11], a different kind of $q$-generalised Hermite polynomials $h_n(w;q)$ were also related to the oscillator quantum group. These polynomials are different from~(\ref{recursion}) (e.g.~they satisfy $h_n(w;q=1)=(1+w)^n$); moreover, the $q$-boson realisation given in~[11] is also quite different from~(\ref{b}--\ref{bd}). \section*{Acknowledgement} The author would like to thank the referee for pointing out references on $q$-special functions. \section*{References} \begin{enumerate} \item Biedenharn LC, {\it J.\ Phys.\ A : Math.\ Gen.} {\bf 22}, L873 (1989) \item Macfarlane AJ, {\it J.\ Phys.\ A : Math.\ Gen.} {\bf 22}, 4581 (1989) \item Gasper G and Rahman M, {\it Basic Hypergeometric Series,} Encyclopedia of Mathematics and its Applications, vol.~{\bf 35}, Cambridge University Press, Cambridge (1990) \item Kulish PP and Damashinsky EV, {\it J.\ Phys.\ A : Math.\ Gen.} {\bf 23}, L415 (1990) \item Manun Yu I, {\it Quantum groups and non-commutative geometry,} Centre de Recherches Math\'ematiques, Montr\'eal, 1988 \item Hong Yan, {\it J.\ Phys.\ A : Math.\ Gen.} {\bf 23}, L1155 (1990) \item Exton H, {\it $q$-Hypergeometric functions and applications,} Ellis Horton Ltd., Chichester (1983) \item Jackson FH, {\it Proc.\ Edin.\ Math.\ Soc.} {\bf 22}, 28 (1904) \item Allaway WR, {\it Canad.\ J.\ Math.} {\bf 32}, 686 (1980) \item Al-Salam WA and Chihara TS, {\it SIAM J.\ Math.\ Anal.} {\bf 7}, 16 (1976) \item Dehesa JS, {\it J.\ Comp.\ Appl.\ Math.} {\bf 5}. 37 (1979) \item Floreanini R and Vinet L, {\it Lett.\ Math.\ Phys.} {\bf 22}, 45 (1991) \end{enumerate} \end{document}