% Latex file % Orthogonal polynomial identities from % tensor product decompositions % J. Van der Jeugt % in Special Functions and Differential Equations % Eds. K. Srinivasa Rao, R. Jagannathan, G. Vanden Berghe % and J. Van der Jeugt, % Allied Publishers, New Delhi, 1998; pp. 171-177. % \documentstyle[11pt]{article} %\pagestyle{empty} \headheight=0mm \headsep=0mm \oddsidemargin=1mm \evensidemargin=1mm \textheight=225mm \textwidth=155mm % \newtheorem{theo}{Theorem} \newtheorem{defi}[theo]{Definition} \newtheorem{lemm}[theo]{Lemma} \newtheorem{coro}[theo]{Corollary} \newtheorem{rema}[theo]{Remark} \newtheorem{prop}[theo]{Proposition} % \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} \def\hy{\hbox{--}} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{1} % % 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\vp{\varphi} \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\Up{\Upsilon} % % some fonts : % NIEUWE FONTS : Blackboard boldface (\Bb) % \font\twelvemsbm=msbm10 at 12 true pt \font\eightmsbm=msbm8 \font\sevenmsbm=msbm7 \newfam\msbmfam \textfont\msbmfam=\twelvemsbm \scriptfont\msbmfam=\eightmsbm \scriptscriptfont\msbmfam=\sevenmsbm \def\Bb#1{{\fam\msbmfam\relax#1}} % Blackboard boldface % % the maths. symbols for natural, integer, real and complex numbers % \def\Nat{\Bb N} \def\Zah{\Bb Z} \def\Real{\Bb R} \def\Q{\Bb Q} \def\C{\Bb C} % % \begin{document} \begin{center} {\Large \bf Orthogonal polynomial identities from\\[3mm] tensor product decompositions }\\[15mm] {\bf J.\ Van der Jeugt\footnote{Research Associate of the Fund for Scientific Research -- Flanders (Belgium).} }\\[1cm] Department of Applied Mathematics and Computer Science,\\ University of Gent, Krijgslaan 281-S9, B-9000 Gent, Belgium.\\[2mm] E-mail : Joris.VanderJeugt@rug.ac.be \end{center} \vskip 15mm \noindent We consider the Lie algebra $su(1,1)$ and its positive discrete series representations. A general self-adjoint element $X$ of $su(1,1)$ then acts as a second order difference operator in a representation. For such a second order difference operator, there exist the associated orthogonal polynomials of the first kind~\cite{Berez}. In the present case, these are Laguerre or Meixner-Pollaczek polynomials. When studying the tensor product of positive discrete series representations, overlap coefficients of generalised eigenvectors of $X$ can again be expressed in terms of orthogonal polynomials; in our case Jacobi or continuous Hahn polynomials. New identities for these orthogonal polynomials, involving the Clebsch-Gordan or Racah coefficients of $su(1,1)$, can then be obtained. The idea originates in work of Granovskii and Zhedanov~\cite{GZ}. The first application to Laguerre polynomials associated with $su(1,1)$, and to Hermite polynomials associated with the oscillator algebra, was given in~\cite{V}. The general case of $su(1,1)$, and of polynomials associated with the quantum algebra $U_q(su(1,1))$, was developed in~\cite{KV1}. Further applications are related to generating functions for orthogonal polynomials~\cite{VJ}, or to bilinear generating functions~\cite{KV2}, see also Section~4. \section{The Lie algebra $su(1,1)$} \def\Zp{\Zah_+} \def\Hi{\ell^2(\Zah_+)} The Lie algebra $su(1,1)$ is generated by $H,B,C$ subject to $$ [H,B] = 2B, \qquad [H,C] = -2C, \qquad [B,C]=H. $$ There is a $\ast$-structure by $H^\ast=H$ and $B^\ast=-C$. The positive discrete series representations $\pi_k$ of $su(1,1)$ are unitary representations labelled by $k>0$. The representation space is $\Hi$ equipped with an orthonormal basis $\{ e^k_n\}_{\{ n\in\Zp\} }$, and the action is given by \beq \begin{array}{ll} \pi_k(H)\, e^k_n &= 2(k+n) \, e^k_n, \\[1mm] \pi_k(B)\, e^k_n &= \sqrt{(n+1)(2k+n)}\, e^k_{n+1},\\[1mm] \pi_k(C)\, e^k_n &= -\sqrt{n(2k+n-1)}\, e^k_{n-1}. \end{array} \label{actionsu11} \eeq The tensor product of two positive discrete series representations decomposes as \beq \pi_{k_1}\otimes \pi_{k_2} = \bigoplus_{j=0}^\infty \pi_{k_1+k_2+j}, \label{tensorproddec} \eeq and the corresponding intertwining operator can be expressed by means of the Clebsch-Gordan coefficients \beq e^{(k_1k_2)k}_n = \sum_{n_1,n_2} C^{k_1,k_2,k}_{n_1,n_2,n}\, e^{k_1}_{n_1}\otimes e^{k_2}_{n_2}. \label{defCGC} \eeq The Clebsch-Gordan coefficients are non-zero only if $n_1+n_2=n+j$, $k=k_1+k_2+j$ for $j,n_1,n_2,n\in\Zp$, and normalised by $\langle e^k_0, e^{k_1}_0\otimes e^{k_2}_j\rangle >0$. For the above results see Vilenkin and Klimyk~\cite[\S 8.7]{VileK}, or~\cite{KV1}. The Clebsch-Gordan coefficients can be expressed in terms of a terminating hypergeometric series ${}_3F_2(1)$ of unit argument, or, equivalently, in terms of a Hahn polynomial. Up to a scalar multiple, the most general self-adjoint element in $su(1,1)$ is of the form \beq X=B-C-\al H,\qquad \al\in\Real, \label{X} \eeq thus $X^*=X$. The action of $X$ on a basis vector $e^k_n$ in the representation $\pi_k$ is then given by \bea &&\pi_k(X) e^k_n = a_{n} e^k_{n+1} + b_n e^k_n + a_{n-1} e^k_{n-1} \label{actionX} \\ &&\hbox{where } a_n=\sqrt{(n+1)(2k+n)} \hbox{ and } b_n=-2\al(k+n). \nn \eea Thus $\pi_k(X)$ act as a recurrence operator, or second order difference operator. Since the coefficients $a_n$ and $b_n$ in (\ref{actionX}) satisfy $a_n>0$ and $b_n\in\Real$, there exist orthogonal polynomials associated with $\pi_k(X)$ defined by~\cite{Berez} \bea &&x p_n(x) = a_{n} p_{n+1}(x) + b_n p_n(x) + a_{n-1} p_{n-1}(x) \label{recurrence} \\ &&\hbox{with } p_{-1}(x)=0, \quad p_0(x)=1. \nn \eea Instead of $xp_n(x)$ in the lhs of (\ref{recurrence}) it is sometimes more convenient to write a linear combination $(ux+v)p_n(x)$; this is a trivial extension. \section{Laguerre and Jacobi polynomials} In this section, we choose the parameter $\al$ in (\ref{X}) as $\al=1$. Consider the classical Laguerre polynomials \beq L^{(\al)}_n(x)={(\al+1)_n \over n!}{\ }_1F_1\left({-n\atop \al+1}; x\right), \eeq with the usual notation for Pochhammer symbols and hypergeometric series, and let for $k>0$, \beq l^{(k)}_n(x) = \sqrt{ n!\over (2k)_n}\, L^{(2k-1)}_n(x). \eeq {}From the three-term recurrence relation for Laguerre polynomials, it follows that the normalised Laguerre polynomials $l^{(k)}_n(x)$ satisfy \beq -x l^{(k)}_n(x) = a_{n} l^{(k)}_{n+1}(x) -2(k+n) l^{(k)}_n(x) + a_{n-1} l^{(k)}_{n-1}(x). \eeq If we denote the orthogonality measure of the normalised Laguerre polynomials $l^{(k)}_n(x)$ by $d\mu_k(x)$, comparing this with (\ref{actionX}) implies~: \begin{prop} $\La : \Hi \rightarrow L^2(\Real,d\mu_k(x))$, mapping $e_n^k$ onto $l^{(k)}_n(x)$, is a unitary mapping intertwining $\pi_k(X)$ acting in $\Hi$ with $M_{-x}$ on $L^2(\Real,d\mu_k(x))$. \end{prop} Herein, $M_g$ denotes multiplication by the function $g$, so $M_g f(x) = g(x)f(x)$. The proposition states that $v^k(x)=\sum_{n=0}^\infty l^{(k)}_n(x) e^k_n$ is a generalised eigenvector for $\pi_k(X)$ for the eigenvalue $-x$. Interesting applications follow from studying the action of $X$ in the tensor product representation $\pi_{k_1}\otimes \pi_{k_2}$. In the tensor product, the action of every element of the Lie algebra, and in particular of $X$, is given by the trivial comultiplication $\De(X)=1\otimes X + X\otimes 1$. Then, we have, \begin{prop} $\Up\colon \Hi\otimes\Hi \to L^2(\Real^2,d\mu_{k_1}(x_1)d\mu_{k_2}(x_2))$, defined by $e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \mapsto l_{n_1}^{(k_1)}(x_1) l_{n_2}^{(k_2)}(x_2)$ is a unitary mapping intertwining $\pi_{k_1}\otimes \pi_{k_2}(\De(X))$ with $M_{-(x_1+x_2)}$. \end{prop} The sum $x_1+x_2$ follows from the trivial comultiplication. A crucial property is that under the mapping of Proposition~2 the ``coupled basis vectors'' $e^{(k_1 k_2)k}_n$ are mapped into a polynomial that factorises. For a proof, see~\cite{KV1}. \begin{prop} For $k=k_1+k_2+j$, $j\in\Zp$, we have \beq \Up e^{(k_1 k_2)k}_n = l^{(k)}_n(x_1+x_2) \, S_j^{(k_1,k_2)}(x_1,x_2), \eeq where \bea S^{(k_1,k_2)}_j(x_1,x_2)&=&(-1)^j \sqrt{ {{ j!}\over{(2k_1)_j(2k_2)_j (2k_1+2k_2+j-1)_j}} }\nn\\ &&\times (x_1+x_2)^j P^{(2k_1-1,2k_2-1)}_j\Bigl( {{x_2-x_1}\over{x_1+x_2}}\bigr). \label{SexplicitL} \eea \end{prop} Herein, $P^{(a,b)}_j(z)=(a+1)_j/j!{\ }_2F_1(-j,j+a+b+1;a+1;(1-z)/2)$ is a classical Jacobi polynomial. A consequence of the above formula is the identity \beq \sum_{n_1,n_2} C^{k_1,k_2,k}_{n_1,n_2,n}\ l^{(k_1)}_{n_1}(x_1) l^{(k_2)}_{n_2}(x_2) = l^{(k)}_n(x_1+x_2)\, S^{(k_1,k_2)}_j(x_1,x_2). \label{cgcident} \eeq Using the explicit expression for $su(1,1)$ Clebsch-Gordan coefficients, and relabelling the parameters, this gives rise to the following identity for Laguerre polynomials~\cite{V}: \begin{theo} \bea &&\sum_{l=0}^{n+j} Q_j(l;a,b,j+n) L_l^{(a)}(x) L_{n+j-l}^{(b)}(y) \nn\\ &&\qquad = {(-1)^j n!j!\over (a+1)_j(n+j)!} (x+y)^j L_n^{(a+b+2j+1)}(x+y) P_j^{(a,b)}({y-x\over y+x}); \label{res1} \eea \end{theo} Herein, we have used the common notation for the Hahn polynomials defined by \beq Q_n(x;a,b,N) = {}_3F_2\left( {{-n,n+a+b+1,-x}\atop{a+1,\ -N}};1\right) \eeq for $N\in\Zp$, $0\leq n\leq N$. For $j=0$, the identity (\ref{res1}) reduces to the classical addition formula for Laguerre polynomials. In a similar way, one can consider the tensor product of three representations $\pi_{k_1}\otimes \pi_{k_2} \otimes \pi_{k_3}$. The Racah coefficients of $su(1,1)$ are transformation coefficients between two orthonormal bases in the tensor product space~: \beq e^{((k_1k_2)k_{12}k_3)k}_n = \sum_{k_{23}} U^{k_1,k_2,k_{12}}_{k_3,k,k_{23}} \ e^{(k_1(k_2k_3)k_{23})k}_n . \eeq Herein, \beas &&k_{12}=k_1+k_2+j_{12}, \qquad k_{23}=k_2+k_3+j_{23}, \\ &&k=k_{12}+k_3+j =k_1+k_{23}+j',\qquad j_{12},j,j_{23},j' \in \Zp,\ \hbox{and}\ j_{12}+j=j_{23}+j'. \eeas Thus the above sum is finite. The identity that can now be obtained is the following~: \beq S_{j_{12}}^{(k_1,k_2)}(x_1,x_2) S_{j}^{(k_{12},k_3)}(x_1+x_2,x_3) = \sum_{j_{23}=0}^{j_{12}+j} U^{k_1,k_2,k_{12}}_{k_3,k,k_{23}}\ S_{j_{23}}^{(k_2,k_3)}(x_2,x_3) S_{j_{12}+j-j_{23}}^{(k_1,k_{23})}(x_1,x_2+x_3) . \eeq Using the explicit expression~(\ref{SexplicitL} for the $S$-function, a convolution identity for Jacobi polynomials involving Racah polynomials is obtained~\cite{V,KV1}. So far, we have considered only the case $\al=1$ in (\ref{X}). In the following section the case $|\al|<1$ is discussed; the case $|\al|>1$ is similar~\cite{KV1}. \section{Meixner-Pollaczek and continuous Hahn polynomials} The Meixner-Polla\-czek polynomials are defined by \beq P_n^{(\la)}(x;\phi) = {{(2\la)_n}\over{n!}} e^{in\phi} \, {}_2F_1\left( {{-n,\la+ix}\atop{2\la}};1-e^{-2i\phi}\right). \eeq For $\la>0$ and $0<\phi<\pi$ these are orthogonal polynomials with respect to a positive measure on $\Real$, see \cite{KoekS}. Let $k>0$; then the orthonormal Meixner-Pollaczek polynomials \beq p_n(x) = p_n^{(k)}(x;\phi) = \sqrt{ {{n!}\over{\Gamma(n+2k)}}} P_n^{(k)}(x;\phi) \eeq satisfy the three-term recurrence relation \beq 2x\, \sin\phi\, p^{(k)}_n(x) = a_n\, p^{(k)}_{n+1}(x) - 2(n+\la)\cos\phi\, p^{(k)}_n(x) + a_{n-1}\, p^{(k)}_{n-1}(x). \eeq Comparing with (\ref{actionX}) leads to identifying $\al$ with $\cos\phi$, and with \beq X_\phi = B-C-\cos\ph H, \eeq we have the following (cfr.\ Proposition~1)~: \begin{prop} Let $d\mu_{k,\phi}(x)$ denote the orthogonality measure of the orthonormal Meixner-Pollaczek polynomials $p_n^{(k)}(x;\phi)$. Then $\Lambda\colon \Hi \to L^2(\Real,d\mu_{k,\phi}(x))$, $e^k_n\mapsto p_n^{(k)}(x;\phi)$, is a unitary mapping intertwining $\pi_k(X_\phi)$ acting in $\Hi$ with $M_{2x\sin\phi}$ on $L^2(\Real,d\mu_{k,\phi}(x))$. \end{prop} Similarly, in the tensor product $\pi_{k_1}\otimes\pi_{k_2}$, \begin{prop} $\Up\colon \Hi\otimes\Hi \to L^2(\Real^2,d\mu_{k_1,\phi}(x_1)d\mu_{k_2,\phi}(x_2))$, defined by $e^{k_1}_{n_1}\otimes e^{k_2}_{n_2} \mapsto p_{n_1}^{(k_1)}(x_1;\phi)p_{n_2}^{(k_2)}(x_2;\phi)$ is a unitary mapping intertwining $\pi_{k_1}\otimes \pi_{k_2}(\De(X_\phi))$ with $M_{2(x_1+x_2)\sin\phi}$. \end{prop} The $S$-function of Proposition~3 now becomes a continuous Hahn polynomial, see~\cite{KV1}~: \begin{prop} For $k=k_1+k_2+j$, $j\in\Zp$, we have \beq \Up e^{(k_1 k_2)k}_n = p^{(k)}_n(x_1+x_2;\phi) \, S_j^{(k_1,k_2)}(x_1,x_2;\phi), \eeq where \bea S^{(k_1,k_2)}_j(x_1,x_2;\phi)&=& (-2\sin\phi)^j \sqrt{ {{j!\, (2j+2k_1+2k_2-1)\Gamma(j+2k_1+2k_2-1)}\over {\Gamma(2k_1+j) \Gamma(2k_2+j)}} } \nn\\ &&\times\ p_j(x_1;k_1, k_2-i(x_1+x_2),k_1,k_2+i(x_1+x_2)). \label{SexplicitMP} \eea \end{prop} Herein, $p_j$ is a continuous Hahn polynomial~\cite{KoekS} \beq p_n(x;a,b,c,d)= i^n {{(a+c)_n(a+d)_n}\over{n!}}\, {}_3F_2\left( {{-n,n+a+b+c+d-1,a+ix}\atop{a+c,\ a+d}};1\right). \eeq The identity (\ref{cgcident}) now gives rise to the following~\cite{KV1} \begin{theo} With the notation for continuous Hahn, Meixner-Pollaczek and Hahn polynomials as above, the following convolution formula holds: \bea &&{{n+j}\choose n} \sum_{l=0}^{n+j} Q_j(l;2k_1-1,2k_2-1,n+j)\, P_l^{(k_1)}(x_1;\phi) \, P_{n+j-l}^{(k_2)}(x_2;\phi) = \label{convol}\\ &&\qquad {{(-2\sin\phi)^j}\over{(2k_1)_j}} \, P_n^{(k_1+k_2+j)}(x_1+x_2;\phi)\, p_j(x_1;k_1, k_2-i(x_1+x_2),k_1,k_2+i(x_1+x_2)). \nn \eea \label{res2} \end{theo} The case $j=0$ gives back the classical convolution identity for the Meixner-Pollaczek polynomials. Formula~(\ref{convol}) also has an interpretation as a connection coefficient formula between orthogonal polynomials in two variables with the same orthogonality measure. \section{Bilinear generating functions} Consider again the polynomials discussed in Section~2. From Proposition~1 it follows that the elements $H$, $B$ and $C$ of $su(1,1)$ have a realisation in $L^2(\Real,d\mu_k(x))$ with standard action~(\ref{actionsu11}) on the basis $l^{(k)}_n(x)$. For fixed $y\in\Real$ and $-1