Contemporary MathematicsVolume 00, 0000Models of q -Algebra Representations:Matrix Elements of Uq(su2)E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEAbstract. This paper continues a study of one and two variable functionspace models of irreducible representations of q-analogs of Lie envelopingalgebras, motivated by recurrence relations satis�ed by q-hypergeometricfunctions. Here we consider the quantum algebra Uq(su2). We show thatvarious q-analogs of the exponential function can be used to mimic theexponential mapping from a Lie algebra to its Lie group and we computethe corresponding matrix elements of the \group operators" on these rep-resentation spaces. This \local" approach applies to more general familiesof special functions, e.g., with complex arguments and parameters, thandoes the quantum group approach. We show that the matrix elementsthemselves transform irreducibly under the action of the quantum algebra.We �nd an alternate and simpler derivation of a q-analog, due to Groza,Kachurik and Klimyk, of the Burchnall-Chaundy formula for the product oftwo hypergeometric functions 2F1. It is interpreted here as the expansionof the matrix elements of a \group operator" (via the exponential mapping)in a tensor product basis in terms of the matrix elements in a reduced basis.1. IntroductionThis paper continues the study of function space models of irreducible rep-resentations of q-algebras [1, 2]. These algebras and models are motivated byrecurrence relations satis�ed by q-hypergeometric functions [3] and our treat-ment is an alternative to the theory of quantum groups. Here, we consider1991Mathematics Subject Classi�cation. Primary 33D55, 33D45; Secondary 17B37, 81R50.Key words and phrases. basic hypergeometric functions, q-algebras, quantum groups,Clebsch-Gordan series.This paper is in �nal form and no version of it will be submitted for publication elsewhere.The second author was supported in part by the National Science Foundation under grantDMS 91{100324The third author was supported by the Study Abroad fellowship of the Government ofIndia. 1

2 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEthe irreducible representations of the quantum algebra Uq(su2). We replace theusual exponential function mapping from the Lie algebra to the Lie group bythe q-exponential mappings Eq and eq. In place of the usual matrix elementson the group (arising from an irreducible representation) which are expressiblein terms of Jacobi polynomials, we �nd eight types of matrix elements express-ible in terms of q-hypergeometric series. These q-matrix elements do not satisfygroup homomorphism properties, so they do not lead to addition theorems inthe usual sense. However, they do satisfy orthogonality relations. Furthermore,in analogy with true group representation theory we can show that each of theeight families of matrix elements determines a two-variable model for irreduciblerepresentations of Uq(su2). In x3 we show how this two-variable model leads toorthogonality relations for one class of matrix elements.In x4 we �nd an alternate and simpler derivation of a q-analog, due to Groza,Kachurik and Klimyk [4], of the Burchnall-Chaundy formula for the product oftwo hypergeometric functions 2F1. It is interpreted here as the expansion of thematrix elements of a \group operator" (via the exponential mapping) in a tensorproduct basis in terms of the matrix elements in a reduced basis.Our approach to the derivation and understanding of q-series identities isbased on the study of q-algebras as q-analogs of Lie algebras, [5, 6]. Essentially,we are attempting to �nd q-analogs of the theory relating Lie algebra and localLie transformation groups [7, 8]. A similar approach has been adopted byFloreanini and Vinet [9-12]. This is an alternative to the elegant papers [13-21] which are based primarily on the theory of quantum groups. The mainjusti�cation of the \local" approach is that it is more general; it applies to moregeneral families of special functions than does the quantum group approach.The notation used for q-series in this paper follows that of Gasper and Rahman[22]. 2. Models of �nite dimensional Uq(su2) representationsThe quantum algebra Uq(su2) is the associative algebra generated by theelements H , E+, E� that obey the commutation relations[H;E+] = E+; [H;E�] = �E�;[E+; E�] = qH � q�Hq 12 � q� 12 :(2.1)Here q is a real parameter such that 0 < q < 1. In the limit as q ! 1 rela-tions (2.1) go to the usual commutation relations for the complexi�cation of theLie algebra su2. Finite dimensional irreducible representations of Uq(su2) aredetermined by the integral or half-integral number u: 2u = 0; 1; 2; � � � . The cor-responding representation D(2u) is de�ned on the (2u+ 1)-dimensional Hilbert

MATRIX ELEMENTS OF Uq(su2) 3space H2u with orthonormal basis fem : m = �u;�u+ 1; � � � ; ug such thatE+em = ([u�m]q [u+m+ 1]q) 12 em+1E�em = ([u+m]q [u�m+ 1]q) 12 em�1(2.2) Hem = memwhere(2.3) [m]q = qm2 � q�m2q 12 � q� 12 = q�m�12 �1� qm1� q � :On this Hilbert space E+ = (E�)� and H� = H . A second convenient basis forH2u is the set ffn : n = 0; 1; � � � ; 2ug such thatE+fn = �q�1[2u� n]qfn+1E�fn = �q[n]qfn�1(2.4) Hfn = (�u+ n)fn:Here(2.5) fn = " (�1)nq( 32�u)n(q; q)n(q�2u; q)n # 12 e�u+n:Since the element(2.6) C = E+E� + qH� 12 + q�H+ 12(q 12 � q� 12 )2commutes with each generator of Uq(su2) it corresponds to a multiple of theidentity operator I on H2u. Indeed,(2.7) C = qu+ 12 + q�u� 12(q 12 � q� 12 )2 I:Now we consider the discrete series "u of in�nite dimensional representationsof Uq(su2). This is de�ned on the Hilbert space H0 with orthogonal basis ffn :n = 0; 1; � � � g such that relations (2.4) hold. Now, however, u is a negative realnumber and jjfnjj =sq( 32�u)n(q; q)n(q�2u; q)n ; n = 0; 1; 2; � � � :On this Hilbert space E+ = �(E�)� and H� = H . (In the limit as q ! 1 theserepresentations correspond to the positive discrete series of unitary irreduciblerepresentations of the Lie algebra su(1; 1), [18, 19].)Note, however, that for each complex number u such that 2u 6= 0; 1; 2; � � � ex-pressions (2.4) de�ne an algebraically irreducible representation "u of Uq(su2) onan in�nite dimensional vector space K consisting of all �nite linear combinationsof the basis vectors ffng.

4 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEWe introduce a convenient one variable model of D(2u). Here the vector spaceH2u consists of polynomials f(z) of maximum order 2u in the complex variablez. The action of Uq(su2) is de�ned by the operatorsE+ =q� 12 z1� q (quT� 12z � q�uT 12z )(2.8) E� = q 32z(1� q) (T 12z � T� 12z )H =� u+ z ddzwhere Tzf(z) = f(qz). The basis functions ffn = zn : n = 0; 1 � � � ; 2ug satisfyrelations (2.4). Similarly, we can introduce a one variable model of "u. A basisfor the vector space consists of the functions ffn(z) = zn : n = 0; 1; 2 � � � g in thecomplex variable z. The action of Uq(su2) on functions f(z) is again de�ned byexpressions (2.8). See [1] for derivations of the inner products corresponding tothese models.In analogy with a standard relationship between special functions and the rep-resentations of Lie groups we shall compute the \matrix elements" of q-analogsof the group operators e�E+e�E� with respect to the ffng basis, in both therepresentations D(2u) and "u. Of course there are many q-analogs of the ex-ponential mapping, none of which have all the properties needed to ensure thatthere is a true \group" associated with the q-algebra. Among the q-analogs weshall limit ourselves to the two that are most important, [22]:(2.9) eq(z) = 1Xk=0 zk(q; q)k ; Eq(z) = 1Xk=0 qk(k�1)=2(q; q)k zk:If z is a complex number, the �rst series converges to 1=(z; q)1 for jzj < 1 andthe second series converges to (�z; q)1 for all z.Using the model (2.4) we can de�ne eight q-analogs of the matrix elements.(e+; e�) : eq(�E+)eq(�E�)fn =Xn0 T (e+;e�)n0n (�; �)fn0 ;(e�; e+) : eq(�E�)eq(�E+)fn =Xn0 T (e�;e+)n0n (�; �)fn0 ;(e+; E�) : eq(�E+)Eq(�E�)fn =Xn0 T (e+;E�)n0n (�; �)fn0 ;(2.10) (e�; E+) : eq(�E�)Eq(�E+)fn =Xn0 T (e�;E+)n0n (�; �)fn0 ;(E+; e�) : Eq(�E+)eq(�E�)fn =Xn0 T (E+;e�)n0n (�; �)fn0 ;

MATRIX ELEMENTS OF Uq(su2) 5(E�; e+) : Eq(�E�)eq(�E+)fn =Xn0 T (E�;e+)n0n (�; �)fn0 ;(E+; E�) : Eq(�E+)Eq(�E�)fn =Xn0 T (E+;E�)n0n (�; �)fn0 ;(E�; E+) : Eq(�E�)Eq(�E+)fn =Xn0 T (E�;E+)n0n (�; �)fn0 :For the representations D(2u) the sum on n0 is from 0 to 2u and n = 0; � � � ; 2u.For the representations "u, the sum on n0 is from 0 to 1 and n = 0; 1; � � � .(Relations (2.10) and the relations to follow are well de�ned as formal powerseries in the variables �, �. We must look at each case separately to determinethe domains of pointwise convergence of the series.)Since E�+ = E� for D(2u) and E�+ = �E� for "u the following relationshipshold: T (e+;e�)n0n (�; �)An0n = T (e+;e�)nn0 (��;��);T (e�;e+)n0n (�; �)An0n = T (e�;e+)nn0 (��;��);T (e+;E�)n0n (�; �)An0n = T (E+;e�)nn0 (��;��);T (e�;E+)n0n (�; �)An0n = T (E�;e+)nn0 (��;��);(2.11) T (E+;E�)n0n (�; �)An0n = T (E+;E�)nn0 (��;��);T (E�;E+)n0n (�; �)An0n = T (E�;E+)nn0 (��;��):Here An0n = q(3=2�u)(n0�n)(q; q)n0(q�2u; q)n(q; q)n(q�2u; q)n0 :Since eq(z)Eq(�z) = 1, we have the identitiesa) X̀T (e+;e�)n0` (�; �)T (E�;E+)`n (��;��) = �n0nb) X̀T (e�;e+)n0` (�; �)T (E+;E�)`n (��;��) = �n0n(2.12) c) X̀T (E�;e+)n0` (�; �)T (E+;e�)`n (��;��) = �n0n:Again, the sum is �nite or in�nite, depending on the representation D(2u) or"u.Using the model (2.4) to compute the matrix elements (which are modelindependent) we obtain the explicit results

6 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEET (e+;e�)n0n (�; �)= (q�2u; q)n0�n0�nq�(n0�n)(n0+n+1)=4qu(n�n0)(q; q)n0�n(q�2u; q)n(1� q)n0�n� 3�1 � q�n; q1�n+2u; 0qn0�n+1 ; q; ��qn�u+1=2(1� q)2 �= (q; q)n(��)n�n0q�(n�n0)(n0+n�5)=4(q; q)n�n0(q; q)n0 (1� q)n�n0� 3�1 q�n0 ; q1�n0+2u; 0qn�n0+1 ; q; ��qn0�u+1=2(1� q)2 !T (e�;e+)n0n (�; �)= (q�2u; q)n0�n0�nq�(n0�n)(n0+n+1)=4qu(n�n0)(q; q)n0�n(q�2u; q)n(1� q)n0�n� 3�1 qn0+1; qn0�2u; 0qn0�n+1 ; q; ��qu�n0+1=2(1� q)2 != (q; q)n(��)n�n0q�(n�n0)(n+n0�5)=4(q; q)n�n0(q; q)n0 (1� q)n�n0� 3�1 � qn+1; qn�2u; 0qn�n0+1 ; q; ��qu�n+1=2(1� q)2 �T (E+;e�)n0n (�; �)(2.13) = (q�2u; q)n0�n0�nq(n0�n)(n0�3n�3)=4qu(n�n0)(q; q)n0�n(q�2u; q)n(1� q)n0�n� 2�1 q�n; q1�n+2uqn0�n+1 ; q; ���qn0�u+1=2(1� q)2 != (q; q)n(��)n�n0q�(n�n0)(n+n0�5)=4(q; q)n�n0(q; q)n0 (1� q)n�n0� 2�1 q�n0 ; q1�n0+2uqn�n0+1 ; q; ���qn0�u+1=2(1� q)2 !

MATRIX ELEMENTS OF Uq(su2) 7T (E�;e+)n0n (�; �)= (q�2u; q)n0�n0�nq�(n0�n)(n0+n+1)=4qu(n0�n)(���qn�u+1=2=(1� q)2; q)1(q; q)n0�n(q�2u; q)n(1� q)n0�n(���qu�n0+1=2=(1� q)2; q)1� 2�1 � q�n; q1�n+2uqn0�n+1 ; q; ���qn�u+1=2(1� q)2 �= (q; q)n(��)n�n0q(n�n0)(n�3n0+3)=4(���qn�u+1=2=(1� q)2; q)1(q; q)n�n0(q; q)n0(1� q)n�n0(���qu�n0+1=2=(1� q)2; q)1� 2�1 � q�n0 ; q1�n0+2uqn�n0+1 ; q; ���qn�u+1=2(1� q)2 �T (E�;E+)n0n (�; �)= (q; q)n(��)n�n0q(n�n0)(n�3n0+3)=4(q; q)n�n0 (q; q)n0(1� q)n�n0� 2�2 qn�2u; qn+1qn�n0+1; 0 ; q; ��qu�n0+1=2(1� q)2 != (q�2u; q)0n�n0�nq(n0�n)(n0�3n�3)=4qu(n�n0)(q; q)n0�n(q�2u; q)n(1� q)n0�n� 2�2� qn0�2u; qn0+1qn0�n+1; 0 ; q; ��qu�n+1=2(1� q)2 �T (E+;E�)n0n (�; �)= (q�2u; q)n0�n0�nq�(n0�n)(n0�3n�3)=4qu(n�n0)(q; q)n0�n(q�2u; q)n(1� q)n0�n� 2�2 q�n; q2u�n+1qn0�n+1; 0 ; q; ��qn0�u+1=2(1� q)2 != (q; q)n(��)n�n0q(n�n0)(n�3n0+3)=4(q; q)n�n0 (q; q)n0(1� q)n�n0� 2�2� q�n0 ; q2u�n0+1qn�n0+1; 0 ; q; ��qn�u+1=2(1� q)2 �For the �nite dimensional representationsD(2u) all of these matrix elements arepolynomials. For the in�nite dimensional representations the elements T (e�;e+)diverge for �� 6= 0, the T (E�;e+) converge for j��qu�n0+1=2=(1� q)2j < 1, andthe remaining matrix elements are entire functions. Identities (2.12) are validfor all �, � corresponding to the �nite dimensional representationsD(2u). Usingthe ratio test to determine the domains of convergence corresponding to "u we�nd that (2.12a) converges for all �, �, (2.12b) diverges, and (2.12c), multipliedby (���qu�n0+1=2=(1� q)2; q)1, converges for all �, �.

8 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEEach of these families of matrix elements determines models of the irreduciblerepresentations D(2u) and "u. This is a consequence of the commutation rela-tions (2.1). To see this we make use of the following formal power series resultsfor linear operators X , Y :Lemma 1. Eq(�X)Y eq(��X) = 1Xn=0 �n(q; q)n [X;Y ]nwhere [X;Y ]0 = Y; [X;Y ]n+1 = X [X;Y ]nqn � [X;Y ]nX; n = 0; 1; � � � :Lemma 2. eq(�X)Y Eq(��X) = 1Xn=0 �n(q; q)n [X;Y ]0nwhere [X;Y ]00 = Y; [X;Y ]0n+1 = X [X;Y ]0n � qn[X;Y ]0nX; n = 0; 1; � � � :As a consequence of these lemmas we havea) Eq(�E�)E+ = E+Eq(�E�)� q1=2�(1� q)2 �q�HEq(�E�)� qHEq(�qE�)�(2.14)b) eq(�E+)E� = E�eq(�E+) + q1=2�(1� q)2 �q�Heq(�E+)� qHeq(�q E+)� :Note also the easily veri�ed identitiesEq(��E+)q�Heq(�qE+) = q�H ;Eq(��qE�)q�Heq(�E�) = q�H :For the matrix elements (e+,e-) the operator identitiesa) eq(�E+)eq(�E�)E� = 1� (I � T�)eq(�E+)eq(�E�)b) eq(�E+)eq(�E�)E+ = 1� (I � T�)eq(�E+)eq(�E�)(2.15) + �q1=2(1� q2) (q�u+n0T�1� � qn0�uT�T�1� )eq(�E+)eq(�E�)c) [H; eq(�E+)eq(�E�)] = (�@� � �@�)eq(�E+)eq(�E�)

MATRIX ELEMENTS OF Uq(su2) 9implya) � q[n]qT (e+;e�)n0;n�1 (�; �) = 1� (I � T�)T (e+;e�)n0n (�; �)b) � q�1[2u� n]qT (e+;e�)n0;n+1 (�; �) =� 1� (I � T�) + �q1=2(1� q)2 (q�u+n0T�1� � qu�n0T�T�1� )� T (e+;e�)n0n (�; �)c) (�u+ n)T (e+;e�)n0n (�; �) = (�u+ n0 + �@� � �@�)T (e+;e�)n0n (�; �)where T�f(�; �) = f(q�; �).For the matrix elements (e-,e+) we �nda) � q[n]qT (e�;e+)n0;n�1 (�; �) =� 1� (I � T�) + �q1=2(1� q)2 (qu�n0T�1� � q�u+n0T�T�1� )� T (e�;e+)n0n (�; �)b) � q�1[2u� n]qT (e�;e+)n0;n+1 (�; �) = 1� (I � T�)T (e�;e+)n0n (�; �)c) (�u+ n)T (e�;e+)n0n (�; �) = (�u+ n0 � �@� + �@�)T (e�;e+)n0n (�; �);for the matrix elements (E+,e-) we �nda) � q[n]qT (E+;e�)n0;n�1 (�; �) = 1� (I � T�)T (E+;e�)n0n (�; �)b) � q�1[2u� n]qT (E+;e�)n0;n+1 (�; �) =� q� (T�1� � I) + �q1=2(1� q)2 (q�u+n0T�1� � qu�n0T�T�1� )�T (E+;e�)n0n (�; �)(2.16)c) (�u+ n)T (E+;e�)n0n (�; �) = (�u+ n0 + �@� � �@�)T (E+;e�)n0n (�; �);for the matrix elements (E-,e+) we �nda) � q[n]qT (E�;e+)n0;n�1 (�; �) =� q� (T�1� � I) + �q1=2(1� q)2 (qu�n0T�1� � q�u+n0T�T�1� )�T (E�;e+)n0n (�; �)b) � q�1[2u� n]qT (E�;e+)n0;n+1 (�; �) = 1� (I � T�)T (E�;e+)n0n (�; �)c) (�u+ n)T (E�;e+)n0n (�; �) = (�u+ n0 � �@� + �@�)T (E�;e+)n0n (�; �);

10 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEfor the matrix elements (E-,E+) we �nda) � q[n]qT (E�;E+)n0;n�1 (�; �) =� q� (T�1� � I) + �q1=2(1� q)2 (qu�n0T�1� T� � q�u+n0T�)�T (E�;E+)n0n (�; �)b) � q�1[2u� n]qT (E�;E+)n0;n+1 (�; �) = q� (T�1� � I)T (E�;E+)n0n (�; �)c) (�u+ n)T (E�;E+)n0n (�; �) = (�u+ n0 � �@� + �@�)T (E�;E+)n0n (�; �);and for the matrix elements (E+,E-) we havea) � q[n]qT (E+;E�)n0;n�1 (�; �) = q� (T�1� � I)T (E+;E�)n0n (�; �)b) � q�1[2u� n]qT (E+;E�)n0;n+1 (�; �) =� q� (T�1� � I) + �q1=2(1� q)2 (q�u+n0T�1� T� � qu�n0T�)�T (E+;E�)n0n (�; �)c) (�u+ n)T (E+;E�)n0n (�; �) = (�u+ n0 + �@� � �@�)T (E+;E�)n0n (�; �):These relations are equivalent to q-di�erence relations satis�ed by variousq-hypergeometric series. Furthermore, it is easy to verify from the series thatthe relations hold also for u and n0 complex. Thus we have a wide variety oftwo variable models of algebraically irreducible representations of Uq(su2). Wenote that this approach is closely related to the factorization method of quantummechanics [23].Let X and Y be linear operators such that Y X = qXY . A straightforwardformal induction argument using this property, [15], [22, page 28], yieldsLemma 3. (Y +X)k = kX̀=0 (q; q)k(q; q)`(q; q)k�`X`Y k�`;eq(X + Y ) = eq(X)eq(Y ); Eq(X + Y ) = Eq(Y )Eq(X):Using these results and iterating relations (2.14) we �ndEq ���q1=2+H(1� q)2 �Eq(�E+)eq(�E�) = eq(�E�)Eq(�E+)Eq ���q1=2�H(1� q)2 � ;Eq ���q1=2�H(1� q)2 �Eq(�E�)eq(�E+) = eq(�E+)Eq(�E�)Eq ���q1=2+H(1� q)2 �

MATRIX ELEMENTS OF Uq(su2) 11or Eq ��q1=2�u+n0(1� q)2 !T (E+;e�)n0n (�; �) = Eq ���q1=2+u�n(1� q)2 �T (e�;E+)n0n (�; �);Eq ��q1=2+u�n0(1� q)2 !T (E�;e+)n0n (�; �) = Eq ���q1=2�u+n(1� q)2 �T (e+;E�)n0n (�; �):Identities (2.12a-c) yield orthogonality and biorthogonality relations forq-hypergeometric functions. For example, (2.12c) can be written in the formX̀ (zq`+1; q)1(�q2uz)`(q�2u; q)`q`(n+n0)� `22 p(`�n0;2u�n0�`)n0 (zq`)p(`�n;2u�n�`)n (zq`)(2.17) = (zq1�n+2u; q)1(�z)nq2un�n=2�3n2=2(q; q)n(q�2u; q)n�nn0 :Here the p(�;�)n (z) = (aq; q)n(q; q)n 2�1� q�n; qn+1abaq ; qz�n = 0; 1; 2; � � �are little q-Jacobi polynomials, [24], where a = q�, b = q� . Note that the littleq-Jacobi polynomials satisfy the identity(q; q)nq 12 (n0�n)( 12�n)(q�+1; q)n(�z)n p(n0�n;�)n (z) = (q; q)n0q 12 (n�n0)( 12�n0)(q�+1; q)n0(�z)n0 p(n�n0;�)n0 (z):Since Fn(y) = q�n`p(`�n;2u�n�`)n (zq`) is a polynomial of order n in the variabley = q�`, we see that expression (2.17) is an orthogonality relation for this set ofpolynomials. See [17] for similar identities associated with q-Bessel functions.A nontrivial extension of identity (2.12c) isX̀T (E�;e+)n0` (�; )T (E+;e�)`n (�;��) = Sn0n( ; �)where the matrix elements Sn0n( ; �) are de�ned byeq(�E�)Eq( E�)fn =Xn0 Sn0n( ; �)fn0 :Explicitly,Sn0n( ; �) = ( �n�n0(q�n;q)n�n0(1�q)n�n0 (q;q)n�n0 q(n�n0)(7+n+n0)=4(� � ; q)n�n0 if n � n00 if n < n0:

12 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEIn the special case =� = �qn0�n+1 we �nd the resultX̀(zqn0+`+2; q)1 (q�2u; q)`(q; q)` (�z)`q`( 32+2u�n)+`2=2p(`�n0;2u�n0�`)n0 (zqn0+`+1)�p(`�n;2u�n�`)n (zqn+`) = (zq2u+2; q)1(�z)nqn( 32+2u)� 32n2(q; q)n(q�2u; q)n�nn0 :(In the case of the Lie algebra of the Euclidean group in the plane, the analogousidentities are the Hansen-Lommel identities for q-Bessel functions, [25, 13, 2].)There are similar extensions of (2.12a-b).3. Orthogonality relations for matrix elementsThe matrix elements of the form (E; e) and (e; E) satisfy orthogonality re-lations analogous to those arising from integration with respect to the Haarmeasure on SU(2) in the q = 1 case. Indeed these relations are directly deriv-able from the q-di�erence relations for the matrix elements T (E+;e�)n0n (�; �) inx2. We follow the derivation in [3] but now make clear the connection to therepresentation theory of Uq(su2).Making the change of variable z = ���qn0�u�1=2=(1� q)2 and factoring outthe dependence of � in the �rst equation (2.13) for T (E+;e�)n0n (�; �) and in therecurrence relations for these expressions, we obtain the (ordinary) di�erenceequations � (a;b)p(�;�)n = �q1�n(1� qn+1ab)p(�+1;�+1)n�1(3.1) ��(aq;bq)p(�+1;�+1)n�1 = �(1� qn)p(�;�)nwhere the p(�;�)n (z) are little q-Jacobi polynomials, a and b are complex numbers,a = q�, b = q� and� (a;b) = 1z (I � Tz); ��(aq;bq) = (z � 1)T�1z + (aq � abq2z):Let Sa;b be the space of all polynomials in z with the complex bilinear form(3.2) (f1; f2)a;b = 12�i IC f1(z)f2(z)wa;b(z) dzz ; f1; f2 2 Sa;bwhere C = fz : jzj = 1 + �; � > 0g and wa;b is a weight function to bedetermined. We consider � and �� as mappings� (a;b) : Sa;b ! Saq;bq ; ��(aq;bq) : Saq;bq ! Sa;band look for a weight function so that �� is the adjoint of � :(3.3) (�f; g)aq;bq = (f; ��g)a;b; 8f 2 Sa;b; g 2 Saq;bq :This yields the necessary and su�cient conditionswaq;bq(z=q) = zq (1� z)wa;b(z); waq;bq(z) = aqz(1� bqz)wa;b(z)

MATRIX ELEMENTS OF Uq(su2) 13with the solution, unique to within a constant multiplier,(3.4) wa;b(z) = ( za ; q)1( qaz ; q)1(zbq; q)1( 1z ; q)1(aq; q)1( 1a ; q)1(�aq; q)1(� 1a ; q)1 :By construction ��� : Sa;b ! Sa;b is self-adjoint and���p(�;�)n = �np(�;�)n ; �n = �q(1� q�n)(1� qn+1ab):Clearly �n 6= �m if n 6= m so we must have(p(�;�)n ; p(�;�)m )a;b = 0 if m 6= n:We have proved orthogonality for the little q-Jacobi polynomials. It remains tocompute the normalization of these functions. Setting f = p(�;�)n , g = p(�+1;�+1)n�1in the adjoint relation (3.3) we obtain the recurrencejjp(�;�)n jj2a;b = q(1� qn+1ab)(1� qn) jjp(�+1;�+1)n�1 jj2aq;bq :From the relation (p(�;�)1 ; p(�;�)0 )a;b = 0 and the explicit expressions for p1, p0 we�nd(3.5) jj1jj2a;bq = (1� bq)(1� abq2) jj1jj2a;b:(Here we use the fact that (1� zbq; 1)a;b = jj1jj2a;bq .)To completely determine the norm we use the second equation (2.13) forT (E+;e�)n0n (�; �). Here the induced action of the recurrence operators on the basisfp(�;�)n g is�(a;b)p(�;�)n = (1� aqn)p(��1;�+1)n ; �(aq�1;bq)p(��1;�+1)n = q�na (1� bqn+1)p(�;�)nwhere �(a;b) = 1� aTz; �(a;b) : Sa;b ! Saq�1;bq ;��(aq�1;bq) = z � 1az T�1z + 1� bqzaz ; ��(aq�1;bq) : Saq�1;bq ! Sa;b:It is straightforward to verify that(3.6) (�f; g)aq�1;bq = (f; ��g)a;b; 8f 2 Sa;b; g 2 Saq�1;bq ;so �� is the adjoint of �. Now set f = p(�;�)0 = 1, g = p(��1;�+1)0 = 1 in (3.6) toobtain(3.7) jj1jj2aq�1;bq = (1� bq)a(1� a) jj1jj2a;b:The solution of the recurrences (3.5) and (3.7) isjj1jj2a;b = (abq2; q)1K(q)(bq; q)1(aq; q)1(� 1a ; q)1(�aq; q)1 :

14 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEEThus we have12�i IC ( za ; q)1( qaz ; q)1( 1a ; q)1( 1z ; q)1(zbq; q)1 dz = (abq2; q)1K(q)(bq; q)1 :To compute K(q) we make particular choices of a and b such that the integralbecomes trivial: a > 1 + � > 1 > qa; abq = 1:It follows easily that K(q) = 1=(q; q)1. Thus we have derived the normalizationfor the little q-Jacobi polynomials entirely by elementary techniques and provedcomplex orthogonality with respect to the bilinear form (3.3), (3.4).To obtain the orthogonality of the fp(�;�)n g with respect to a real positivede�nite inner product we expand the integral expression for the complex bilinearform (p(�;�)n ; p(�;�)m )a;b by residues at the poles z = qk, k = 0; 1; 2; � � � of theweight function. The straightforward computation yields(3.8)Z 10 p(�;�)n (x)p(�;�)m (x)��;�(x) dqx = (aq)n(abqn+1; q)1(aq; q)n(q; q)1(1� q)(bqn+1; q)1(aq; q)1(q; q)n(1� abq2m+1)�m;nwhere Z 10 f(x) dqx = (1� q) 1Xk=0 f(qk)qkand ��;�(x) = x�(xq; q)1=(bxq; q)1. See [26-28] for related results on orthogo-nality of q-series and associated polynomials of a discrete variable.4. A tensor product identity for Uq(su2) representationsGiven the irreducible representations D(2u1) and D(2u2) on the spaces H2u1and H2u2 , respectively, we de�ne the tensor product representation D(2u1) D(2u2) of Uq(su2) on the space H2u1 H2u2 by the operators [5],F+ =�(E+) = E+ q 12H + q� 12H E+F� =�(E�) = E� q 12H + q� 12H E�(4.1) L =�(H) = H I + I H:The operators F�, L satisfy the same commutation relations as the operatorsE�, H :(4.2) [L; F�] = �F�; [F+; F�] = qL � q�Lq 12 � q� 12 :

MATRIX ELEMENTS OF Uq(su2) 15From Lemma 3 we have the formal identityEq(�F+)eq(�F�) = Eq(�E+ q 12H)Eq(�q� 12H E+)� eq(�E� q 12H)eq(�q� 12H E�)= Eq(�E+ q 12H)eq(�E� q 12H)�Eq(�q� 12H E+)eq(�q� 12H E�):Thus,(4.3) T (E+;e�)n0m0;nm(�; �) = T (E+;e�);u1n0n (�qm02 ; �qm02 )T (E+;e�);u2m0m (�q�n2 ; �q�n2 )where the functions T (E+;e�)n0m0;nm(�; �) are the matrix elements of the operatorEq(�F+)eq(�F�) in the tensor product basis ffu1n fu2m g. In other words, forthis operator the matrix elements in a tensor product basis actually factor.It is well known that the tensor product decomposes into a direct sum ofirreducible representations [5, 29, 1],D(2u1)D(2u2) �= u1+u2Xv=ju1�u2j�D(2v):Choosing canonical orthonormal basis vectors fevm : m = v; v � 1; � � � ;�vg forthe irreducible subspace corresponding to the representationD(2v) we de�ne theClebsch-Gordan coe�cients by the expansion formula(4.4) evm = Xn1;n2 � u1 u2 vn1 n2 m�q eu1n1 eu2n2 :A generating function for the coe�cients is given by [29, 1]xu3+u2�u12 (x3x2 qu1�u2�u3 ; q)u2+u3�u1xu3+u1�u23 (x1x3 qu2�u3�u1 ; q)u3+u1�u2� xu1+u2�u31 (x2x1 q1+u3�u1�u2 ; q)u1+u2�u3= (�1)u3+u2�u1q 12 (u1+u2�u3)+u3(u1+u2)� 12 (u21+u22+u23)�� (�1)u1+u2�u3(q; q)u1+u2�u3(q�u1�u2�u3�1; q)u1+u2�u3(q�2u1 ; q)u1+u2�u3(q�2u2 ; q)u1+u2�u3 � 12 �Xn1;n2 xu3�m3 xu1+n11 xu2+n22 q 12 [m(u2�u1+u3)+n1(u2�u3+u1)+n2(u3�u1+u2+2)]� (�1)u3+m � (q�2u3 ; q)u3+m(q�2u1 ; q)u1+n1(q�2u2 ; q)u2+n2(q; q)u3+m(q; q)u1+n1(q; q)u2+n2(�1)u1+u2�u3 � 12 � u1 u2 u3n1 n2 m �q :It follows that(4.5) � u1 u2 vn1 n2 m �q =

16 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEE� (q�2u1 ; q)u1+u2�v(q�2u2 ; q)u1+u2�v(q�2u1 ; q)u1+n1(q�2u2 ; q)u2+n2(q; q)u1+u2�v(q�u1�u2�v�1; q)u1+u2�v(q�2v ; q)v+m(q; q)u2+n2 � 12� [(q; q)v+m(q; q)u1+n1 ] 12 (qn1+v�u2+1; q)1(q; q)1(q�2u1 ; q)u1+u2�v� q 12 (u1�u2)(u1+u2+n1�n2)q 12 [�u21+u22+2u1u2�2u2v�n1u1+n2u1�n1u2�n2u2]� 3�2� qv�u1�u2 ; qu1�u2+v+1; q�u2�n2qv+n1�u2+1; q�2u2 ; q; q�v+m� :Note that the transformation [1]xj ! x�1j ; uj ! uj ; q ! q�1; j = 1; 2; 3followed by a multiplication by x2u11 x2u22 x2u33 maps the generating function to(�1)u1+u2+u3q2(u21+u22+u23)� 12 (u1+u2+u3)2+ 32u3� 12u1� 12u2times itself. This leads to the transformation formula3�2� q�n; b; cd; e ; q; deqnbc � = (e=c; q)n(e; q)n 3�2� q�n; c; d=bd; cq1�n=e ; q; q� ;[22], so that the following alternate expression for the coe�cients holds [4, 29,30].(4.6) � u1 u2 vn1 n2 m �q =� [u1 � u2 + v]![v +m]![v �m]![u1 � n1]![2v + 1][u1 + n1]![u1 + u2 � v + 1]![u1 + u2 � v]![v + u2 � u1]![u2 + n2]![u2 � n2]!� 12�q� 14 (v+u1�u2+1)(u1+u2�v)+ 12 (v�u2)n2� 12n1u2 (�1)v�u1�u2 [2u2]![v � u2 � n1]!�3�2� qv�u1�u2 ; qu1�u2+v+1; q�u2+n2qv�u2�n1+1; q�2u2 ; q; q� ;provided v = u1 + u2; u1 + u2 � 1; � � � ; ju1 � u2j, ni = ui; ui � 1; � � � ;�ui andm = n1 + n2. Otherwise, the coe�cients vanish. Here,[k]! = q�k(k�1)=4(1� q)k (q; q)k :De�ning normalized matrix elements for the representation D(2u) byS(E+;e�)n1n2 (�; �) =< Eq(�E+)eq(�E�)en1 ; en2 >= " (�1)n�n0(q; q)n0(q�2u; q)nq(3=2�u)(n�n0)(q; q)n(q�2u; q)n0 # T (E+;e�)n0n (�; �);(4.7)

MATRIX ELEMENTS OF Uq(su2) 17where n1 = �u+ n, n2 = �u+ n0, we note from (4.4), (4.6) and (4.7) that thefollowing identity, relating the matrix elements of the operator Eq(�F+)eq(�F�)in two distinct orthonormal bases, must hold:S(E+;e�)m1m2;n1n2(�; �) =Xv � u1 u2 vn1 n2 n1 + n2 �q S(E+;e�);vm1+m2;n1+n2(�; �)� � u1 u2 vm1 m2 m1 +m2 �q ;or, 2�1 � qa; qbqc ; q;x� 2�1� q�; q�q ; q; qa+b�cx� =1Xr=0 qr(r+c�1) (qa; q)r(qb; q)r(q ; q)rxr(q; q)r(qc; q)r(qc+ +r�1; q)r�(4.8) 3�2 � q�r; q�; q1�r�cq ; q1�r�a ; q; q� 3�2 � q�r; q� ; q1�r�cq ; q1�r�b ; q; q�� 2�1� qa+�+r; qb+�+rqc+ +2r ; q;x� :The result (4.8) is established, intitially, only for negative integer values of theparameters, corresponding to the �nite dimensional irreducible representationsof Uq(su2). However, using the same arguments as in the interpretation of (2.12)we can establish it for all complex values of the parameters for which both sidesof the equation are well de�ned. This result was �rst derived by Groza, Kachurikand Klimyk, [4] who used an argument in the spirit of quantum groups. Theproof here is much simpler and the interpretation of the result is somewhatdi�erent.As a �nal note we remark that Biedenharn and Tarlini, [31], have shown howto extend the notion of tensor operators for a Lie algebra to q-tensor operatorsfor a quantum algebra in such a way that a generalized Wigner-Eckart theoremholds. This relates to our q-algebra models.References1. E.G. Kalnins, H.L. Manocha and W. Miller (1992), Models of q-algebra representations:Tensor products of special unitary and oscillator algebras, J. Math. Phys. 33, 2365{2383.2. E.G. Kalnins, S. Mukherjee and W. Miller (1993), Models of q-algebra representations:The group of plane motions, SIAM J. Math. Anal. (to appear).3. W. Miller, Jr. (1989), in q-Series and Partitions, D. Stanton, ed., IMA Volumes in Math-ematics and its Applications, Vol. 18, Springer-Verlag, New York, pp. 191{212.4. V.A. Groza, I.I. Kachurik and A.U. Klimyk (1990), On matrix elements and Clebsch-Gordan coe�cients of the quantum algebra Uq(SU2), J. Math. Phys. 31, 2769{2780.5. M. Jimbo (1985), A q-di�erence analogue of U(g) and the Yang-Baxter equation, Lettersin Mathematical Physics 10, 63{69.6. A.K. Agarwal, E.G. Kalnins, and W. Miller (1987), Canonical equations and symmetrytechniques for q-series, SIAM J. Math. Anal. 18, 1519-1538.

18 E.G. KALNINS, WILLARD MILLER, JR. AND SANCHITA MUKHERJEE7. W. Miller (1968), Lie Theory and Special Functions, Academic Press, New York.8. N. Ja. Vilenkin (1968), Special Functions and the Theory of Group Representations, Amer-ican Mathematical Society, Providence, Rhode Island.9. R. Floreanini and L. Vinet (1990), q-Orthogonal polynomials and the oscillator quantumgroup, INFN preprint, Trieste AE-90/23.4. R. Floreanini and L. Vinet (1991), Quantum algebras and q-special functions, LettersMath. Physics 22, 45{54.11. R. Floreanini and L. Vinet (1991), Addition formulas for q-Bessel functions, Universit�e deMontr�eal preprint, Montr�eal UdeM-LPN-TH60.12. L. Floreanini and L. Vinet (1990), q-Analogues of the parabose and parafermi oscillatorsand representations of quantum algebras, J. Phys. A23, L1019{L1023.13. H.T. Koelink (1991), On quantum groups and q-special functions, thesis University ofLeiden.14. H.T. Koelink and T.H. Koornwinder (1989), The Clebsch-Gordan coe�cients for the quan-tum group S�U(2) and q-Hahn polynomials, Proc. Koninklijke Nederl. Akad. Wetenschap-pen, Series A 92, 443{456.15. T.H. Koornwinder (1989), Representations of the twisted SU(2) quantum group and someq-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Proc. Ser A92, 97{117.16. T.H. Koornwinder (1991), The addition formula for little q-Legendre polynomials and theSU(2) quantum group, SIAM J. Math. Anal. 22, 295{301.17. T.H. Koornwinder and R.F. Swarttouw (1993), On q-analogues of the Fourier and Hankeltransforms group, (to appear in Trans. American Math. Soc.).18. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Saburi and K. Ueno (1990), Unitaryrepresentations of the quantum groups SUq(1; 1): Structure of the dual space of Uq(sl(2)),Lett. Math. Phys. 19, 187{194.19. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Saburi and K. Ueno (1990), Unitaryrepresentations of the quantum groups SUq(1; 1): II. Matrix elements of unitary represen-tations and the basic hypergeometric functions, Lett. Math. Phys. 19, 194{204.20. L.L. Vaksman and L.I. Korogodski�i (1989), An algebra of bounded functions on the quan-tum group of the motions of the plane, and q-analogues of Bessel functions, Soviet Math.Dokl. 39, 173{177.21. S. L. Woronowicz (1987), Twisted SU(2) group. An example of a noncommutative di�er-ential calculus, Publ. RIMS, Kyoto Univ. 23, 117{181.22. G. Gasper and M. Rahman (1990), Basic Hypergeometric Series, Cambridge UniversityPress, Cambridge.23. W. Miller (1970), Lie theory and q-di�erence equations, SIAM J. Math. Anal. 1, 171{188.24. G.E. Andrews and R. Askey (1985), Classical Orthogonal Polynomials, Lecture Notes #1171, Springer-Verlag, New York, Berlin, 36{62.25. H.T. Koelink (1991), Hansen-Lommel orthogonality relations for Jackson's q-Bessel func-tions, Report W-91-11 University of Leiden.26. A. F. Nikiforov, S.K. Suslov, and V.B. Uvarov (1985), Classical Orthogonal Polynomialsof a Discrete Variable, Nauka, Moscow (in Russian).27. A.F. Nikiforov and S.K. Suslov (1986), Classical orthogonal polynomials of a discretevariable on nonuniform lattices, Lett. Math. Phys. 11, 27{34.28. E.G. Kalnins and W. Miller (1989), Symmetry techniques for q-series: Askey-Wilson poly-nomials, Rocky Mtn. J. Math. 19, 223{230.29. I.I. Kachurik and A.U. Klimyk (1989), On Clebsch-Gordan coe�cients of quantum algebraUq(SU2), Preprint, Inst. for Theor. Phys., Kiev.30. N. Ja. Vilenkin and A. U. Klimyk (1992), Representations of Lie Groups and Special Func-tions, Volume 3 (Chapter 14) (russian translation), Kluwer, Dordrecht, The Netherlands.31. L.C. Biedenharn and M. Tarlini (1990), On q-tensor operators for quantum groups, Lettersin mathematical Physics 20, 271{278.Department of Mathematics and Statistics, University of Waikato, Hamilton,

MATRIX ELEMENTS OF Uq(su2) 19New ZealandSchool of Mathematics and Institute for Mathematics and its Applications,University of Minnesota, Minneapolis, Minnesota 55455.E-mail address: miller@ima.umn.eduInstitute for Mathematics and its Applications, University of Minnesota, Min-neapolis, Minnesota 55455.