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Representations of the q -deformed algebras U q ( so 3 ) and U q ( so 5 ) and q - orthogonal polynomials Alexander Rozenblyum Citation: Journal of Mathematical Physics 46, 123508 (2005); doi: 10.1063/1.2146192 View online: http://dx.doi.org/10.1063/1.2146192 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/46/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Jordan algebras and orthogonal polynomials J. Math. Phys. 52, 103512 (2011); 10.1063/1.3653482 Center of quantum algebra U q ′ ( so 3 ) J. Math. Phys. 52, 043521 (2011); 10.1063/1.3579992 Automorphisms of the Heisenberg–Weyl algebra and d -orthogonal polynomials J. Math. Phys. 50, 033511 (2009); 10.1063/1.3087425 Orthogonal polynomials from Hermitian matrices J. Math. Phys. 49, 053503 (2008); 10.1063/1.2898695 Quasiexactly solvable problems and the dual (q-)Hahn polynomials J. Math. Phys. 41, 569 (2000); 10.1063/1.533143 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 202.28.191.34 On: Sat, 20 Dec 2014 02:55:42

Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

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Page 1: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

Representations of the q -deformed algebras U q ( so 3 ) and U q ( so 5 ) and q -orthogonal polynomialsAlexander Rozenblyum Citation: Journal of Mathematical Physics 46, 123508 (2005); doi: 10.1063/1.2146192 View online: http://dx.doi.org/10.1063/1.2146192 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/46/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Jordan algebras and orthogonal polynomials J. Math. Phys. 52, 103512 (2011); 10.1063/1.3653482 Center of quantum algebra U q ′ ( so 3 ) J. Math. Phys. 52, 043521 (2011); 10.1063/1.3579992 Automorphisms of the Heisenberg–Weyl algebra and d -orthogonal polynomials J. Math. Phys. 50, 033511 (2009); 10.1063/1.3087425 Orthogonal polynomials from Hermitian matrices J. Math. Phys. 49, 053503 (2008); 10.1063/1.2898695 Quasiexactly solvable problems and the dual (q-)Hahn polynomials J. Math. Phys. 41, 569 (2000); 10.1063/1.533143

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Page 2: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

Representations of the q-deformed algebras Uq„so3…

and Uq„so5… and q-orthogonal polynomialsAlexander RozenblyumMathematics Department, New York City College of Technology, City Universityof New York, Brooklyn, New York 11201

�Received 3 June 2005; accepted 28 October 2005; published online 29 December 2005�

Orthogonal polynomials related to irreducible representations of the classical typeof the q-deformed algebras Uq�so3� and Uq�so5� are investigated. The main methodconsists in the diagonalization of corresponding infinitesimal operators �generators�of representations. For the algebra Uq�so3� this method leads to q-analogs ofKrawtchouk polynomials. The properties of these polynomials are considered, theq-difference equation, the recurrence and explicit formulas. For the algebraUq�so5�, the diagonalization process of generators of representations leads to theconnection with some class of orthogonal polynomials in two discrete variables.These variables are the so-called q-numbers �n�, where �n�= �qn−q−n� / �q−q−1�.The introduced polynomials can be considered as two-dimensional q-analogs ofKrawtchouk polynomials. The q-difference equation of the Sturm-Liouvilletype for these polynomials is constructed. The corresponding eigenvalues areinvestigated including the explicit formulas for their multiplicities. The structure ofpolynomial solutions is described. © 2005 American Institute of Physics.�DOI: 10.1063/1.2146192�

I. INTRODUCTION

It is well known that representation theory of the group SO�3� of the rotations in three-dimensional Euclidean space relates to different types of classical special functions includingJacobi, Krawtchouk, and Meixner polynomials �see, for example, Refs. 1–3�. It is shown in Ref.4 that representations of the group SO�n� for n�5 �and other high-dimensional groups� lead toorthogonal polynomials in many discrete and continuous variables. In particular, representations ofthe group SO�5� are related to orthogonal polynomials in two variables that can be considered astwo-dimensional analogs of the Krawtchouk and Hermite polynomials.

The aim of the present paper is to describe in a similar manner the relations between therepresentations of the q-deformed algebras Uq�so3� and Uq�so5� and some classes of orthogonalpolynomials in one and two discrete variables. As for the classical groups SO�n� and U�n�, studiedin Refs. 4 and 5, the main method consists in the diagonalization of corresponding infinitesimaloperators �generators� of representations. To do this, we construct the realization of representationspace as the space of polynomials in q-numbers �n�. Orthogonal polynomials we consider hererelate to eigenvectors of generators of representation and can be treated as one- and two-dimensional q-analogs of Krawtchouk polynomials. The results obtained for the group Uq�so3�allow the construction in explicit form of the matrix of the operator connecting the bases in whichtwo generators of a representation are diagonal.

II. REPRESENTATIONS OF THE ALGEBRA Uq„SO3… AND q-KRAWTCHOUKPOLYNOMIALS

Algebra Uq�so3� is the q-deformation of the universal enveloping algebra U�so3�. AlgebraUq�so3� and its irreducible representations of the classical type �i.e., representations that are

JOURNAL OF MATHEMATICAL PHYSICS 46, 123508 �2005�

46, 123508-10022-2488/2005/46�12�/123508/14/$22.50 © 2005 American Institute of Physics

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Page 3: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

q-deformations of irreducible representations of the algebra so3� were constructed in Ref. 6. Itcontains �similar to the algebra so3� three generators J12

q , J23q , and J13

q that satisfy the relations:

�J12q ,J23

q �q � q1/2 · J12q · J23

q − q−1/2 · J23q · J12

q = J13q ,

�J23q ,J13

q �q = J12q , �J13

q ,J12q �q = J23

q .

Let T be an irreducible representation of the algebra Uq�so3�. Then T is defined by a non-negative integer or a half-integer l �the highest weight of the representation�. The representationspace V contains the basis ��m�, m=−l , . . . , l, in which the operators Ijk

q =T�Jjkq � act by the formulas

I12q �m = i�m��m, i = �− 1, m = − l, . . . ,l ,

I13q �m = iq1/2�qmA�m��m+1 + q−mA�m − 1��m−1� ,

I23q �m = A�m��m+1 − A�m − 1��m−1, �1�

where

A�m� = �m��m + 1��2m��2m + 2�

�l − m��l + m + 1�1/2

. �2�

Here the q-numbers �n� are defined as

�n� = �qn − q−n�/�q − q−1� =sinh�n ln q�sinh�ln q�

.

If k=0, we assume that �2k� / �k�=2.The basis ��m� consists of the eigenvectors of the operator I12

q . Similarly, one can consideranother basis in representation space V consisting of the eigenvectors of the operator I23

q . We willconstruct the matrix connecting these two bases, i.e., we will diagonalize the operator I23

q . Themethod used is similar to the one developed for the classical groups SO�n� and U�n� �Refs. 4 and5� and is based on the realization of the representation space as a space of polynomial functions ofa discrete variable. This approach leads to a class of orthogonal polynomials that can be consid-ered as q-analogs of the Krawtchouk polynomials.7,8 We will assume that the highest weight l ofthe representation T is a non-negative integer.

Let L be the space of all complex-valued functions defined on the lattice �−l , l�. Thendim L=2l+1, and L is isomorphic to the representation space V. This isomorphism allows treatingvectors from V as functions of a discrete variable m that runs through the lattice �−l , l�. It followsfrom �1� that operator I23

q acts in the space L by the formula

�I23q f��m� = − A�m�f�m + 1� + A�m − 1�f�m − 1�, f � L . �3�

Let P̃�m� be an eigenfunction of the operator I23q in the space L with the eigenvalue �,

�I23q P̃��m� = � · P̃�m�, m = − l, . . . ,l . �4�

Let us make the following substitutions in Eq. �4�:

P̃�m� = im���m�P�m� , �5�

where

��m� =�2m��m�

1

�l − m�!�l + m�!, �6�

123508-2 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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Page 4: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

�n�! = �n��n − 1� . . . �2��1�, �0�! = 1.

Below we will also use the notations

�2n�!! = �2n��2n − 2� . . . �2�, �2n − 1�!! = �2n − 1��2n − 3� . . . �1� .

The substitution �5� is equivalent to replacing the operator I23q with the operator

J = A−1I23q A ,

where A is the multiplication operator, �Af��m�= im���m�f�m�.Then the function P�m� satisfies the equation �JP��m�=� · P�m�. Using �3� and �2�, this

equation can be written as

�m��2m�

��l − m�P�m + 1� + �l + m�P�m − 1�� = �P�m�, � = �i , �7�

or

1

�qm + q−m��q − q−1���ql−m − q−l+m�P�m + 1� + �ql+m − q−l−m�P�m − 1�� = �P�m� .

Let us denote by Aq the operator from the left-hand part of Eq. �7�,

�Aqf��m� =�m�

�2m���l − m�f�m + 1� + �l + m�f�m − 1�� .

The following statement can be proved.Proposition 1: For any n=0,1 , . . . ,

Aq��m�n� = �l − n��m�n + Qn−2��m�� ,

where Qn−2�t� is a polynomial of t of the degree n−2, n�2; Q−1�t��Q−2�t��0.It follows from Proposition 1 that the operator Aq acts in the space P��m�� of all polynomials

of �m�, and has in this space the eigenvalues of the form

� = �n = �l − n� =ql−n − q−l+n

q − q−1 , n = 0,1, . . . .

Accordingly, the operator I23q is diagonalizable in the representation space V and has eigenvalues,

� = �n = i�n − l� = iqn−l − q−n+l

q − q−1 , n = 0,1, . . . ,2l .

Proposition 1 allows one to realize the representation space V as the set of all polynomials in �m�,defined on the lattice �−l , l�. This realization shows that Eq. �7� has polynomial solutions. Moreexactly, the following statement is true:

Proposition 2: For any n=0,1 , . . . , the equation

�l − m�P�m + 1� + �l + m�P�m − 1� =�2m��m�

�l − n�P�m� �8�

has a solution Pn�m� which is a polynomial of �m� of the degree n.If we let q→1, then Eq. �8� is transformed into the following:

123508-3 Representations of q-deformed algebras J. Math. Phys. 46, 123508 �2005�

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Page 5: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

�l − m�P�m + 1� + �l + m�P�m − 1� = 2�l − n�P�m� .

This equation describes the Krawtchouk polynomials.7 It relates to eigenvectors of the generatorof representation of the group SO�3�.4 We can treat the solutions of Eq. �8� as q-analogs of theKrawtchouk polynomials.

Similar to the classical case, Eq. �8� can be written in terms of finite differences,

�l + m�� � P�m� −�2l��l�

�m��P�m� =�2m��m�

��l − n� − �l��P�m� ,

where ��f��m�= f�m+1�− f�m�, ��f��m�= f�m�− f�m−1�.Equation �8� can also be written in self-adjoint form

�m��2m�

�l − m�!�l + m�!�� 1

�l − m�!�l + m − 1�!� P�m�� = ��l − n� − �l��P�m� .

The following theorem can be proved.Theorem 1: Solutions of Eq. �8� are described by the formula

Pn�m� = k=0

n

�− 1�k �n�!�l − m�!�l + m�!�k�!�n − k�!�l − m − k�!�l + m − n + k�!

= an�m�n + . . . , n = 0,1, . . . . �9�

The leading coefficient an of the polynomial Pn�m� is

an =�2l��l�

�2l − 2��l − 1�

¯

�2l − 2n + 2��l − n + 1�

= �k=0

n−1

�ql−k + q−l+k� .

Formula �9� can also be written in the following symbolic form. Denote

r = �m + l�, s = �m − l� .

Define the symbolic powers

r�k� = �m + l��m + l − 1� ¯ �m + l − k + 1�, r�0� = 1,

s�k� = �m − l��m − l + 1� ¯ �m − l + k − 1�, s�0� = 1.

Then

Pn�m� = �r + s��n� = k=0

n �n

k�r�k�s�n−k�,

where � nk� are q-binomial coefficients,

�n

k� =

�n�!�k�!�n − k�!

.

Here are some examples of polynomials Pn�m�,

P0�m� � 1, P1�m� = �m + l� + �m − l� =�2l��l�

�m� ,

123508-4 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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Page 6: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

P2�m� = �m + l��m + l − 1� + �2��m + l��m − l� + �m − l��m − l + 1�

=�2l��2l − 2�

�l��l − 1��m�2 − �2l� ,

P3�m� = �m + l��m + l − 1��m + l − 2� + �3��m + l��m + l − 1��m − l�

+ �3��m + l��m − l��m − l + 1� + �m − l��m − l + 1��m − l + 2�

=�2l��2l − 2��2l − 4�

�l��l − 1��l − 2��m�3 −

�2l��2l − 2��l��l − 1�

��l + 1� + 2�l − 1���m� .

Below is a list of some properties of polynomials Pn�m�.Proposition 3: Polynomials Pn�m� defined in �9�, satisfy the recurrence relation

Pn+1�m� =�2l − 2n�

�l − n��m�Pn�m� − �n��2l − n + 1�Pn−1�m�, n = 0,1, . . . ; P−1�m� � 0.

Proposition 4: Polynomials Pn�m� are orthogonal on the lattice �−l , l� with the weight �6�,

m=−l

l

Pn�m�Pk�m���m� = 0 if n � k .

Proposition 5: The norm of the polynomial P0�m��1 is equal to

�P0� = m=−l

l�2m��m�

1

�l − m�!�l + m�!1/2

=2�2l�!!

�l�!� �l�

�2l��2l�!. �10�

Formula �10� can be derived from the following formula for the sum of q-binomial coefficients:

k=0

n�n�!

�k�!�n − k�!= �

k=1

n�n + 1 − 2k�

��n + 1 − 2k�/2�= �

k=1

n

�1 + q2k−n−1� . �11�

Formula �11� can also be represented in the form

k=0

n�n�!

�k�!�n − k�!= �2 �n − 1�!!

��n − 1�/2�!2

if n is odd,

�n − 1�!!�1/2��3/2� . . . ��n − 1�/2�

2

if n is even.�Propositions 3 and 5 allow to calculate the norm of the polynomial Pn�m� for arbitrary n.

Proposition 6: The norm of the polynomial Pn�m� defined in �9�, is equal to

�Pn� =2�2l�!!

�l�!� �l − n�

�2l − 2n��n�!

�2l − n�!, n = 0,1, . . . ,2l .

Proposition 7: “Middle” polynomials Pl�m� can also be represented by the formula

Pl�m� =�2l�!!�l�! �

k=1

l

�m − l − 1 + 2k�, �Pl� =�2l�!!�2

�l�!.

Proposition 8: Polynomial Pn�m� has the same parity as its power n,

Pn�− m� = �− 1�nPn�m� .

Proposition 9: Normalized polynomials Pn�m� have the properties

123508-5 Representations of q-deformed algebras J. Math. Phys. 46, 123508 �2005�

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Page 7: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

P2l−n�m� = �− 1�l+mPn�m� ,

Pl+m�n − l� = �− 1�l+m+n� ��m���n − l�

Pn�m� ,

m = − l, . . . ,l; n = 0,1, . . . ,2l; ��m� is defined in �6� .

Propositions 8 and 9 allow to reduce the calculation of polynomials Pn�m� to the values of nand m from 0 to l such that n�m.

Consider the �2l+1�� �2l+1� matrix Pnorm, consisting �in columns� of the normalized poly-nomials Pn�m�,

Pnorm =�P0�− l� P1�− l� ¯ P2l�− l�P0�− l + 1� P1�− l + 1� ¯ P2l�− l + 1�

¯

P0�l� P1�l� ¯ P2l�l�� .

Then the first column of the matrix Pnorm consists of elements

P0�m� ��l�!

2�2l�!!��2l��2l�!

�l�, m = − l, . . . ,l .

The middle column consists of elements

Pl�m� =�2

2 �k=1

l

�m − l + 2k − 1�

= ��− 1��l−m�/2�2

2�l − m − 1�!!�l + m − 1�!! if l − m is even,

0 if l − m is odd.�

Here m=−l , . . . , l; �−1�!!=1.The last column consists of elements

P2l�m� = �− 1�l+m �l�!2�2l�!!

��2l��2l�!�l�

, m = − l, . . . ,l .

The first row consists of elements

Pn�− l� = �− 1�n �l�!�2l − 1�!!2

��2l − 2n��l − n�

1

�n�!�2l − n�!, n = 0,1, . . . ,2l .

The middle row consists of elements

Pn�0� = ��− 1�n/2 �n − 1�!!�l�!2�2l − n�!!

��2l − 2n��l − n�

�2l − n�!�n�!

if n is even,

0 if n is odd.�

The last row consists of elements

123508-6 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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Page 8: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

Pn�l� =�l�!�2l − 1�!!

2��2l − 2n�

�l − n�1

�n�!�2l − n�!, n = 0,1, . . . ,2l .

If we multiply mth row �m=−l , . . . , l� of the matrix Pnorm by im���m�, where ��m� is defined

in �6�, we will get unitary matrix P̃ consisting �in columns� of the orthonormal eigenvectors

P̃0 , P̃1 , . . . , P̃2l of the operator I23q . This matrix connects bases in which operators I12

q and I23q are

diagonal. Eigenvector P̃n corresponds to the eigenvalue �n− l�i of the operator I23q �n

=0,1 , . . . ,2l�.Example: Let l=2. Then

I23q =�

0 − 1 0 0 0

1 0 − a 0 0

0 a 0 − a 0

0 0 a 0 − 1

0 0 0 1 0� , a =��3�

2,

I12q = diag�− �2�i,− i,0,i,�2�i� .

We have I12q = P̃−1I23

q P̃, where P̃ is the unitary matrix,

P̃ =�b c − d c b

− ci ci 0 − ci ci

d 0 e 0 d

ci ci 0 − ci − ci

b − c − d − c b� , P̃−1 = P̃* = P̃

¯ t,

b = −1

2�2�, c =

1

2, d =

�2�3�2�2�

, e = −1

�2�.

III. REPRESENTATIONS OF THE ALGEBRA Uq„SO5… AND q-KRAWTCHOUKPOLYNOMIALS IN TWO VARIABLES

A. Finite-difference equation related to eigenvectors of the infinitesimal operator

Algebra Uq�son� for arbitrary n and its irreducible representations of the classical type werestudied in Ref. 9. Let us rewrite corresponding formulas for n=5. Finite-dimensional irreduciblerepresentations of the algebra Uq�so5� are given by two integral or half-integral numbers n1 and n2

�highest weight�, such that n1�n2�0. We will consider the case of integers. The representationspace V has the dimension

dim V =�2n1 + 3��2n2 + 1��n1 − n2 + 1��n1 + n2 + 2�

6.

The q-analog of Gel’fand-Tsetlin basis in the representation space corresponds to successivereduction of the representation of Uq�so5� to subalgebras Uq�so4�, Uq�so3�, and Uq�so2�. The basisvectors � can be enumerated by the tableaux

= lk

m1,m2 ,

where the components of are integers that satisfy the conditions

123508-7 Representations of q-deformed algebras J. Math. Phys. 46, 123508 �2005�

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n1 � m1 � n2 � m2 � − n2,

m1 � l � m2 � − l ,

l � k � − l . �12�

Vectors � are eigenvectors of the representation’s generator I1,2 which corresponds to the rotationin the plane �e1 ,e2� in the five-dimensional space. The corresponding eigenvalues are =k

= i�k� , k=−n1 , . . . ,n1.Let I4,5 be the generator in the representation space V corresponding to the rotation in the

plane �e4 ,e5�. Then the operator I4,5 acts on the basic vector

� = � lk

m1,m2by the formula

I4,5��m1,m2

l

k� = Ix,y

n1,n2,l��m1 + 1,m2

l

k� − Ix−1,y

n1,n2,l��m1 − 1,m2

l

k�

+ Iy,xn1,n2,l��m1,m2 + 1

l

k� − Iy−1,x

n1,n2,l��m1,m2 − 1

l

k� . �13�

Here

x = m1 + 1, y = m2,

Ix,yn1,n2,l = �x��x + 1��n1 + x + 2��n1 − x + 1��x + n2 + 1��x − n2��x + l + 1��x − l�

�2x��2x + 2��x + y��x − y��x + y + 1��x − y + 1� 1/2

. �14�

We consider the problem of diagonalization of the operator I4,5. Obviously, the operator I4,5

has the same eigenvalues k= i�k�, k=−n1 , . . . ,n1, as the operator I1,2. It is not difficult to showthat the multiplicity dim k of the eigenvalue k in the representation space V is described by theformula

dim k = ��n1 − n2 + 1�n22 − k2 +

�2n2 + 1��n1 − n2 + 2�2

if �k� � n2,

1

2�2n2 + 1��n1 − �k� + 1��n1 − �k� + 2� if n2 � �k� � n1.�

k = − n1, . . . ,n1.

Let us fix the parameters l and k such that �k�� l�n1 and consider the subspace Wkl �V

spanned by the vectors

� = � lk

m1,m2 .

As it is seen from inequalities �12�, the parameters m1 and m2 run over the intervals

123508-8 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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max�n2,l� � m1 � n1,

�15��m2� � min�n2,l� .

Therefore the subspace Wkl has the dimension

dim Wkl = �n1 − max�n2,l� + 1��2 min�n2,l� + 1� .

For fixed l �0� l�n1�, all subspaces Wkl �k=−l , . . . , l� are isomorphic and the representation space

V is decomposed into the direct sum of the subspaces Wkl ,

V = l=0

n1

k=−l

l

Wkl .

It is obvious from �13� that all subspaces Wkl are invariant with respect to the operator I4,5.

Therefore, the problem of diagonalization of the operator I4,5 in the representation space V can bereduced to its diagonalization in each subspace Wk

l . Let us denote basic vectors � belonging to thesubspace Wk

l by ��x ,y�. Here the parameters x and y are defined in �14� and according to �15� runthrough the set of integer points of the rectangle

� = ��x,y��− c � y � c � b � x � a� , �16�

where

a = n1 + 1,

b = max�n2,l� + 1,

c = min�n2,l� .

Operator I4,5 acts in the space Wkl by the formula

I4,5��x,y� = Ax,ya,b,c��x + 1,y� − Ax−1,y

a,b,c ��x − 1,y� + Ay,xa,b,c��x,y + 1� − Ay−1,x

a,b,c ��x,y − 1� ,

where

Ax,ya,b,c = �x��x + 1��a − x��a + x + 1��x − b + 1��x + b��x − c��x + c + 1�

�2x��2x + 2��x + y��x − y��x + y + 1��x − y + 1� 1/2

.

Similar to the case of the group Uq�so3�, we can construct the following functional realizationof the space Wk

l . Consider the space L��� of all complex-valued functions in two discrete variablesx and y, defined on the lattice �16�. Obviously, the space L��� is isomorphic to the space Wk

l , andthe operator I4,5 acts in L��� by the formula

∀ f � L��� ,

�I4,5f��x,y� = − Ax,ya,b,cf�x + 1,y� + Ax−1,y

a,b,c f�x − 1,y� − Ay,xa,b,cf�x,y + 1� + Ay−1,x

a,b,c f�x,y − 1� .

Therefore, we can reduce the diagonalization problem of the operator I4,5 to that in the spaceL���. Let Q�L��� be an eigenfunction of the operator I4,5 with the eigenvalue ,

�I4,5Q��x,y� = · Q�x,y�, �x,y� � � . �17�

Let us make the following substitution in Eq. �17�,

123508-9 Representations of q-deformed algebras J. Math. Phys. 46, 123508 �2005�

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Page 11: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

Q�x,y� = ix+y���x,y�P�x,y� ,

where

��x,y� =�2x��2y��x − y��x + y��x − c − 1�!�x + b − 1�!�b − y − 1�!�b + y − 1�!�x��y��a − x�!�a + x�!�a − y�!�a + y�!�x + c�!�y + c�!�x − b�!�c − y�!

. �18�

Then the function P�x ,y� satisfies the equation

�x��2x�

��a − x��x + b��x − c�P�x + 1,y� + �x + a��x − b��x + c�P�x − 1,y��

+�y�

�2y���a − y��y + b��c − y�P�x,y + 1� + �y + a��b − y��y + c�P�x,y − 1��

= ���x�2 − �y�2�P�x,y�, � = · i . �19�

This equation can also be written in terms of finite differences in self-adjoint form using theoperators

��xf��x,y� = f�x + 1,y� − f�x,y�,��xf��x,y� = f�x,y� − f�x − 1,y� ,

and similarly �y and �y. Equation �19� takes the form

�x��2x�

�a + x�!�a − x�!�x − b�!�x + c�!�x + b − 1�!�x − c − 1�!

�x� �x + b − 1�!�x − c − 1�!�a + x − 1�!�a − x�!�x − b − 1�!�x + c − 1�!

�xP�x,y��+

�y��2y�

�a + y�!�a − y�!�c + y�!�c − y�!�b + y − 1�!�b − y − 1�!

�y� �b + y − 1�!�b − y�!�a + y − 1�!�a − y�!�c + y − 1�!�c − y�!

�yP�x,y��= ���x�2 − �y�2�P�x,y�, � = � − �a − b + c� . �20�

If we let q→1 �case of the classical group SO�5��, Eq. �20� becomes

�a + x�!�a − x�!�x − b�!�x + c�!�x + b − 1�!�x − c − 1�!

�x� �x + b − 1�!�x − c − 1�!�a + x − 1�!�a − x�!�x − b − 1�!�x + c − 1�!

�xP�x,y��+

�a + y�!�a − y�!�c + y�!�c − y�!�b + y − 1�!�b − y − 1�!

�y� �b + y − 1�!�b − y�!�a + y − 1�!�a − y�!�c + y − 1�!�c − y�!

�yP�x,y��= 2��x2 − y2�P�x,y� . �21�

This equation describes eigenvectors of infinitesimal operators of irreducible representations of thegroup SO�5�. As is shown in Ref. 4, Eq. �21� can be considered as a two-dimensional analog of theequation for Krawtchouk polynomials. In turn, Eq. �20� can be treated as two-dimensionalq-analog of the equation for Krawtchouk polynomials.

Let us make the following substitution in Eq. �21�: x=h−1x1, y=h−1x2, where x1 and x2 are newvariables, h 0. If we let h→0, and a ,b ,c→� such that ah2→1, bh→, b−c→s, then thediscrete equation �21� is transformed into the following differential equation:

1

x22 − x1

2ex12�x1

2 − 2�−s �

�x1�e−x1

2�x1

2 − 2�s+1�P�x1,x2��x1

�+

1

x12 − x2

2ex22�x2

2 − 2�−s �

�x2�e−x2

2�x2

2 − 2�s+1�P�x1,x2��x2

� = 2� · P�x1,x2� . �22�

Equation �22� can be considered as a two-dimensional analog of the equation for Hermite poly-

123508-10 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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nomials. The explicit formulas for the complete set of polynomial solutions of Eq. �22� were foundin Ref. 4.

B. Orthogonality of solutions of Eq. „19…

Let A be the operator

AP�x,y� = u1�x,y��x�v1�x,y��xP�x,y�� + u2�x,y��y�v2�x,y��yP�x,y�� ,

� be the lattice

� = � � x � �,� � y � �� ,

and ��x ,y� be a positive function on �. Consider the Euclidean space L���� of all functions on �with the scalar product

�f ,g� = x=

y=�

f�x,y�g�x,y���x,y� .

Proposition 10: Suppose that

�1� ��x ,y�u1�x ,y� does not depend on x.�2� ��x ,y�u2�x ,y� does not depend on y.�3� v1� ,y��v1��+1,y��v2�x ,���v2�x ,�+1��0.

Then the operator A acts in the space L����, and is self-adjoint in it:

�Af ,g� = �f ,Ag�, ∀ f , g � L���� .

This proposition can be applied to Eq. �19�. Let B be the operator in the left part of �19�. Byrepresenting Eq. �19� in self-adjoint form �20�, it is easy to verify that the operator

A = ��x�2 − �y�2�−1B �23�

satisfies the conditions of Proposition 10, where the weight function ��x ,y� is defined in �18�, andthe lattice � is in �16�. Therefore, the solutions of Eq. �19� corresponding to distinct eigenvaluesare orthogonal on the lattice �16� with the weight �18�.

C. Spectrum and structure of the eigenfunctions of Eq. „19…

Let A be the operator defined in �23�.Proposition 11: Let m and n be non-negative integers, having the same parity, and m�n. Then

A��x�m�y�n + �x�n�y�m� = �n��x�m�y�n + �x�n�y�m� + Q�x,y� ,

where �n= �a−b+c−n�, Q is a symmetric polynomial in �x� and �y�, such that all powers of �x�and �y� have the same parity as n, and are less than n. In particular, if n=0 or n=1, then Q�0.

Consider the liner space S of all symmetric polynomials in �x� and �y� such that ∀f �S anymonomial containing in f has powers of �x� and �y� of the same parity. It follows from Proposition11 that the operator A acts in the space S.

Proposition 12: Operator A is diagonalizable in the space S and has in it the eigenvalues

�n = �a − b + c − n�, n = 0,1, . . . .

The multiplicity of �n is equal to the integer part of �n+2� /2.Let S��� be the set of all polynomials from the space S restricted onto the lattice �16�.Proposition 13: The space S��� is isomorphic to the space L��� of all functions on the lattice

�.

123508-11 Representations of q-deformed algebras J. Math. Phys. 46, 123508 �2005�

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Page 13: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

Realization of the space L��� as a space of symmetric polynomials S��� allows one to provethe following main result regarding the multiplicities of eigenvalues of Eq. �19� and operator I4,5.

Theorem 2: In the space L���, Eq. �19� has 2�a−b+c�+1 distinct eigenvalues of the form

� = �n = �a − b + c − n�, n = 0,1, . . . ,2�a − b + c� .

The multiplicity of the values �n is equal to the number of integer points �r ,s� on the line r+2s=n that lie in the rectangle

� = ��r,s��0 � r � 2c, 0 � s � a − b� . �24�

The explicit formulas for the multiplicities dim �n are as follows. If c�a−b then

dim �n =��n + 2

2� if 0 � n � 2c ,

c +�− 1�n + 1

2if 2c � n � 2�a − b� ,

a − b + c − � n − 1

2� if 2�a − b� � n � 2�a − b + c� .

�Here �k� means the integer part of the number k. The multiplicity of eigenvalues �n can berepresented by the diagram

If c�a−b then

dim �n =��n + 2

2� if 0 � n � 2�a − b� ,

a − b + 1 if 2�a − b� � n � 2c ,

a − b + c − � n − 1

2� if 2c � n � 2�a − b + c� .�

The multiplicity of eigenvalues �n can be represented by the diagram

Proof: Consider the following two-parametric family of symmetric polynomials in �x� and �y�:

f �r,s��x,y� = ��r,s��x� · ��r,s��y� ,

where

123508-12 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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Page 14: Representations of the q-deformed algebras U[sub q](so[sub 3]) and U[sub q](so[sub 5]) and q-orthogonal polynomials

��r,s��x� = �x��1−�− 1�r�/2�m=1

k

��x�2 − �c − m + 1�2��n=1

s

��x�2 − �a − n + 1�2� ,

�r ,s���, k is an integral part of r /2. It is obvious that f �r,s��S��� and the highest degrees of �x�and �y� in f �r,s� are equal to r+2s. It is not difficult to show that all polynomials f �r,s� are linearindependent, and, therefore, form a basis in the space S���. It follows from Proposition 11 that

Af �r,s� = �r+2sf�r,s� + g ,

where g is a polynomial from S��� having the highest degree of �x� less than r+2s. From here, foreach polynomial f �r,s� we can put into correspondence the eigenfunction P�r,s� of the operator Awith the eigenvalue �r+2s of the form

P�r,s� = f �r,s� + Q , �25�

where Q�S���, and the highest degree of �x� in Q is less than r+2s. All eigenfunctions P�r,s� arelinear independent and form a basis in S��� when parameters r and s run through the lattice �24�.Polynomials P�r,s� in �25� can be described by the projection operator,

P�r,s��x,y� = �k=0

r+2s−1A − �kE

�r+2s − �kf �r,s��x,y�, �r,s� � � , �26�

where E is an identical operator. All eigenfunctions �26� correspond to the same eigenvalue �n ifr+2s=n. As follows from �24�, the maximum value of n is 2�a−b+c�. Theorem 2 is proved.

Corollary: Generator I4,5 of the representation of the algebra Uq�so5� has the following dis-tinct eigenvalues in the subspace Wk

l :

= n = �n − n1 + �n2 − l�� · i, n = 0,1, . . . ,2�n1 − �n2 − l�� .

The multiplicity of n is the same as multiplicity of �n in Theorem 2.One possible basis of solutions of Eq. �19� is indicated in �26�. The general structure of

solutions and another bases are described by the following theorem.Theorem 3: In the space L���, Eq. �19� has �a−b+1��2c+1� linearly independent solutions.

The solutions are symmetric polynomials in �x� and �y�, which form a basis in the space S���.Solutions belonging to different eigenvalues are orthogonal on the lattice �16� with the weight�18�. A basis of solutions of Eq. �19� can be obtained by the orthogonalization of the sequence

Another basis can be constructed by the orthogonalization of the sequence

ACKNOWLEDGMENT

This work was supported in part by a grant from The City University of New York PSC-CUNY Research Award Program Grant No. 66523-00 35.

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4 A. V. Rozenblyum, Acta Appl. Math. 29, 171 �1992�.

123508-13 Representations of q-deformed algebras J. Math. Phys. 46, 123508 �2005�

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123508-14 Alexander Rozenblyum J. Math. Phys. 46, 123508 �2005�

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