Representations of the Quantum Algebra su_q(1,1) and Discrete q-Ultraspherical Polynomials

We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthonormal basis in the representation space.


V. Groza
For positive values of l the representations T + l are * -representations. For studying discrete qultraspherical polynomials we use the representations T + l for which q 2l−1 = −a, a > 0. They are not * -representations. But we shall use operators of these representations which are symmetric or self-adjoint. Note that q l is a pure imaginary number.
2 Discrete q-ultraspherical polynomials and their duals There are two types of discrete q-ultraspherical polynomials [4]. The first type, denoted as C (a) n (x; q), a > 0, is a particular case of the well-known big q-Jacobi polynomials. For this reason, we do not consider them in this paper. The second type of discrete q-ultraspherical polynomials, denoted asC where A n = 1 + aq n+1 1 + aq 2n+1 , C n = A n − 1 = aq n+1 (1 − q n ) 1 + aq 2n+1 .
Note that A n ≥ 1 and, hence, coefficients in the recurrence relation (2) satisfy the conditions A n C n+1 > 0 of Favard's characterization theorem for n = 0, 1, 2, . . .. This means that these polynomials are orthogonal with respect to a positive measure. Orthogonality relation for them is derived in [4]. We give here an approach to this orthogonality by means of operators of representations T + l of su q (1, 1). Dual to the polynomials C  n (µ(x; a)|q), where µ(x; a) = q −x + aq x+1 . These polynomials are a particular case of the dual big q-Jacobi polynomials, studied in [5], and we do not consider them. Dual to the polynomialsC (a) n (x; q) are the polynomials For a > 0 these polynomials satisfy the conditions of Favard's theorem and, therefore, they are orthogonal. We derive an orthogonality relation for them by means of operators of representations T + l of su q (1, 1).

Representation operators I and J
Let T + l be the irreducible representation of su q (1, 1) with lowest weight l such that q 2l−1 = −a, a > 0 (note that a can take any positive value). We consider the operator of the representation T + l , where α = (a 2 /q) 1/2 (1 − q) and We have the following formula for the symmetric operator I: I |n = a n |n + 1 + a n−1 |n − 1 , The operator I is bounded. We assume that it is defined on the whole representation space H. This means that I is a self-adjoint operator. Actually, I is a Hilbert-Schmidt operator since a n+1 /a n → q 1/2 when n → ∞. Thus, a spectrum of I is simple (since it is representable by a Jacobi matrix with a n = 0), discrete and have a single accumulation point at 0 (see [6, Chapter VII]).
To find eigenvectors ψ λ of the operator I, Iψ λ = λψ λ , we set Acting by I upon both sides of this relation, one derives n β n (λ)(a n |n + 1 + a n−1 |n − 1 ) = λ β n (λ)|n , where a n are the same as in (5). Collecting in this identity factors at |n with fixed n, we obtain the recurrence relation for the coefficients β n (λ): a n β n+1 (λ) + a n−1 β n−1 (λ) = λβ n (λ). Making the substitution we reduce this relation to the following one It is the recurrence relation (2) for the discrete q-ultraspherical polynomialsC n (λ; q) and For the eigenfunctions ψ λ (x) we have the expansion Since a spectrum of the operator I is discrete, only a discrete set of these functions belongs to the Hilbert space H and this discrete set determines the spectrum of I. We intend to study the spectrum of I. It can be done by using the operator In order to determine how this operator acts upon the eigenvectors ψ λ , one can use the qdifference equation for the discrete q-polynomials polynomials. Multiply both sides of (8) by d n |n , where d n are the coefficients ofC n (λ; q) in the expression (6) for the coefficients β n (λ), and sum up over n. Taking into account formula (7) and the fact that J |n = (q −n − a 2 q n+1 ) |n , one obtains the relation which is used below.
4 Spectrum of I and orthogonality of discrete q-ultraspherical polynomials Let us analyse a form of the spectrum of I by using the representations T + l of the algebra su q (1, 1) and the method of paper [7]. If λ is a spectral point of I, then (as it is easy to see from (9)) a successive action by the operator J upon the eigenvector ψ λ leads to the vectors ψ q m λ , m = 0, ±1, ±2, . . .. However, since I is a Hilbert-Schmidt operator, not all these points can belong to the spectrum of I, since q −m λ → ∞ when m → ∞ if λ = 0. This means that the coefficient λ −2 (λ 2 − a 2 q 2 ) at ψ q −1 λ in (9) must vanish for some eigenvalue λ. There are two such values of λ: λ = aq and λ = −aq. Let us show that both of these points are spectral points of I. We havẽ n (aq; q) = 2 φ 1 q −n , a 2 q n+1 ; −aq; q, q = a 2 q n(n+1) .
It is made in the same way as in the case of big q-Jacobi polynomials in paper [7]. The above inequality shows that ψ aq 2 is an eigenvector of I and the point aq 2 belongs to the spectrum of I. Setting λ = aq 2 in (9) and acting similarly, one obtains that ψ aq 3 is an eigenvector of I and the point aq 3 belongs to the spectrum of I. Repeating this procedure, one sees that ψ aq n , n = 1, 2, . . ., are eigenvectors of I and the set aq n , n = 1, 2, . . ., belongs to the spectrum of I. Likewise, one concludes that ψ −aq n , n = 1, 2, . . ., are eigenvectors of I and the set −aq n , n = 1, 2, . . ., belongs to the spectrum of I. Let us show that the operator I has no other spectral points.
The vectors ψ aq n and ψ −aq n , n = 1, 2, . . ., are linearly independent elements of the representation space H. Suppose that aq n and −aq n , n = 1, 2, . . ., constitute the whole spectrum of I. Then the set of vectors ψ aq n and ψ −aq n , n = 1, 2, . . ., is a basis of H. Introducing the notations Ξ n := ψ aq n+1 and Ξ ′ n := ψ −aq n+1 , n = 0, 1, 2, . . ., we find from (9) that As we see, the matrix of the operator J in the basis Ξ n , Ξ ′ n , n = 0, 1, 2, . . ., is not symmetric, although in the initial basis |n , n = 0, 1, 2, . . ., it was symmetric. The reason is that the matrix M := (a mn a ′ m ′ n ′ ) with entries where β m (dq n+1 ), d = ±a, are coefficients (6) in the expansion is not unitary. (This matrix M is formed by adding the columns of the matrix (a ′ m ′ n ′ ) to the columns of the matrix (a mn ) from the right.) It maps the basis {|n } into the basis {ψ aq n+1 , ψ −aq n+1 } in the representation space. The nonunitarity of the matrix M is equivalent to the statement that the basis Ξ n , Ξ n , n = 0, 1, 2, . . ., is not normalized. In order to normalize it we have to multiply Ξ n by appropriate numbers c n and Ξ ′ n by numbers c ′ n . Let Ξ n = c n Ξ n ,Ξ ′ n = c ′ n Ξ n , n = 0, 1, 2, . . . , be a normalized basis. Then the operator J is symmetric in this basis and has the form The symmetricity of the matrix of the operator J in the basis {Ξ n ,Ξ ′ n } means that for coefficients c n we have the relation . The relation for c ′ n coincides with this relation. Thus, This means that c n = C q n (−a 2 q 2 ; q 2 ) n (q 2 ; q 2 ) n 1/2 , c ′ n = C ′ q n (−a 2 q 2 ; q 2 ) n (q 2 ; q 2 ) n 1/2 , where C and C ′ are some constants.

V. Groza
Therefore, in the expansionŝ Ξ n ≡ mâ mn |m ,Ξ n (x) ≡ mâ ′ mn |m the matrixM := (â mnâ ′ mn ) with entriesâ mn = c n β m (aq n ) andâ ′ mn = c n β m (cq n ) is unitary, provided that the constants C and C ′ are appropriately chosen. In order to calculate these constants, one can use the relations ∞ m=0 |â mn | 2 = 1 and ∞ m=0 |â ′ mn | 2 = 1 for n = 0. Then these sums are multiples of the sum in (10), so we find that The orthogonality of the matrixM ≡ (â mnâ ′ mn ) means that Substituting the expressions forâ mn andâ ′ mn into (12), one obtains the relation This identity must give an orthogonality relation for the discrete q-ultraspherical polynomials C m (y; q). An only gap, which appears here, is the following. We have assumed that the points aq n+1 and −aq n+1 , n = 0, 1, 2, . . ., exhaust the whole spectrum of the operator I. If the operator I would have other spectral points x k , then on the left-hand side of (13) would appear other summands µ x kC m ′ (x k ; q), which correspond to these additional points. Let us show that these additional summands do not appear. For this we set m = m ′ = 0 in the relation (13) with the additional summands. This results in the equality In order to show that k µ x k = 0, take into account formula (2.10.13) in [1]. By means of this formula it is easy to show that the relation (14) without the summand k µ x k is true. Therefore, in (14) the sum k µ x k does really vanish and formula (13) gives an orthogonality relation for the discrete q-ultraspherical polynomials. The relation (13) and the results of Chapter VII in [6] shows that the spectrum of the operator I coincides with the set of points aq n+1 , −aq n+1 , n = 0, 1, 2, . . ..
This orthogonality relation coincides with the sum of two orthogonality relations (9) and (10) in [4]. The orthogonality measure in (15) is not extremal since it is a sum of two extremal measures.