Matrix-valued little q-Jacobi polynomials

arXiv:1308.2540v2 [math.CA] 5 Sep 2014 Matrix-Valued Little q-Jacobi Polynomials Noud Aldenhoven ∗1, Erik Koelink †1, and Ana M. de los R´ıos ‡2 1Radboud Universiteit, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands. 2Departamento de An´alisis Matem´atico, Universidad de Sevilla, Apdo (P. O. BOX) 1160, 41080 Sevilla, Spain Dedicated to Dick Askey on the occasion of his 80th birthday with admiration for Dick’s achievements in special functions and mathematics education Abstract Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2×2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao’s matrix-valued hypergeometric differential operator. 1 Introduction Matrix-valued orthogonal polynomials were originally introduced by M.G.Kre˘ın in 1949, initially studying the corresponding moment problem, see references in [3, 5], and to study differential operators and their deficiency indices, see also [25]. Since then an increasing amount of authors are contributing to build up a general theory of matrix-valued orthogonal polynomials (see for example [7, 14, 17, 23, 27], etc.). In the study of matrix-valued orthogonal polynomials the general theory deals with obtaining appropriate analogues of classical results known for (scalar-valued) orthogonal polynomials, and many results and proofs have been generalized in this direction, see [6, 7] and the overview paper [5]. But also new features that do not hold in the scalar theory have been discovered, like the existence of different second order differential equations satisfied by a family of matrix orthogonal polynomials, see [10, 23]. The theory of matrix-valued orthogonal polynomials has also turned out to be a fruitful tool in the solution of higher order recurrence relations, see [12, 15]. For orthogonal polynomials the theory is complemented by many explicit families of orthogonal polynomials, notably the ones in the Askey scheme and its q-analogue, see [21, 22], which have turned out to be very useful in many different contexts, such as mathematical physics, representation theory, combinatorics, number theory, etc. The orthogonal polynomials in the (q-)Askey scheme are characterized by being eigenfunctions of a suitable second order differential or difference operator, so that all these families correspond to solutions of a bispectral problem. E.g., for the Jacobi polynomials this is the hypergeometric differential operator and for the little q-Jacobi polynomials this is the q-hypergeometric difference operator, see also [13, 20]. This is closely related to Bochner’s 1929 classification theorem of second order differential operators having polynomial eigenfunctions, see [20] for extensions and references. For matrix-valued orthogonal polynomials there is no classification result of such type known, so that we have to study the properties of specific examples of families of matrix-valued orthogonal polynomials. Already many examples are known, either from scratch, [11], related to representation theory ∗n.aldenhov[email protected] †e.koeli[email protected] ‡[email protected] 1 e.g. [16, 23, 24] or motivated from spectral theory [15]. In most of these papers, the matrix-valued orthogonal polynomials are eigenfunctions of a second order matrix-valued differential operator, so that these polynomials are usually considered as matrix-valued analogues of suitable polynomials from the Askey-scheme. The matrix-valued differential operator is often of the type of the matrix-valued hypergeometric differential operator of Tirao [28] and this makes it possible to express matrix-valued orthogonal polynomials in terms of the matrix-valued hypergeometric functions, see e.g. [24] for an example. More recently, in [2] the step has been made to use matrix-valued difference operators and consider corresponding matrix-valued orthogonal polynomials as eigenfunctions. Again these matrix-valued orthogonal polynomials can be seen as analogues of orthogonal polynomials from the Askey-scheme. In this paper, motivated by [2], we study a specific case of matrix-valued orthogonal polynomials which are analogues of the little q-Jacobi polynomials, moving from analogues of classical discrete orthogonal polynomials to orthogonal polynomials on a q-lattice. As far as we are aware, these matrix-valued orthogonal polynomials are a first example of the matrix-valued analogue of a family of polynomials in the q-Askey scheme. An essential ingredient in the study of these matrix-valued little q-Jacobi polynomials is the second order q-difference operator (3.2). In particular this gives the possibility to introduce and employ matrix-valued basic hypergeometric series in the same spirit as Tirao [28], which differs from the approach of Conflitti and Schlosser [4]. The content of the paper are as follows. In Section 2 we recall the basics of the (scalar-valued) little q-Jacobi polynomials and the general theory of matrix-valued orthogonal polynomials. In Section 3 we study the matrix-valued second order q-difference equations as well as under which conditions such an operator is symmetric for a suitable matrix-weight function. In Section 4 we study the relevant qanalogue of Tirao’s [28] matrix-valued hypergeometric functions. In Section 5 the 2 ×2-matrix-valued little q-Jacobi polynomials are studied in detail. In particular, we give explicit orthogonality relations, the moments, the matrix-valued three-term recurrence relation, expressions in terms of the matrix-valued basic hypergeometric function, the link to the scalar little q-Jacobi polynomials, and the Rodrigues formula for these family of polynomials. It would be interesting to find a group theory interpretation of these matrix-valued little q-Jacobi polynomials along the lines of [16, 23, 24] in the quantum group setting. 2 Preliminaries 2.1 Basic hypergeometric functions We recall some of the definitions and facts about basic hypergeometric functions, see Gasper and Rahman [13]. We fix 0 < q < 1. For a∈Cthe q-Pochhammer symbol is defined recursively by (a;q)0= 1 and (a;q)n= (1 −aqn−1)(a;q)n−1,(a;q)−n=1 (aq−n;q)n , n ∈N={0,1,2,...}. The infinite q-Pochhammer symbol is defined as (a;q)∞= ∞ Y k=0 (1 −aqk). For a1,...,aℓ∈Cwe use the abbreviation (a1, a2, ..., aℓ;q)n=Qℓ i=1(ai;q)n. The basic hypergeometric series r+1φrwith parameters a1,...,ar+1, b1,...,br∈C, base qand variable zis defined by the series r+1φra1, a2,...,ar+1 b1, b2,...,br ;q, z= ∞ X k=0 (a1, a2,...,ar+1;q)k (q;q)k(b1, b2,...br;q)k zk,|z|<1. The q-derivative Dqof a function fat z6= 0 is defined by (Dqf)(z) = f(z)−f(qz) (1 −q)z, 2 and (Dqf)(0) = f′(0), provided that f′(0) exists. Two useful formulas are the q-Leibniz rule [13, Exercise 1.12.iv] Dn q(fg)(z) = n X k=0 n kq Dn−k qf(qkz)Dk qg(z),(2.1) and the formula (Dn qf)(z) = 1 (1 −q)nq(n 2)zn n X j=0 (−1)n−jn jq q(n−j 2)f(qjz),(2.2) where the q-binomial coefficient n kqis given by n kq =(q;q)n (q;q)k(q;q)n−k . The q-integral of a function fis defined as Z1 0 f(z)dqz= (1 −q) ∞ X k=0 f(qk)qk, whenever the series converges. The q-analogue of the fundamental theorem of calculus states Z1 0Dqf(z)dqz=f(qx)|0 x=∞=f(1) −f(0),(2.3) whenever all the limits converge. 2.2 The little q-Jacobi polynomials Let 0 < a < q−1and b < q−1. The little q-Jacobi polynomials are the polynomials defined by pn(z;a, b;q) = 2φ1q−n, abqn+1 aq ;q, qz.(2.4) The little q-Jacobi polynomials have been introduced by Andrews and Askey [1], see also [13, §7.3] and [21, §14.12]. These polynomials satisfy the following orthogonality relation hpm(z, a, b;q), pn(z, a, b;q)i= ∞ X k=0 (aq)k(bq;q)k (q;q)k pm(qk;a, b;q)pn(qk;a, b;q) (2.5) =(abq2;q)∞ (aq;q)∞ (1 −abq)(aq)n (1 −abq2n+1) (q, bq;q)n (aq, abq;q)n δm,n =hn(a, b;q)δm,n, where δm,n is the Kronecker delta function and hn(a, b;q)>0. If we need to emphasize the dependence on aand bwe write h·,·i(a,b). The moments of the little q-Jacobi polynomials are given by mn(a, b) = hzn,1i(a,b)=(abqn+2;q)∞ (aqn+1;q)∞ .(2.6) The sequence of the little q-Jacobi polynomials satisfies the three term recurrence relation −zpn(z;a, b;q) = Anpn+1(z;a, b;q)−(An+Cn)pn(z;a, b;q) + Cnpn−1(z;a, b;q) (2.7) with An=qn(1 −aqn+1)(1 −abqn+1) (1 −abq2n+1)(1 −abq2n+2), Cn=aqn(1 −qn)(1 −bqn) (1 −abq2n)(1 −abq2n+1). They are also eigenfunctions of the second order q-difference operator λnpn(z) = a(bq −z−1)(E1pn)(z)−((abq + 1) −(1 + a)z−1)(E0pn)(z) + (1 −z−1)(E−1pn)(z),(2.8) where λn=q−n(1 −qn)(1 −abqn+1), pn(z) = pn(z;a, b;q) and Eℓare the q-shift operators defined by (Eℓp)(z) = p(qℓz). 3 2.3 Matrix-valued orthogonal polynomials We review here some basic concepts of the theory of matrix-valued orthogonal polynomials, also see [5, 18, 26]. A matrix-valued polynomial of size N∈Nis a polynomial whose coefficients are elements of MatN(C). If no confusion is possible we will omit the size parameter Nand write P[z] for the space of matrix polynomials with coefficients in MatN(C) and Pn[z] for polynomials in P[z] of degree at most n. The orthogonality will be with respect to a N×Nweight matrix W, that is a matrix of Borel measures supported on a common set of the real line S, such that the following is satisfied: 1. for any Borel set A⊆Sthe matrix W(A) = RAdW (z) is positive semi-definite, 2. Whas finite moments of every order, i.e. RSzndW (z) is finite for all n≥0, 3. if Pis a matrix-valued polynomial with non-singular leading coefficient then RSP(z)dW (z)P∗(z) is also non-singular. A weight matrix Wdefines a matrix-valued inner product on the space P[z] by hP, Qi=ZS P(z)dW (z)Q∗(z)∈MatN(C). Note that for every matrix-valued polynomial Pwith non-singular leading coefficient, hP, P iis positive definite. A sequence of matrix-valued polynomials (Pn)n≥0is called orthogonal with respect to the weight matrix Wif 1. for every n≥0 we have dgr(Pn) = nand Pnhas non-singular leading coefficient, 2. for every m, n ≥0 we have hPm, Pni= Γmδm,n, where Γmis a positive definite matrix. Given a weight matrix Wthere always exists a unique sequence of polynomials (Pn)n≥0orthogonal with respect to Wup to left multiplication of each Pnby a non-singular matrix, see [5, Lemma 2.2 and Lemma 2.7] or [18]. We say that a matrix-valued orthogonal polynomials sequence (Pn)n≥0is orthonormal if Γn=Ifor all n≥0. We call (Pn)n≥0monic if every Pnis monic, i.e. the leading coefficient of Pnis the identity matrix. A weight matrix Wwith support Sis said to be reducible to scalar weights if there exists a non-singular matrix K, independent of z, and a diagonal matrix D(z) = diag(w1(z), w2(z),...,wN(z)) such that for all z∈S W(z) = K D(z)K∗. In this case the orthogonal polynomials with respect to W(z) are of the form Pn(z) =      pn,1(z) 0 ··· 0 0pn,2(z)··· 0 . . .. . ..... . . 0 0 ··· pn,N (z)      K−1 where (pn,i)nare the orthogonal polynomials with respect to Di,i(z) = wi(z) for i= 1,...,N. Therefore weight matrices that reduce to scalar weights can be viewed as a set of independent scalar weights, so they are not interesting for the theory of matrix orthogonal polynomials. In this paper Sis countable and assuming additionally that W(a) = Ifor some a∈S, by [19, Theorem 4.1.6] the weight matrix W can be reduced to scalar weights if and only if W(x)W(y) = W(y)W(x) for all x, y ∈S, also see [2, p. 43]. In the rest of this paper we only consider weight matrices such that dW (z) = 1 1−qW(z)dqzand we assume that W(qn)>0 for all n∈N. These weight matrices are called q-weight matrices or just q-weights. The matrix-valued inner product defined by such a q-weight is of the form hP, QiW=1 1−qZ1 0 P(z)W(z)Q∗(z)dqz= ∞ X n=0 qnP(qn)W(qn)(Q(qn))∗,(2.9) whenever the series converges termwise. 4 3q-Difference operators In order to study matrix-valued analogues of the little q-Jacobi polynomials appearing in the q-Askey scheme we focus our attention on operators of the form D=E−1F−1+E0F0+E1F1,(3.1) where Fℓ(z) are matrix-valued polynomials in z−1satisfying certain degree conditions assuring the preservation of the polynomials, cf (2.8). In particular we are interested in operators having families of matrixvalued polynomials as eigenfunctions, (DPn)(z) = Pn(q−1z)F−1(z) + Pn(z)F0(z) + P(qz)F1(z) = ΛnPn(z).(3.2) It is important to notice that the coefficients Fℓappear on the right whereas the eigenvalue matrix Λn appears on the left, cf. [8]. 3.1 q-Difference operators preserving polynomials Suppose that there is a family of solutions of matrix-valued orthogonal polynomials to (3.2), then D preserves polynomials and does not raise the degree of a polynomial. Theorem 3.1 characterizes the q-difference operators with polynomial coefficients in z−1preserving polynomials of degree nfor all n. Theorem 3.1 is an analogue of [2, Lemma 2.2] and [9, Lemma 3.2], where the proof is a slight adaptation of [2, Lemma 2.2] and [9, Lemma 3.2]. Theorem 3.1. Let D= r X ℓ=s EℓFℓ, Fℓ∈Pn[z−1] with r, s integers such that s≤r. The following conditions are equivalent: 1. D:Pn[z]→Pn[z] for all n≥0. 2. Fℓ(z)∈Pr−s[z−1] for ℓ=s,...,r and Pr ℓ=sqℓkFℓ(z)∈Pk[z−1] for k= 0,...,r−s. To prove Theorem 3.1 we use Lemma 3.2. Lemma 3.2. Let r, s and nbe integers such that s≤rand 0 ≤n. Let Gk(z) be matrix-valued polynomials in z−1of degree at most nfor k= 0,...,r−s. The system of linear equations r X ℓ=s qℓkFℓ(z) = Gk(z),0≤k≤r−s, determines the functions Fℓ(z), ℓ=s, . . . , r, uniquely as polynomials in z−1of degree at most n. The proof is straightforward, see [9, Lemma 2.1], using the Vandermonde matrix. Proof of Theorem 3.1. First we prove 1 ⇒2. For k= 0,...,r −s, let Gk(z) = Pr ℓ=sqℓkFℓ(z). If 0≤n≤r−s, write P(z) = znI. Then (DP )(z) = r X ℓ=s (qℓz)nFℓ(z) = zn r X ℓ=s qℓnFℓ(z) = znGn(z)∈Pn[z]. Since DP is a polynomial with dgr(DP )≤dgr(P), Gnis a polynomial in z−1of degree at most nfor 0≤n≤r−s. By Lemma 3.2 Fℓare actually polynomials in z−1of degree at most r−s. To prove 2 ⇒1, consider Gk(z) = Pr ℓ=sqkℓFℓ(z), for k≥0. Now Gkis a matrix-valued polynomial in z−1. If 0 ≤k≤r−sthen, by 2, dgr(Gk)≤kand if k≥r−s+ 1 then dgr(Gk)≤r−s. Now put P(z) = znIso DP (z) = r X ℓ=s (qℓz)nFℓ(z) = zn r X ℓ=s qℓnFℓ(z). If 0 ≤n≤r−swe have that Pr ℓ=sqℓnFℓ(z) is a polynomial in z−1of degree at most n. Hence DP is a polynomial of degree at most n. On the other hand if n≥r−sthen Pr l=sqℓnFℓ(z) is a matrix-valued polynomial in z−1of degree at most r−s. Hence DP is also a polynomial in zof degree at most n. 5 3.2 The symmetry equations Symmetry is a key concept when looking for weight matrices having matrix-valued orthogonal polynomials as eigenfunctions of a suitable q-difference operator. In Definition 3.3 we use the notation (2.9). Definition 3.3. An operator D:P[z]→P[z] is symmetric with respect to a weight matrix Wif hDP, QiW=hP, DQiWfor all P, Q ∈P[z]. Theorem 3.4 is a well-known result relating symmetric operators and matrix-valued orthogonal polynomials, see [8, Lemma 2.1] and [18, Proposition 2.10 and Corollary 4.5]. Theorem 3.4. Let Dbe a q-difference operator preserving P[z], so that dgr(DP )≤dgr(P) for any P∈P[z]. If Dis symmetric with respect to a weight matrix Wthen there exists a sequence of matrixvalued orthonormal polynomials (Pn)n≥0and a sequence of Hermitian matrices (Λn)n≥0such that DPn= ΛnPn,∀n≥0.(3.3) Conversely if W(z) is a weight matrix and (Pn)n≥0a sequence of matrix-valued orthonormal polynomials such that there exists a sequence of Hermitian matrices (Λn)n≥0satisfying (3.3), then Dis symmetric with respect to W. It should be observed that Theorem 3.5 is an analogue of a similar statement for differential operators in [8, 18]. Also note that the q-difference operator Dhas has polynomial coefficients in z−1(instead of z), so that the essential condition is the preservation of the space of polynomials instead of the degree condition, which is the essential condition in [8, 18]. Theorem 3.5 is an analogue of [11, Theorem 3.1] and [17, Section 4] for symmetric second order differential operators. Theorem 3.5. Let Dbe a q-difference operator preserving P[z] of the form (3.1) with Fℓmatrix-valued polynomials in z−1. Let Wbe a q-weight matrix as in (2.9). Suppose that the coefficients Fℓand the weight matrix Wsatisfy the following equations F0(qx)W(qx) = W(qx)F0(qx)∗, x ∈N,(3.4) F1(qx−1)W(qx−1) = qW (qx)F−1(qx)∗, x ∈N\{0},(3.5) and the boundary conditions W(1)F−1(1)∗= 0,(3.6) q2xF1(qx)W(qx)→0,as x→ ∞, qxF1(qx)W(qx)−W(qx)F1(qx)∗→0,as x→ ∞. Then the q-difference operator Dis symmetric with respect to W. Proof. We assume that the operator Dand the weight matrix Wsatisfy the symmetry equations (3.4), (3.5) and the boundary conditions (3.6). For an integer M > 0, we consider the truncated inner product hP, QiM W= M X x=0 qxP(qx)W(qx)Q∗(qx). It is clear that hP, QiM W→ hP, QiWas M→ ∞ for P, Q ∈P[z]. Then hDP, QiM W= M X x=0 qx(DP )(qx)W(qx)Q∗(qx) = M X x=0 qxP(qx+1)F1(qx) + P(qx)F0(qx) + P(qx−1)F−1(qx)W(qx)Q∗(qx) 6 By a straightforward and careful computation using (3.4), (3.5) and the first boundary condition of (3.6) we have hDP, QiM W− hP, DQiM W=qMP(qM+1)F1(qM)W(qM)Q∗(qM)−qMP(qM)W(qM)F∗ 1(qM)Q∗(qM+1). Write P(z) = P0+zP1(z) and Q(z) = Q0+zQ1(z) so that hDP, QiM W− hP, DQiM W=qMP0F1(qM)W(qM)−W(qM)F1(qM)∗Q∗ 0+ remainder where the remainder consists of terms of the form q2MR(qM)F1(qM)W(qM)S(qM) or its adjoint for suitable matrix-valued polynomials Rand S. Taking M→ ∞ and using the last two boundary conditions of (3.6) we get the result. 4 A matrix-valued q-hypergeometric equation Motivated by Tirao [28] we define a matrix-valued analogue of the basic hypergeometric series. This definition is different from that given by Conflitti and Schlosser [4], where some additional factorization is assumed. Consider the following q-difference equation on row-vector-valued functions F:C→CN. F(q−1z)(R1+zR2) + F(z)(S1+zS2) + F(qz)(T1+zT2) = 0.(4.1) where R1, R2, S1, S2, T1, T2∈MatN(C). The case N= 1 is the scalar hypergeometric q-difference equation, see [13, Exercise 1.13]. Let Fbe a solution of (4.1) of the form F(z) = zµG(z) where G(z) = ∞ X k=0 Gkzk, G06= 0, Gk∈CN The Frobenius method gives the recursions 0 = G0q−µR1+S1+qµT1, 0 = Gkq−k−µR1+S1+qk+µT1+Gk−1q−k−µ+1R2+S2+qk+µ−1T2, k ≥1. The first equation implies det(q−µR1+S1+qµT1) = 0 and G0∗∈ker(q−¯µR∗ 1+S∗ 1+q¯µT∗ 1). The solution of the indicial equation det(q−µR1+S1+qµT1) = 0 is the set of exponents E. For each µ∈Ewe write dµ= dim(ker(q−¯µR∗ 1+S∗ 1+q¯µT∗ 1)) for the multiplicity of the exponent µ. In order to have analytic solutions of (4.1) we require that 0 ∈E. Moreover we assume that the multiplicity for 0 is maximal, d0=N, which implies S1=−R1−T1. Under this assumption E={µ: det(q−µR1−T1) = 0} ∪ {0}. Since we are only interested in polynomial solutions, we only consider expansions around z= 0, but we can also study solutions at ∞in a similar fashion. We specialize to the case R1=−R2=I; F(q−1z)(1 −z) + F(z)(−I−T1+zS2) + F(qz)(T1+zT2) = 0.(4.2) For any G0∈CN,G(z) = P∞ k=0 Gkzkis a solution of (4.2) if and only if 0 = Gk(q−k−1)I+ (qk−1)T1+Gk−1−q−k+1I+S2+qk−1T2, k ≥1. Assuming σ(T1)∩q−N=∅, the coefficients are Gk=qk (q;q)k G0 k −→ Y i=1 I−qi−1S2−q2i−2T2I−qiT1−1, k ≥1, where k −→ Y i=1 Ai=A1A2. . . Akis an ordered product. We summarize this discussion with Definition 4.1 and Theorem 4.2. 7 Definition 4.1. Let A, B, C ∈MatN(C) where σ(C)∩q−N\{0}=∅. Define (A, B;C;q)0=I, (A, B;C;q)k= (A, B;C;q)k−1I−qk−1A−q2k−2BI−qkC−1, k ≥1. Define the function 2η1by 2η1A, B C;q, z= ∞ X n=0 (A, B;C;q)n zn (q;q)n .(4.3) Now (4.3) converges for |z|<1 in the norm of MatN(C). Theorem 4.2. Let A, B, C ∈MatN(C) such that σ(C)∩q−N\{0}=∅. F(z) = F02η1A, B C;q, qz, F 0∈CNas row-vector,(4.4) is a solution of the matrix-valued q-difference equation F(q−1z)(1 −z) + F(z) (−I−C+zA) + F(qz) (C+zB) = 0,(4.5) with condition F(0) = F0. Conversely, any analytic solution Faround z= 0 of (4.5) with initial condition F(0) 6= 0 is of the form (4.4). 5 The 2×2matrix-valued little q-Jacobi polynomials Based on [2, Theorem 4.2] we present a method to construct q-difference operators Dand weight matrices Wsatisfying the symmetry equations (3.4). By applying this method we construct a matrix analogue of the scalar little q-Jacobi polynomials, and we give explicit expressions of the matrix-valued little q-Jacobi polynomials in terms of the scalar ones. We also show how these matrix polynomials can be written as a matrix-valued q-hypergeometric function, motivated by the work of Tirao [28]. 5.1 The construction Lemma 5.1 is an adapted version of [2, Theorem 4.2]. We omit the proof here because it is completely analogous to that in [2]. Lemma 5.1. Let sbe a scalar function satisfying s(qx)6= 0 for x∈N\{0}. Assume that F1and F−1 are matrix-valued polynomials satisfying F1(qx−1)F−1(qx) = q|s(qx)|2I, ∀x∈N\{0}.(5.1) Let Tbe a solution of the q-difference equation T(qx−1) = s(qx)−1F−1(qx)T(qx), x ∈N\{0}, T (1) = I. (5.2) Then the q-weight defined by W(qx) = T(qx)T(qx)∗satisfies the symmetry equation F1(qx−1)W(qx−1) = qW (qx)F−1(qx)∗, x ∈N\{0}. We are now ready to introduce 2×2 matrix-valued orthogonal polynomials related to a specific q-difference operator. Theorem 5.2. Assume aand bsatisfy 0 < a < q−1and b < q−1. For v∈Cdefine matrices K=0v(1 −q)(q−1−a) 0 0 , M =1v 0 0, A =elog(q)M=q−v(1 −q) 0 1 . 8 The q-difference operator given by D=E−1F−1(z) + E0F0(z) + E1F1(z), F−1(z) = (z−1−1)A−1, F0(z) = K−z−1(A−1+aA), F1(z) = (az−1−abq)A, (5.3) is symmetric with respect to the matrix-valued inner product (2.9) where the weight matrix is given by W(qx) = ax(bq;q)x (q;q)x Ax(A∗)x,(5.4) Moreover, the monic orthogonal polynomials (Pn)n≥0with respect to Wsatisfy DPn= ΛnPn, with eigenvalues Λn=−q−n−1−abqn+2 v(1 −q)(abqn+1 −q−1−n+q−1−a) 0−q−n−abqn+1 .(5.5) Remark 5.3. These polynomials are a matrix-valued analogues of the little q-Jacobi polynomials and for v6= 0 they cannot be reduced to scalars. This follows from W(q0) = Iand for v6= 0 and x, y ∈N\{0} with x6=ywe have W(qx)W(qy)6=W(qy)W(qx), as can be checked by substituting (5.9) in (5.4). Proof. To prove the theorem we first prove that Dpreserves polynomials and then apply Theorem 3.5 to see that the operator is symmetric with respect to W. We proceed in three steps. Step 1. Dpreserves polynomials and degree. As polynomials in z−1, dgr(Fi) = 1 <2, so that condition 2 for k= 1,2 in Theorem 3.1 is satisfied. Because dgr(F0+F1+F−1) = 0 condition 2 is also satisfied for k= 0 from which we conclude from Theorem 3.1 that D:Pn[z]→Pn[z]. Step 2. Symmetry equations. F1(qx−1)W(qx−1) = qW (qx)F−1(qx)and F0(qx)W(qx) = W(qx)F0(qx)∗ Consider the function s(qx) = q−xpa(1 −qx)(1 −bqx). The function ssatisfies s(qx)6= 0 because 0< a < q−1and b < q−1, and by a direct computation we see that with this choice of s, (5.1) is satisfied. The solution of (5.2) in this case is given by T(qx) = sax(bq;q)x (q;q)x Ax, By Lemma 5.1 we can conclude that the symmetry equation, F1(qx−1)W(qx−1) = qW (qx)F−1(qx) holds for W(qx) = T(qx)T(qx)∗and x= 1,2,... . Note that T(qx) is invertible for all x∈N. The symmetry equation (3.4) is equivalent to showing that the matrix T(qx)−1F0(qx)T(qx) is Hermitian. 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