PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 6, June 2012, Pages 2075–2089 S 0002-9939(2011)11066-9 Article electronically published on October 19, 2011 BASIC HYPERGEOMETRIC FUNCTIONS AND ORTHOGONAL LAURENT POLYNOMIALS MARISA S. COSTA, EDUARDO GODOY, REGINA L. LAMBLÉM, AND A. SRI RANGA (Communicated by Walter Van Assche) Abstract. A three-complex-parameter class of orthogonal Laurent polynomi- als on the unit circle associated with basic hypergeometric or q-hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2Φ1(q−n, qb+1; q−c+b−n; q, q−c+d−1z)}∞n=0, where 0 < q < 1 and the complex parameters b, c and d are such that b �= −1,−2, . . ., c− b+ 1 �= −1,−2, . . ., Re(d) > 0 and Re(c− d+ 2) > 0. Ex- plicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived. 1. Introduction Given the double sequence {μn}∞n=−∞ of complex numbers, let the linear func- tional M on the space of Laurent polynomials be defined by (1.1) M[z−n] = μn, n = 0,±1,±2, . . . . The functional M can be referred to as a moment functional. Let Dn, n = 0, 1, . . ., be the associated Toeplitz determinants as given by: D−1 = 1, D0 = μ0 and Dn = ∣∣∣∣∣∣∣∣∣ μ0 μ−1 · · · μ−n μ1 μ0 · · · μ−n+1 ... ... ... μn μn−1 · · · μ0 ∣∣∣∣∣∣∣∣∣ , n ≥ 1. We consider the sequence of polynomials {Qn} that satisfies M[z−sQn(z)] = ρn δn,s, 0 ≤ s ≤ n, n ≥ 1, where Qn, for any n ≥ 0, is a monic polynomial of degree n. If the moment functional M is such that Dn �= 0, n ≥ 0, then we will refer to it as a semi-definite Received by the editors September 7, 2010 and, in revised form, February 11, 2011. 2010 Mathematics Subject Classification. Primary 33D15, 42C05; Secondary 33D45. Key words and phrases. Basic hypergeometric functions, continued fractions, orthogonal Lau- rent polynomials, Szegő polynomials. This work was partially support by the joint project CAPES(Brazil)/DGU(Spain). The second author’s work was also supported by the European Community fund FEDER. The third and fourth authors have also received other funds from CNPq, CAPES and FAPESP of Brazil for this work. c©2011 American Mathematical Society 2075 Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2076 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA moment functional. In this case it is easily seen that the sequence of polynomials {Qn} exists uniquely and that Qn(z) = 1 Dn−1 ∣∣∣∣∣∣∣∣∣∣∣ μ0 μ−1 · · · μ−n μ1 μ0 · · · μ−n+1 ... ... ... μn−1 μn−2 · · · μ−1 1 z · · · zn ∣∣∣∣∣∣∣∣∣∣∣ and ρn = M[z−nQn(z)] = Dn Dn−1 . There have been different nomenclatures used with respect to such polynomials in recent years. The polynomials Qn are related to orthogonal Laurent polynomials considered by, for example, Hendriksen and van Rossum [11] and Jones and Thron [15], in the sense that the Laurent polynomials B2n(z) = z−nQ2n(z), B2n+1(z) = z−nQ2n+1(z), n ≥ 0, satisfy the orthogonality relations M[Bn(z)Bm(z)] = δn,mρ̃n, n,m = 0, 1, 2, . . . . With the monic polynomials {Q̂n} given by Q̂n(z) = 1 Dn−1 ∣∣∣∣∣∣∣∣∣∣∣ μ0 μ1 · · · μn μ−1 μ0 · · · μn−1 ... ... ... μ−n+1 μ−n+2 · · · μ1 1 z · · · zn ∣∣∣∣∣∣∣∣∣∣∣ , n ≥ 1, we obtain the biorthogonality relations M[Q̂m(1/z)Qn(z)] = δn,mρn, n,m = 0, 1, 2, . . . . Hence, Zhedanov [32] calls such polynomials Laurent biorthogonal. With respect to the moment functional L[zn] = M[z−n] = μn, n = 0,±1,±2, . . ., the reciprocal polynomials Q• n(z) = znQn(1/z) satisfy the orthogonality relations L[z−n+sQ• n(z)] = δn,sρn, 0 ≤ s ≤ n. Polynomials satisfying such orthogonality relations have been referred to as L-orthogonal polynomials in some earlier contri- butions, including [1], of one of the present authors. We remark that Zhedanov [32] uses the definition L[zn] = μn, n = 0,±1,±2, . . . , for his moment functional and L[Q̂m(z)Qn(1/z)] = δn,mρn, n,m = 0, 1, 2, . . . . In a recent manuscript [17], {Qn} has been called a sequence of monic Szegő type polynomials when M is such that Dn �= 0 and μ−n = μn for n ≥ 0. In this case the Zhedanov [32] biorthogonality can be written as M[Qm(1/z)Qn(z)] = δn,mρn, n,m = 0, 1, 2, . . . . However, if M is such that Dn > 0 and μ−n = μn, n ≥ 0, then this moment functional is known as a positive definite moment functional and the sequence of polynomials {Sn} = {Qn} are known as monic Szegő polynomials. Now we must have M[f ] = ∫ C f(z)dμ(z), where μ(z) = μ(eiθ) is a positive measure on the unit circle C = {z = eiθ : 0 ≤ θ ≤ 2π}. Since the integration is along the unit circle,∫ C z −jSn(z)dμ(z) = ∫ C z̄ jSn(z)dμ(z) and the associated sequence of monic Szegő polynomials {Sn} are usually defined by∫ C Sm(z)Sn(z)dμ(z)= ∫ 2π 0 Sm(eiθ)Sn(e iθ)dμ(eiθ)=κ−2 n δnj , m, n = 0, 1, 2, . . . , where κ−2 n = ‖Sn‖2 = ∫ C |Sn(z)|2dμ(z). With his publications [28] and [29], Szegő introduced these orthogonal polynomi- als on the unit circle in the early 20th century. Many interesting results on these Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2077 polynomials can be found in his classical book [30], the first edition of which was published in 1939. Since then, these polynomials which bear the name of Szegő were extensively studied by many. We cite, for example, [5], [6], [7], [10], [19], [20], [22], [25] and [27] as some of the very recent contributions. The recent publications of the two excellent volumes [23] and [24] by Simon have given a boost to the in- terest in studying these polynomials. We also cite the recent book [13] by Ismail containing a nice chapter on these orthogonal polynomials on the unit circle. Some information on the Szegő polynomials with respect to the measure dμ(eiθ) = [e−θ]η[sin2(θ/2)]λdθ, defined for η, λ ∈ R and λ > −1/2, are provided in [27]. It was shown that these Szegő polynomials are constant multiples of the hyperge- ometric polynomials 2F1(−n, b + 1; b + b̄ + 1; 1 − z), n ≥ 1, where η = Im(b) and λ = Re(b). Results used in [27] have an important root in the paper [11] of Hendriksen and van Rossum, where these authors look at T-fractions and orthogonal Lau- rent polynomials originating from three-term contiguous relations satisfied by the hypergeometric functions 2F1(a, b; c; z). In this paper, using a three-term contiguous relation satisfied by q-hypergeo- metric functions 2Φ1(q a, qb; qc; q, z), we obtain information on the three-parameter class of orthogonal Laurent polynomials {z−�n/2�Q (b,c,d) n (z)}∞n=0 on the unit circle C = {z = eiθ : 0 < θ < 2π}, where the monic polynomials Q (b,c,d) n , n ≥ 0, are given by Q(b,c,d) n (z) = (qc−b+1; q)n (qb+1; q)n qn(b−d+1) 2Φ1(q −n, qb+1; q−c+b−n; q, q−c+d−1z), with 0 < q < 1 and the three complex parameters b, c and d are such that b �= −1,−2, . . ., c − b + 1 �= −1,−2, . . . , Re(d) > 0 and Re(c + 2 − d) > 0. The orthogonality is with respect to the semi-definite moment functional M(b,c,d) given by M(b,c,d)[f(z)] = τ (b,c) 2π ∫ 2π 0 f(eiθ) (q−b+deiθ; q)∞(qb−d+1e−iθ; q)∞ (qdeiθ; q)∞(qc+2−de−iθ; q)∞ dθ. Here the constant τ (b,c), defined in Theorem 3.5, is such that M(b,c,d)[1] = 1. By considering separately the real and imaginary parts of b, c and d, and neglecting Im(d), which induces only a rotation, we can also consider {z−�n/2�Q (b,c,d) n (z)}∞n=0 as a five-real-parameter class of orthogonal Laurent polynomials. The class of polynomials considered here is somewhat different and broader than the class of orthogonal Laurent polynomials {z−�n/2�Pn(−z, α, β)}∞n=0 that follows from Pastro [21], where Pn(z, α, β) = 2Φ1(q̃ −n, q̃α; q̃2−β−n; q̃, q̃−β+3/2z), with |q̃| < 1, α > 1/2 and β > 1/2. Pastro shows that the polynomials Pn(−z, α, β) are the Laurent biorthogonal polynomials with respect to the semi-definite moment functional given by M̃(α,β)[f(z)] = ∫ 2π 0 f(eiθ) (q̃1/2eiθ; q̃)∞(q̃1/2e−iθ; q̃)∞ (q̃α−1/2eiθ; q̃)∞(q̃β−1/2e−iθ; q̃)∞ dθ. These Pastro polynomials can be considered as belonging to a class with the three real parameters α, β and, say, ϑ if one assumes the q̃ to be such that q̃ = |q̃|eiϑ. An Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2078 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA explicit expression for the moments M̃(α,β)[z−n] is found in Vinet and Zhedanov [31]. Note that the moment functional M̃(α,β) can only be made positive definite with the choice −1 < q̃ < 1 and α = β > 1/2. Thus with this choice, Pastro [21] recovers the class of real Szegő polynomials previously described by Askey [4, p. 806]. By a special choice of the parameters b, c and d we also obtain in the present manuscript information regarding the class of (complex and real) Szegő polynomials S (λ,η,φ) n characterized by the reflection coefficients a(λ,η,φ)n = (qλ+iη; q)n (qλ+1−iη; q)n qn[ 1 2−i(η+φ)], n ≥ 1, where λ, η, φ ∈ R and λ > −1/2. The parameter φ, which comes from Im(d), induces only a rotation and can be made equal to zero without any loss of generality. The polynomials obtained by taking η = φ = 0 and λ > −1/2, for example, coincide with the real Szegő polynomials of [21] and [4], obtained when 0 < q̃ < 1 and α = β > 1/2. The paper is organized as follows. In Section 2 we present some fundamental results on three-term recurrence relations, continued fractions and basic hypergeo- metric functions, which we will be using in later sections. In Section 3 we define the monic q-hypergeometric polynomials Q (b,c,d) n (z) and obtain their orthogonality and asymptotic properties. In Section 4, in addition to discussing when the polynomials Q (b,c,d) n (z) coincide with the Szegő polynomials S (λ,η,φ) n mentioned above, we also obtain explicitly the associated Szegő function. 2. Some preliminary results Let {Qn} be the sequence of polynomials given by the three-term recurrence relation (2.1) Qn+1(z) = ( z + βn+1 ) Qn(z)− αn+1zQn−1(z), n ≥ 1, with Q0(z) = 1 and Q1(z) = z + β1. Lemma 2.1. Let β1 �= 0 and αn+1 �= 0 for n ≥ 1. Given any sequence {hn} of arbitrary complex numbers hn (or complex functions hn(z)), let the sequence of functions {Gn(hn; z)} be such that G1(h1; z) = β1 z + β1 − h1 and Gn(hn; z) = β1 z + β1 − α2z z + β2 − · · · − αnz z + βn − hn , n ≥ 2. Then Gn(hn; z)−Gn(0; z) = β1α2α3 · · ·αnhnz n−1 Qn(z)[Qn(z)− hnQn−1(z)] . Proof. Let the sequence of polynomials {Rn} be such that Rn+1(z) = ( z + βn+1 ) Rn(z)− αn+1zRn−1(z), n ≥ 1, with R0(z) = 0, R1(z) = β1. Then from basic results on continued fractions (see, for example, [14, 18]) Gn(hn; z)−Gn(0; z) = Rn(z)− hnRn−1(z) Qn(z)− hnQn−1(z) − Rn(z) Qn(z) , n ≥ 1. Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2079 Hence, Gn(hn; z)−Gn(0; z) = hn[Rn(z)Qn−1(z)−Qn(z)Rn−1(z)] Qn(z)[Qn(z)− hnQn−1(z)] , n ≥ 1. Therefore, the lemma follows from Rn(z)Qn−1(z) − Qn(z)Rn−1(z) = β1α2α3 · · · αnz n−1. � As a particular case of this lemma, if one takes hn = αn+1z/(z + βn+1), then (2.2) Gn+1(0; z)−Gn(0; z) = β1α2α3 · · ·αn+1 Qn(z)Qn+1(z) zn, n ≥ 1. Lemma 2.2. In the three-term recurrence relation (2.1), if βn �= 0 and αn+1 �= 0, n ≥ 1, then there exists a semi-definite moment functional M such that the polynomials Qn satisfy M[z−sQn(z)] = δn,sρn, 0 ≤ s ≤ n, n ≥ 1, where ρn = α2 · · ·αn+1 β2 · · ·βn+1 . Moreover, the associated moments μn = M[z−n], n = 0,±1,±2, . . . are such that L0(z) = ∑∞ j=0 μjz j, L∞(z) = − ∑∞ j=1 μ−jz −j, where (2.3) L0(z)−Gn(0; z) = ρn 1 Qn(0) zn +O(zn+1), L∞(z)−Gn(0; z) = ρnQn+1(0) 1 zn+1 +O( 1 zn+2 ). Proof. First note that Qn(0) = β1β2 · · ·βn �= 0. Now from (2.2) by considering the expansions of Gn(0; z) about the origin and infinity there exist power series L0(z) = ∑∞ j=0 μjz j and L∞(z) = − ∑∞ j=1 μ−jz −j such that (2.3) holds. With respect to these power series coefficients, if we define the moment functional M by (1.1), then the lemma follows from the linear system on the coefficients of Qn and Rn obtained from (2.3). � For a, b, c ∈ C, c �= 0,−1,−2, . . . and 0 < |q| < 1, the 2Φ1 q-hypergeometric or the 2Φ1 basic hypergeometric function (hypergeometric function with base q) may be defined by 2Φ1(q a, qb; qc; q, z) = ∞∑ n=0 (qa; q)n (q b; q)n (qc; q)n (q; q)n zn, for |z| < 1 and by analytic continuation for other values of z ∈ C. Here, (qa; q)0 = 1 and (qa; q)n = (1− qa)(1− qa+1) · · · (1− qa+n−1), n ≥ 1. For more information regarding q-hypergeometric functions, we refer to, for ex- ample, Andrews, Askey and Roy [2], Gasper and Rahman [8], Koekoek and Swart- touw [16] and Slater [26]. Two “distinct” q-hypergeometric functions 2Φ1(q a1 , qa2 ; qa3 ; q, z) and 2Φ1(q ã1 , qã2 ; qã3 ; q, z) may be called contiguous if |ai − ãi| = 0 or 1 for at least one i ∈ {1, 2, 3}. There are interesting relations between contiguous q-hypergeometric func- tions called contiguous relations. Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2080 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA Lemma 2.3. If c �= 0,−1,−2, . . . , then 2Φ1(q a, qb+1; qc; q, z) = ( 1 + 1− qa−b 1− qc qbz ) 2Φ1(q a+1, qb+1; qc+1; q, z) − (1− qa+1) (1− qc−b) (1− qc) (1− qc+1) qbz 2Φ1(q a+2, qb+1; qc+2; q, z). Proof. From contiguous relations obtained by Heine (see [8, p. 22]), we consider the following: 2Φ1(q a+1, qb+1; qc+1; q, z) = 2Φ1(q a+1, qb; qc; q, z) + (1− qa+1) (1− qc−b) (1− qc) (1− qc+1) qbz 2Φ1(q a+2, qb+1; qc+2; q, z), 2Φ1(q a+1, qb; qc; q, z) = 2Φ1(q a, qb+1; qc; q, z) − (1− qa−b) (1− qc) qbz 2Φ1(q a+1, qb+1; qc+1; q, z), which hold for c �= 0,−1,−2, . . . . Substitution for 2Φ1(q a+1, qb; qc; q, z) in the first relation using the other gives the required result. � We will be using the q-binomial theorem (see [16, Eq. (0.5.2)]) (2.4) 2Φ1(q a, qc; qc; q, z) = 1Φ0(q a; q, z) = (qaz; q)∞ (z; q)∞ , which holds for c �= 0,−1,−2, . . . and |z| < 1, and the Heine transformation formula (see [16, Eq. (0.6.3)]) (2.5) 2Φ1(q a, qb; qc; q, z) = (qa+b−cz; q)∞ (z; q)∞ 2Φ1(q −a+c, q−b+c; qc; q, qa+b−cz), which holds for c �= 0,−1,−2, . . . and |z| < min{1, |qc−a−b|}. We will also be needing the polynomial identities (see [16, Eq. (0.6.19)]) (2.6) 2Φ1(q −n, qb; qc; q, z) = (qb; q)n (qc; q)n q−n(n+1)/2(−z)n 2Φ1(q −n, q−c−n+1; q−b−n+1; q, qc−b+n+1z−1), for n ≥ 0, which hold when c �= 0,−1,−2, . . . and b �= −n+1,−n+2,−n+3, . . . . 3. q-orthogonal Laurent polynomials From now on we restrict the value of q to be such that 0 < q < 1. Then for any b ∈ C we have qb = qb̄ and |qb| = qRe(b). With b, c, d ∈ C and c− b+ 1 �= 0,−1,−2, . . . , let F (b,c,d) n (z) = 2Φ1(q n+1, q−b; qc−b+n+2; q, qdz) 2Φ1(qn, q−b; qc−b+n+1; q, qdz) , n ≥ 0. Then from Lemma 2.3, F (b,c,d) n (z) = 1 1 + g (b,c,d) n+1 z − f (b,c,d) n+2 zF (b,c,d) n+1 (z) , n ≥ 0, Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2081 where g(b,c,d)n = 1− qb+n 1− qc−b+n q−b+d−1, f (b,c,d) n+1 = (1− qn) (1− qc+n+1) (1− qc−b+n) (1− qc−b+n+1) q−b+d−1, for n ≥ 1. This leads to the continued fraction expansion (3.1) F (b,c,d) 0 (z) = 1 1 + g (b,c,d) 1 z − f (b,c,d) 2 z 1 + g (b,c,d) 2 z − · · · − f (b,c,d) n z 1 + g (b,c,d) n z − f (b,c,d) n+1 zF (b,c,d) n (z) . Also assuming b �= −1,−2, . . . , this can be written in the equivalent form (3.2) F (b,c,d) 0 (z) = β (b,c,d) 1 z + β (b,c,d) 1 − α (b,c,d) 2 z z + β (b,c,d) 2 − · · · − α (b,c,d) n z z + β (b,c,d) n − α (b,c,d) n+1 z F (b,c,d) n (z) β (b,c,d) n+1 , where β (b,c,d) n = 1/g (b,c,d) n and α (b,c,d) n+1 = f (b,c,d) n+1 /(g (b,c,d) n g (b,c,d) n+1 ), n ≥ 1. Theorem 3.1. With b �= −1,−2, . . . and c− b+ 1 �= −1,−2, . . ., let the sequence of monic polynomials {Q(b,c,d) n } be given by (3.3) Q (b,c,d) n+1 (z) = (z + β (b,c,d) n+1 )Q (b,c,d) n (z)− α (b,c,d) n+1 z Q (b,c,d) n−1 (z), n ≥ 1, with Q (b,c,d) 0 (z) = 1 and Q (b,c,d) 1 (z) = z + β (b,c,d) 1 , where β(b,c,d) n = 1− qc−b+n 1− qb+n qb−d+1, α (b,c,d) n+1 = (1− qn) (1− qc+n+1) (1− qb+n) (1− qb+n+1) qb−d+1, n ≥ 1. Then the polynomials Q (b,c,d) n satisfy the orthogonality relations (3.4) M(b,c,d)[z−sQ(b,c,d) n (z)] = δn,sρ (b,c) n , 0 ≤ s ≤ n, n ≥ 1, with respect to the semi-definite moment functional M(b,c,d)[z−j ] = (q−b; q)j (qc−b+2; q)j qjd, j = 0,±1,±2, . . . . Here, ρ(b,c)n = α (b,c,d) 2 · · ·α(b,c,d) n+1 β (b,c,d) 2 · · ·β(b,c,d) n+1 = (q; q)n (q c+2; q)n (qb+1; q)n (qc−b+2; q)n . Proof. We first prove the theorem for c−b+1 �= 0,−1,−2, . . . and b �= −1,−2, . . . . With these restrictions β (b,c,d) n �= 0 and α (b,c,d) n+1 �= 0, n ≥ 1, and hence from Lemma 2.2 there exists a semi-definite moment functional such that (3.4) holds. To obtain the values of μ (b,c,d) j = M(b,c,d)[z−j ], j = 0,±1,±2, . . . , let us consider the functions G(b,c,d) n (z) = β (b,c,d) 1 z + β (b,c,d) 1 − α (b,c,d) 2 z z + β (b,c,d) 2 − · · · − α (b,c,d) n z z + β (b,c,d) n , n ≥ 1. Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2082 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA Then from Lemma 2.1 and the continued fraction expansion (3.2), F (b,c,d) 0 (z)−G (b,c,d) n (z) = β (b,c,d) 1 α (b,c,d) 2 · · ·α(b,c,d) n α (b,c,d) n+1 znF (b,c,d) n (z) Q (b,c,d) n (z) [ β (b,c,d) n+1 Q (b,c,d) n (z)− α (b,c,d) n+1 z F (b,c,d) n (z)Q (b,c,d) n−1 (z) ] = ρ(b,c)n 1 Q (b,c,d) n (0) zn +O(zn+1), for n ≥ 1. Since F (b,c,d) 0 (z) = 2Φ1(q, q −b; qc−b+2; q, qdz), from the latter part of Lemma 2.2, μ (b,c,d) j = (q−b; q)j (qc−b+2; q)j qjd, j = 0, 1, 2, . . . , thus giving the results for the positive moments. From (3.1), by realizing that g (c−b,c,c+2−d) n = β (b,c,d) n and f (c−b,c,c+2−d) n+1 = α (b,c,d) n+1 , n ≥ 1, we also obtain the continued fraction expansion β (b,c,d) 1 z F (c−b,c,c+2−d) 0 (z−1) = β (b,c,d) 1 z + β (b,c,d) 1 − α (b,c,d) 2 z z + β (b,c,d) 2 − · · · − α (b,c,d) n z z + β (b,c,d) n − α (b,c,d) n+1 F (c−b,c,c+2−d) n (z−1) 1 . Hence, again from Lemma 2.1, β (b,c,d) 1 z F (c−b,c,c+2−d) 0 (z−1)−G(b,c,d) n (z) = β (b,c,d) 1 α (b,c,d) 2 · · ·α(b,c,d) n α (b,c,d) n+1 zn−1F (c−b,c,c+2−d) n (z−1) Q (b,c,d) n (z) [ Q (b,c,d) n (z)− α (b,c,d) n+1 F (c−b,c,c+2−d) n (z−1)Q (b,c,d) n−1 (z) ] = ρ(b,c)n Q (b,c,d) n+1 (0) 1 zn+1 +O( 1 zn+2 ), for n ≥ 1. Since F (c−b,c,c+2−d) 0 (z−1) = 2Φ1(q, q −c+b; qb+2; q, qc+2−dz−1), from the latter part of Lemma 2.2, μ (b,c,d) −j = (q−c+b−1; q)j (qb+1; q)j qj(c+2−d), j = 1, 2, 3, . . . . Thus, using (a, q)n = (a; q)∞/(aqn; q)∞, for n = 0,±1,±2, . . . , we also obtain the results for the negative moments. This concludes the theorem for c − b + 1 �= 0,−1,−2, . . . and b �= −1,−2, . . . . Now to extend the results for c − b + 1 �= −1,−2, . . . and b �= −1,−2, . . ., we need to prove the theorem for c− b+ 1 = 0 and b �= −1,−2, . . . . If b �= −1,−2, . . . , then β (b,b−1,d) 1 = 0 and β (b,b−1,d) n+1 = α (b,b−1,d) n+1 �= 0 for n ≥ 1. Hence, Q (b,b−1,d) n (z) = zn, n ≥ 0 and M(b,b−1,d)[z−sQ(b,b−1,d) n (z)] = M(b,b−1,d)[zn−s] = (q−b; q)−n+s (q; q)−n+s q(−n+s)d. Since (q−b; q)−n+s (q; q)−n+s q(−n+s)d = 0 if s < n and ρ(b,b−1) n = (q−b; q)0 (q; q)0 q(0)d = 1, Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2083 the validity of the theorem when c − b + 1 = 0 and b �= −1,−2, . . . is confirmed. This concludes the theorem. � The same explicit expression for the moments, when the moment functional is considered as in Pastro [21], is obtained in [31]. From the three-term recurrence relation (3.3) it follows that Q(b,c,d) n (0) = β (b,c,d) 1 β (b,c,d) 2 · · ·β(b,c,d) n = (qc−b+1; q)n (qb+1; q)n qn(b−d+1), n ≥ 1. Theorem 3.2. Let b �= −1,−2, . . . and c− b+ 1 �= −1,−2, . . . . Then a) lim n→∞ β(b,c,d) n = qb−d+1, lim n→∞ α(b,c,d) n = qb−d+1, b) lim n→∞ q−n(b−d+1)Q(b,c,d) n (0) = (1− q)−c+2b Γq(b+ 1) Γq(c− b+ 1) , c) lim n→∞ ρ(b,c)n = Γq(b+ 1) Γq(c− b+ 2) Γq(c+ 2) . Proof. Part a) of this theorem is clear. To obtain parts b) and c) we use the definition Γq(x) = (q; q)∞ (qx; q)∞ (1− q)1−x of the q-gamma function. � Theorem 3.3. Let b �= −1,−2, . . . and c− b+1 �= −1,−2, . . . . Then the monic polynomials Q (b,c,d) n , n ≥ 0, given by the recurrence relation (3.3) have the explicit representation (3.5) Q(b,c,d) n (z) = (qc−b+1; q)n (qb+1; q)n qn(b−d+1) 2Φ1(q −n, qb+1; q−c+b−n; q, q−c+d−1z). Proof. From (3.3) it is easily verified that the reciprocal (or inverse) polynomials Q∗(b,c,d) n (z) = znQ (b,c,d) n (1/z̄) and Q•(b,c,d) n (z) = znQ(b,c,d) n (1/z), n ≥ 0, satisfy the three-term recurrence relations (3.6) Q ∗(b,c,d) n+1 (z) = (1 + β (b̄,c̄,d̄) n+1 z)Q ∗(b,c,d) n (z)− α (b̄,c̄,d̄) n+1 z Q ∗(b,c,d) n−1 (z), Q •(b,c,d) n+1 (z) = (1 + β (b,c,d) n+1 z)Q •(b,c,d) n (z)− α (b,c,d) n+1 z Q •(b,c,d) n−1 (z), n ≥ 1, with Q ∗(b,c,d) 0 (z) = Q •(b,c,d) 0 (z) = 1, Q ∗(b,c,d) 1 (z) = 1 + β (b̄,c̄,d̄) 1 z and Q •(b,c,d) 1 (z) = 1 + β (b,c,d) 1 z. This means that Q ∗(b,c,d) n (z) = 2Φ1(q −n, qc̄−b̄+1; q−b̄−n; q, q−d̄+1z), Q •(b,c,d) n (z) = 2Φ1(q −n, qc−b+1; q−b−n; q, q−d+1z), n ≥ 1, which we can easily verify from Lemma 2.3. Hence, application of the transforma- tion (2.6) in Q •(b,c,d) n , for example, gives the required results of the theorem. � Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2084 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA Note that by the application of the transform (2.5) in Q ∗(b,c,d) n , for example, we can also write that (q−d̄+1z; q)∞ (qc̄−d̄+2z; q)∞ Q∗(b,c,d) n (z) = 2Φ1(q −b̄, q−c̄−n−1; q−b̄−n; q, qc̄−d̄+2z), n ≥ 1, provided that |z| < q−Re(c−d+2). This can also be directly verified from Lemma 2.3 and (3.6). Theorem 3.4. Let b �= −1,−2, . . . , c− b+ 1 �= −1,−2, . . . and σ = min{q−Re(c−d+2), q−Re(b−d+1)}. Then uniformly on compact subsets of |z| < σ, lim n→∞ Q∗(b,c,d) n (z) = (qc̄−d̄+2z; q)∞ (qb̄−d̄+1z; q)∞ . Proof. Since lim n→∞ (q−c̄−n−1; q)j (q−b̄−n; q)j q(c̄−d̄+2)j = q(b̄−d̄+1)j , using Lebesgue’s dominated convergence theorem and then (2.4), we obtain lim n→∞ 2Φ1(q −b̄, q−c̄−n−1; q−b̄−n; q, qc̄−d̄+2z) = 1Φ0(q −b̄; q, qb̄−d̄+1z)= (q−d̄+1z; q)∞ (qb̄−d̄+1z; q)∞ , uniformly on compact subsets of |z| < σ. Thus, the result of the theorem follows. � Theorem 3.5. In addition to b �= −1,−2, . . . and c − b + 1 �= −1,−2, . . . , if one also assumes that Re(c+ 2) > Re(d) > 0, then the polynomials Q (b,c,d) n , n ≥ 0, given by (3.5), satisfy the orthogonality rela- tions τ (b,c) 2πi ∫ C z−sQ(b,c,d) n (z) (q−b+dz; q)∞(qb−d+1/z; q)∞ (qdz; q)∞(qc−d+2/z; q)∞ 1 z dz = δn,sρ (b,c) n , 0 ≤ s ≤ n. Here, ρ (b,c) n are as in Theorem 3.1 and τ (b,c) = (q; q)∞(qc+2; q)∞ (qc−b+2; q)∞(qb+1; q)∞ . Proof. Let us consider the following identity of Ramanujan: ∞∑ −∞ (α; q)n (β; q)n xn = (q; q)∞ ( βα ; q)∞ (αx; q)∞ ( q αx ; q)∞ (β; q)∞ ( q α ; q)∞ ( β αx ; q)∞ (x; q)∞ , which holds for |βα−1| < |x| < 1. Simple proofs of this identity can be found in [3] and [12]. In our case, since 0 < q < 1, with the assumptions of the theorem if we take x = qdz, α = q−b and β = qc−b+2, then (3.7) ∞∑ −∞ (q−b; q)n (qc−b+2; q)n qndzn = τ (b,c) (q−b+dz; q)∞(qb−d+1/z; q)∞ (qdz; q)∞(qc+2−d/z; q)∞ , which holds for |qc+2−d| < |z| < |q−d|, where |qc+2−d| < 1 and |q−d| > 1. Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2085 Hence, multiplying by z−j−1 and integrating along the unit circle we obtain from Laurent’s theorem (q−b; q)j (qc−b+2; q)j qjd= τ (b,c) 2πi ∫ C z−j−1 (q −b+dz; q)∞(qb−d+1/z; q)∞ (qdz; q)∞(qc+2−d/z; q)∞ dz, j = 0,±1,±2, . . . . Thus, the moment functional in Theorem 3.1 satisfies (3.8) M(b,c,d)[z−j ] = τ (b,c) 2πi ∫ C z−j−1 (q −b+dz; q)∞(qb−d+1/z; q)∞ (qdz; q)∞(qc+2−d/z; q)∞ dz, for j = 0,±1,±2, . . . , which completes the proof of the theorem. � As a particular case, letting b = 0 and c+ 2 �= 0,−1,−2, . . . we have β (0,c,d) 1 = 1−qc+1 1−q q−d+1 and β (0,c,d) n+1 = α (0,c,d) n+1 = 1− qc+n+1 1− qn+1 q−d+1, n ≥ 1. Moreover, μ (0,c,d) 0 = 1, μ (0,c,d) j = 0 and μ (0,c,d) −j = (q−c−1; q)j (q; q)j qj(c+2−d), j = 1, 2, . . . . Furthermore, the following corollary holds. Corollary 3.5.1. If Re(c + 2) > Re(d) > 0, then the sequence of polynomials {Q(0,c,d) n } given by Q(0,c,d) n (z) = (qc+1; q)n (q; q)n qn(−d+1) 2Φ1(q −n, q; q−c−n; q, q−c+d−1z), n ≥ 1, apart from satisfying the three-term recurrence relation Q(0,c,d) n+1 (z) = (z + 1− qc+n+1 1− qn+1 q−d+1)Q(0,c,d) n (z)− 1− qc+n+1 1− qn+1 q−d+1 zQ(0,c,d) n−1 (z), for n ≥ 1, with Q(0,c,d) 0 (z) = 1 and Q(0,c,d) 1 (z) = z + 1−qc+1 1−q q−d+1, satisfies the orthogonality relations 1 2πi ∫ C z−sQ(0,c,d) n (z) (q−d+1/z; q)∞ (qc+2−d/z; q)∞ 1 z dz = δn,s, 0 ≤ s ≤ n. Moreover, uniformly on compact subsets of |z| < min{q−Re(c−d+2), q−Re(−d+1)}, lim n→∞ Q∗(0,c,d) n (z) = (qc̄−d̄+2z; q)∞ (q−d̄+1z; q)∞ . As another particular case, letting c = b and b+ 1 �= 0,−1,−2, . . . , we have β(b,b,d) n = α (b,b,d) n+1 = 1− qn 1− qb+n qb−d+1, n ≥ 1. Moreover, μ (b,b,d) j = (q−b; q)j (q2; q)j qjd, j = 0,±1,±2, . . . . Furthermore, the following corollary can be stated. Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2086 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA Corollary 3.5.2. If b + 1 �= 0 and Re(b+ 2) > Re(d) > 0, then the sequence of polynomials {Q(b,b,d) n } given by Q(b,b,d) n (z) = (q; q)n (qb+1; q)n qn(b−d+1) n∑ j=0 (qb+1; q)j (q; q)j q−j(b−d+1)zj , n ≥ 1, satisfies the orthogonality relations 1 2πi (1− q) (1− qb+1) ∫ C z−sQ(b,b,d) n (z) (q−b+dz; q)∞ (qdz; q)∞ z − qb−d+1 z2 dz = δn,s, 0 ≤ s ≤ n. Moreover, uniformly on compact subsets of |z| < q−Re(b−d+1), lim n→∞ Q∗(b,b,d) n (z) = 1 (1− qb̄−d̄+1z) . 4. q-Szegő polynomials From (3.8) the moment functional M(b,c,d) is easily seen to be positive definite if b �= −1,−2, . . ., c− b+1 �= −1,−2, . . . , Re(c+2) > Re(d) > 0, −b+ d = b− d+1 and d = c+ 2− d. That is, with the restrictions c = b+ b̄− 1, d+ d̄ = b+ b̄+ 1 and Re(b) > −1/2, the moment functional M(b,c,d) is positive definite, and hence the polynomials Q (b,c,d) n are the associated Szegő polynomials. Hence, setting b = λ− iη, c = 2λ− 1 and d = 1 2 + λ+ iφ, if λ > −1/2, our special case of Ramanujan identity (3.7) becomes ∞∑ −∞ (q−λ+iη; q)n (qλ+1+iη; q)n qn( 1 2+λ+iφ)zn = τ̃ (λ,η) (q 1 2+i(η+φ)z; q)∞ (q 1 2−i(η+φ)/z; q)∞ (q 1 2+λ+iφz; q)∞ (q 1 2+λ−iφ/z; q)∞ , which holds for qλ+1/2 < |z| < q−λ−1/2, where τ̃ (λ,η) = (q; q)∞ (q2λ+1; q)∞ (q1+λ+iη; q)∞ (q1+λ−iη; q)∞ . This means that we can write M(λ−iη, 2λ−1, λ+iφ+1/2)[z−j ] = (q−λ+iη; q)j (qλ+1+iη; q)j qj( 1 2+λ+iφ) = ∫ C z−jdμ(λ,η,φ)(z) = ∫ 2π 0 e−ijθω(λ,η,φ)(θ) dθ, for j = 0,±1,±2, . . . , where ω(λ,η,φ)(θ)dθ = dμ(λ,η,φ)(eiθ), with dμ(λ,η,φ)(z) dz = τ̃ (λ,η) 2πi 1 z (q 1 2+i(η+φ)z; q)∞ (q 1 2−i(η+φ)/z; q)∞ (q 1 2+λ+iφz; q)∞ (q 1 2+λ−iφ/z; q)∞ and (4.1) ω(λ,η,φ)(θ) = τ̃ (λ,η) 2π (q 1 2+i(η+φ)eiθ; q)∞ (q 1 2−i(η+φ)e−iθ; q)∞ (q 1 2+λ+iφeiθ; q)∞ (q 1 2+λ−iφe−iθ; q)∞ . As expected, ω(λ,η,φ)(θ) is a positive weight function in [0, 2π]. Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2087 Adopting the notation S (λ,η,φ) n (z) = Q (λ−iη, 2λ−1, 1 2+λ+iφ) n (z) we can write the following: S (λ,η,φ) n+1 (z) = ( z + 1− qλ+iη+n 1− qλ−iη+n+1 q 1 2−i(η+φ) ) S(λ,η,φ) n (z) − (1− qn)(1− q2λ+n) (1− qλ−iη+n)(1− qλ−iη+n+1) q 1 2−i(η+φ) z S (λ,η,φ) n−1 (z), n ≥ 1, with S (λ,η,φ) 0 (z) = 1 and S (λ,η,φ) 1 (z) = ( z + 1−qλ+iη 1−qλ−iη+1 q 1 2−i(η+φ) ) . Moreover, (4.2) S (λ,η,φ) n (0) = (qλ+iη; q)n (q1+λ−iη; q)n qn[ 1 2−i(η+φ)], n ≥ 1. Hence, in particular, using Theorems 3.3 and 3.5 we have Theorem 4.1. If λ, η, φ ∈ R and λ > −1/2, then the polynomials S(λ,η,φ) n (z) = (qλ+iη; q)n q n[ 12−i(η+φ)] (qλ+1−iη; q)n 2Φ1(q −n, qλ+1−iη; q−λ−n+1−iη; q, q 1 2−λ+iφz) are the monic Szegő polynomials satisfying∫ 2π 0 S (λ,η,φ) n (eiθ)S(λ,η,φ) m (eiθ)ω(φ,η,λ)(θ)dθ = [κ(λ,η) n ]−2δn,m, n,m = 0, 1, 2, . . . , with respect to the weight function ω(λ,η,φ)(θ) given by (4.1). Here, [κ(λ,η) n ]−2 = ρ(λ−iη,2λ−1) n = (q; q)n (q 2λ+1; q)n (qλ+1+iη; q)n (qλ+1−iη; q)n . Moreover, these polynomials satisfy the Szegő recurrence relation S ∗(λ,η,φ) n (z) = a (λ,η,φ) n zS (λ,η,φ) n−1 (z) + S ∗(λ,η,φ) n−1 (z), n ≥ 1, where the reflection (or Verblunsky) coefficients a (λ,η,φ) n = S (λ,η,φ) n (0) are given by (4.2). Now using Theorem 3.4 we can state the following. Let λ, η, φ ∈ R, λ > −1/2 and σ = min{q−1/2, q−λ−1/2}. Then uniformly on compact subsets of |z| < σ, (4.3) lim n→∞ S∗(λ,η,φ) n (z) = (qλ+ 1 2+iφz; q)∞ (q 1 2+i(η+φ)z; q)∞ . Moreover, ∞∑ n=1 |a(λ,η,φ)n |2 = |1−qλ+iη|2 ∞∑ n=1 qn |1− qn+λ+iη|2 ≤ |1−qλ+iη|2 ∞∑ n=1 qn (1− qn+λ)2 < ∞. This last result means that the Szegő condition 1 2π ∫ 2π 0 log(ω(λ,η,φ)(θ))dθ > −∞ holds and we can now give an expression for the associated Szegő function D(λ,η,φ)(z) = exp ( 1 4π ∫ 2π 0 eiθ + z eiθ − z log(ω(λ,η,φ)(θ))dθ ) . Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2088 M.S. COSTA, E. GODOY, R.L. LAMBLÉM, AND A. SRI RANGA Theorem 4.2. If λ, η, φ ∈ R and λ > −1/2, then for |z| < 1, D(λ,η,φ)(z) = √ Γq(2λ+ 1) Γq(λ+ 1− iη) Γq(λ+ 1 + iη) (q 1 2+i(η+φ)z; q)∞ (qλ+ 1 2+iφz; q)∞ . Proof. Since, κ (λ,η) n S ∗(λ,η,φ) n → [D(λ,η,φ)(z)]−1 for |z| < 1 (see [23, p. 144]), the result follows from part c) of Theorem 3.2 and from (4.3). � Acknowledgements The authors would like to thank Professor Zhedanov for valuable bibliographic information. The authors would also like to thank Professor Ismail for pointing out that the orthogonality relation in Theorem 4.1 is a special case of a biorthogonality relation involving polynomials attributed in his book [13] to Pastro [21]. References [1] E.X.L. de Andrade, C.F. Bracciali and A. Sri Ranga, Another connection between orthogo- nal polynomials and L-orthogonal polynomials, J. Math. Anal. Appl., 330 (2007), 114-132. MR2298161 (2008f:42026) [2] G.E. Andrews, R. Askey and R. Roy, “Special Functions”, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2000. MR1688958 (2000g:33001) [3] R. Askey, Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), 346-359. MR567718 (82g:01030) [4] R. Askey (editor), “Gábor Szegő: Collected Papers. Volume 1”, Contemporary Mathemati- cians, Birkhäuser, Boston, 1982. MR674482 (84d:01082a) [5] A. Cachafeiro, F. Marcellán and C. Prez, Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein-Szegö measure, Adv. Comput. Math., 26 (2007), 81-104. MR2350346 (2008m:33032) [6] R. Cruz-Barroso, P. González-Vera and F. Perdomo-Ṕıo, Quadrature formulas associated with Rogers-Szegő polynomials, Comput. Math. Appl., 57 (2009), 308-323. MR2488385 (2009k:65040) [7] L. Daruis, O. Nj̊astad, and W. Van Assche, Szegő quadrature and frequency analysis, Elec- tron. Trans. Numer. Anal., 19 (2005), 48-57. MR2149269 (2006e:41057) [8] G. Gasper and M. Rahman, “Basic Hypergeometric Series”, Cambridge Univ. Press, Cam- bridge, 1990. MR1052153 (91d:33034) [9] Ya.L. Geronimus, “Orthogonal Polynomials”, Amer. Math. Soc. Transl., Ser. 2, vol. 108, American Mathematical Society, Providence, RI, 1977. [10] L. Golinskii and A. Zlatoš, Coefficients of orthogonal polynomials on the unit circle and higher-order Szegő theorems, Constr. Approx., 26 (2007), 361-382. MR2335688 (2008k:42080) [11] E. Hendriksen and H. van Rossum, Orthogonal Laurent polynomials, Indag. Math. (Ser. A), 48 (1986), 17-36. MR834317 (87j:30008) [12] M.E.H. Ismail, A simple proof of Ramanujan’s 1Ψ1 sum, Proc. Amer. Math. Soc., 63 (1977), 185-186. MR0508183 (58:22695) [13] M.E.H. Ismail, “Classical and Quantum Orthogonal Polynomials in One Variable”, Cam- bridge Univ. Press, Cambridge, 2005. MR2191786 (2007f:33001) [14] W.B. Jones and W.J. Thron, “Continued Fractions. Analytic Theory and Applications”, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley, Reading, MA, 1980. MR595864 (82c:30001) [15] W.B. Jones and W.J. Thron, Survey of continued fraction methods of solving moment prob- lems, in: Analytic Theory of Continued Fractions, Lecture Notes in Math. 932, Springer, Berlin, 1981. MR690450 (84b:30002) [16] R. Koekoek and R. Swarttouw, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue”, Reports of the Faculty of Technical Mathematics and Informatics 98-17, Delft University of Technology, Delft, 1998. [17] R.L. Lamblém, J.H. McCabe, M.A. Piñar and A. Sri Ranga, Szegő type polynomials and para-orthogonal polynomials, J. Math. Anal. Appl., 370 (2010), 30-41. MR2651127 Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use http://www.ams.org/mathscinet-getitem?mr=2298161 http://www.ams.org/mathscinet-getitem?mr=2298161 http://www.ams.org/mathscinet-getitem?mr=1688958 http://www.ams.org/mathscinet-getitem?mr=1688958 http://www.ams.org/mathscinet-getitem?mr=567718 http://www.ams.org/mathscinet-getitem?mr=567718 http://www.ams.org/mathscinet-getitem?mr=674482 http://www.ams.org/mathscinet-getitem?mr=674482 http://www.ams.org/mathscinet-getitem?mr=2350346 http://www.ams.org/mathscinet-getitem?mr=2350346 http://www.ams.org/mathscinet-getitem?mr=2488385 http://www.ams.org/mathscinet-getitem?mr=2488385 http://www.ams.org/mathscinet-getitem?mr=2149269 http://www.ams.org/mathscinet-getitem?mr=2149269 http://www.ams.org/mathscinet-getitem?mr=1052153 http://www.ams.org/mathscinet-getitem?mr=1052153 http://www.ams.org/mathscinet-getitem?mr=2335688 http://www.ams.org/mathscinet-getitem?mr=2335688 http://www.ams.org/mathscinet-getitem?mr=834317 http://www.ams.org/mathscinet-getitem?mr=834317 http://www.ams.org/mathscinet-getitem?mr=0508183 http://www.ams.org/mathscinet-getitem?mr=0508183 http://www.ams.org/mathscinet-getitem?mr=2191786 http://www.ams.org/mathscinet-getitem?mr=2191786 http://www.ams.org/mathscinet-getitem?mr=595864 http://www.ams.org/mathscinet-getitem?mr=595864 http://www.ams.org/mathscinet-getitem?mr=690450 http://www.ams.org/mathscinet-getitem?mr=690450 http://www.ams.org/mathscinet-getitem?mr=2651127 BASIC HYPERGEOMETRIC ORTHOGONAL LAURENT POLYNOMIALS 2089 [18] L. Lorentzen and H. Waadeland, “Continued Fractions with Applications”, Studies in Com- putational Mathematics, vol. 3, North-Holland, Amsterdam, 1992. MR1172520 (93g:30007) [19] A.L. Lukashov and F. Peherstorfer, Zeros of polynomials orthogonal on two arcs of the unit circle, J. Approx. Theory, 132 (2005), 42-71. MR2110575 (2006g:42045) [20] A. Mart́ınez-Finkelshtein, K.T.-R. McLaughlin and E.B. Saff, Szegő orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics, Constr. Approx., 24 (2006), 319–363. MR2253965 (2007e:42029) [21] P.I. Pastro, Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl., 112 (1985), 517-540. MR813618 (87c:33015) [22] J. Petronilho, Orthogonal polynomials on the unit circle via a polynomial mapping on the real line, J. Comput. Appl. Math., 216 (2008), 98-127. MR2421843 (2009e:42054) [23] B. Simon, “Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory”, Ameri- can Mathematical Society Colloquium Publications, vol. 54, part 1, American Mathematical Society, Providence, RI, 2004. MR2105088 (2006a:42002a) [24] B. Simon, “Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory”, Ameri- can Mathematical Society Colloquium Publications, vol. 54, part 2, American Mathematical Society, Providence, RI, 2004. MR2105089 (2006a:42002b) [25] B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging, 1 (2007), 713-772. MR2350223 (2008k:31003) [26] L.J. Slater, “Generalized Hypergeometric Functions”, Cambridge Univ. Press, Cambridge, 1966. MR0201688 (34:1570) [27] A. Sri Ranga, Szegő polynomials from hypergeometric functions, Proc. Amer. Math. Soc., 138 (2010), 4259-4270. MR2680052 [28] G. Szegő, Über Beiträge zur theorie der toeplitzschen formen, Math. Z., 6 (1920), 167-202. MR1544404 [29] G. Szegő, Über Beiträge zur theorie der toeplitzschen formen, II, Math. Z., 9 (1921), 167-190. MR1544462 [30] G. Szegő, “Orthogonal Polynomials”, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR0372517 (51:8724) [31] L. Vinet and A. Zhedanov, Spectral transformations of the Laurent biorthogonal polynomials, II. Pastro polynomials, Canad. Math. Bull., 44 (2001), 337-345. MR1847496 (2002g:33022) [32] A. Zhedanov, The “classical” Laurent biorthogonal polynomials, J. Comput. Appl. Math., 98 (1998), 121-147. MR1656982 (99k:33065) Pós-Graduação em Matemática, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil E-mail address: isacosta.mat@bol.com.br Departamento de Matemática Aplicada II, E.T.S.I. Industriales, Universidade de Vigo, Campus Lagoas-Marcosende, 36310 Vigo, Spain E-mail address: egodoy@dma.uvigo.es Pós-Graduação em Matemática, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil E-mail address: parareginae@hotmail.com Departamento de Ciências de Computação e Estat́ıstica, IBILCE, UNESP-Universi- dade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil E-mail address: ranga@ibilce.unesp.br Licensed to Universidade Estadual Paulista "J·lio de Mesquita Filho". Prepared on Thu Jul 11 09:01:02 EDT 2013 for download from IP 200.145.3.34. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use http://www.ams.org/mathscinet-getitem?mr=1172520 http://www.ams.org/mathscinet-getitem?mr=1172520 http://www.ams.org/mathscinet-getitem?mr=2110575 http://www.ams.org/mathscinet-getitem?mr=2110575 http://www.ams.org/mathscinet-getitem?mr=2253965 http://www.ams.org/mathscinet-getitem?mr=2253965 http://www.ams.org/mathscinet-getitem?mr=813618 http://www.ams.org/mathscinet-getitem?mr=813618 http://www.ams.org/mathscinet-getitem?mr=2421843 http://www.ams.org/mathscinet-getitem?mr=2421843 http://www.ams.org/mathscinet-getitem?mr=2105088 http://www.ams.org/mathscinet-getitem?mr=2105088 http://www.ams.org/mathscinet-getitem?mr=2105089 http://www.ams.org/mathscinet-getitem?mr=2105089 http://www.ams.org/mathscinet-getitem?mr=2350223 http://www.ams.org/mathscinet-getitem?mr=2350223 http://www.ams.org/mathscinet-getitem?mr=0201688 http://www.ams.org/mathscinet-getitem?mr=0201688 http://www.ams.org/mathscinet-getitem?mr=2680052 http://www.ams.org/mathscinet-getitem?mr=1544404 http://www.ams.org/mathscinet-getitem?mr=1544462 http://www.ams.org/mathscinet-getitem?mr=0372517 http://www.ams.org/mathscinet-getitem?mr=0372517 http://www.ams.org/mathscinet-getitem?mr=1847496 http://www.ams.org/mathscinet-getitem?mr=1847496 http://www.ams.org/mathscinet-getitem?mr=1656982 http://www.ams.org/mathscinet-getitem?mr=1656982 1. Introduction 2. Some preliminary results 3. q-orthogonal Laurent polynomials 4. q-Szego polynomials Acknowledgements References