7 The structure relation for Askey-Wilson 0 0 polynomials 2 n a Tom H. Koornwinder J 3 2 Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht24, 1018 TV Amsterdam, The Netherlands ] A [email protected] C This paper is dedicated to Nico Temme on the occasion of his 65th birthday. . h t a m Abstract [ AnexplicitstructurerelationforAskey-Wilsonpolynomialsisgiven. 3 This involves a divided q-difference operator which is skew symmetric v withrespecttotheAskey-Wilsoninnerproductandwhichsendspoly- 3 nomials of degree n to polynomials of degree n+1. By specialization 0 of parameters and by taking limits, similar structure relations, as well 3 as lowering and raising relations, can be obtained for other families in 1 0 theq-AskeyschemeandtheAskeyscheme. Thisisexplicitlydiscussed 6 for Jacobi polynomials, continuous q-Jacobi polynomials, continuous 0 q-ultraspherical polynomials, and for big q-Jacobi polynomials. An / h already known structure relation for this last family can be obtained t from the new structure relation by using the three-term recurence re- a m lation and the second order q-difference formula. The results are also putintheframeworkofamoregeneraltheory. Theirrelationshipwith : v earlier work by Zhedanov and Bangerezako is discussed. There is also i X a connection with the string equation in discrete matrix models and with the Sklyanin algebra. r a 1 Introduction Oneofthemanywaystocharacterizeafamily{p (x)}ofclassicalorthogonal n polynomials (Jacobi, Laguerre and Hermite polynomials) is by a structure relation π(x)p′ (x) = a p (x)+b p (x)+c p (x), (1.1) n n n+1 n n n n−1 where π(x) is a fixed polynomial (necessarily of degree ≤ 2). In view of the three-term recurrence relation this is equivalent to a characterization by a lowering relation π(x)p′ (x) = (α x+β )p (x)+γ p (x) (1.2) n n n n n n−1 1 or to a characterization by a raising relation π(x)p′ (x) = (α˜ x+β˜ )p (x)+γ˜ p (x). (1.3) n n n n n n+1 Theloweringrelation(1.2)wasgivenin[7,10.7(4)](in[7,10.8(15)]explicitly for Jacobi polynomials), where it was attributed to Tricomi (1948). The characterization of classical orthogonal polynomials by their property (1.2) (and thus equivalently by their property (1.1)) was first given by Al-Salam & Chihara [2]. See also [16] for a characterization by (1.1). Note that the lowering and raising relations (1.2), (1.3) are different from the more familiar shift operator relations, where not only the degree is lowered or raised, but also the parameters are shifted. These formulas are well known for all orthogonal polynomials in the (q-)Askey scheme, see for instance[14]. Orthogonal polynomials satisfying the more general structure relation n+t π(x)p′ (x) = a p (x) (π(x) a polynomial; s,t independent of n). n n,j j j=n−s X (1.4) are called semi-classical, see Maroni [17]. According to [2] the question to characterize orthogonal polynomials satisfying (1.4) was first posed by R. Askey. Garc´ıa, Marcell´an & Salto [8] characterized discrete classical orthogonal polynomials in the Hahn class (Hahn, Krawtchouk, Meixner and Charlier polynomials) by a structure relation similar to (1.1), with the derivative replacedbythedifferenceoperator(∆f)(x) := f(x+1)−f(x). NextMedem, A´lvarez-Nodarse &Marcell´an [18](see also[3]) characterized theorthogonal polynomials in the q-Hahn class by a structure relation obtained from (1.1) by replacing the derivative by the q-derivative D , where q f(x)−f(qx) (D f)(x)= D f(x) := . (1.5) q q,x (1−q)x (cid:0) (cid:1) Here a family of orthogonal polynomials p (x) is in the q-Hahn class if the n polynomials (D p )(x) are again orthogonal. q n Variants of lowering and raising relations (1.2), (1.3) are scattered over the literature. See a brief survey in [15]. Note in particular the lowering and raising relations for A type Macdonald polynomials given by Kirillov & n Noumi [13]. TheA case yields lowering and raising relations for continuous 1 q-ultraspherical polynomials. A lowering relation for continuous q-Jacobi polynomials was given by Ismail [12, Theorem 15.5.2]. For Askey-Wilson polynomials, a structure relation was given in a very implicit way by Zhedanov [24] (see my discussion in Remark 2.6), while a loweringandraisingrelationwasgivenbyBangerezako[5](seemydiscussion 2 after (4.13)). Themain result of the present paper gives a structure relation for Askey-Wilson polynomials in the form Lp = a p +c p , (1.6) n n n+1 n n−1 where L is a linear operator which is skew symmetric with respect to the inner product h.,.i for which the Askey-Wilson polynomials are orthogonal: hLf,gi = −hf,Lgi. (1.7) This operator is explicitly given by (Lf)[z] := (1−az)(1−bz)(1−cz)(1−dz)z−2f[qz] (cid:16) −(1−a/z)(1−b/z)(1−c/z)(1−d/z)z2f[q−1z] (z−z−1)−1. (1.8) (cid:17) It sends symmetric Laurent polynomials of degree n to symmetric Laurent polynomialsofdegreen+1. Byspecializationandlimittransitionastructure relation of the form (1.6) with L satisfying (1.7) can be obtained for all families of orthogonal polynomials in the q-Askey scheme and the Askey scheme (a list of these families is given in [14]). For instance, for Jacobi (α,β) polynomials P (x) we get for the operator L: n (Lf)(x) := (1−x2)f′(x)− 1 α−β+(α+β +2)x f(x) (1.9) 2 = (1−x)−12α+21(1+x)−21(cid:0)β+21 d (1−x)12α+12(1(cid:1)+x)21β+21 f(x) dx (cid:16) (1(cid:17).10) = ((DX −XD)f)(x), (1.11) where X is multiplication by x and D is a second order differential oper- ator having the Jacobi polynomials as eigenfunctions (see (3.6)). It will turn out that the form (1.10) of L, involving something close to the square root of the weight function, can also be realized higher up in the (q-)Askey scheme,notablyintheAskey-Wilsoncase(1.8). Itwillalsoturnoutthatthe form (1.11) of L, i.e., as a commutator of X and a second order differential or (q-)difference operator D having the orthogonal polynomials as eigen- functions, persists in the (q-)Askey scheme. In fact, there is an essentially one-to-one relationship between operators L and D. For those families where a structure relation had been given earlier, one can relate that formula to (1.6) by use of the three-term recurrence relation, and sometimes also of the second-order (q-)difference equation. The general theory of the structure relation of the form (1.6) with skew symmetric L will be given in §2. This theory is easy and elegant. The coefficients in the resulting structure relation and in the lowering and rais- ing relations are very close to the coefficients in the three-term recurrence 3 relation. In §2 I will also discuss the relationship with bispectral problems (Gru¨nbaum and Haine), Zhedanov’s algebra, and the string equation in the context of discrete matrix models. The case of Jacobi polynomials will be discussed in §3. The main result, the Askey-Wilson case, is the topic of §4. Here also a connection with the Sklyanin algebra will be made. In §5 this is specialized to the case of continuous q-Jacobi polynomials and we show that it has the results for Jacobi polynomials as a limit case. A further spe- cialization to continuous q-ultraspherical polynomials is given in §6, and the resulting structure relation is related to another one obtained from results in [15]. Finally, in §7, we take the limit of the Askey-Wilson case to the case of big q-Jacobi polynomials, and we relate the resulting structure relation to the one in [18]. Acknowledgement I thank G. Bangerezako, L. Haine, H. Rosengren, P. van Moerbeke and the referees for helpful comments. Conventions Throughout assume that 0 < q < 1. For (q-)Pochhammer symbols and (q-)hypergeometric series use the notation of Gasper & Rahman [9]. Sym- metricLaurentpolynomialsf[z]= n c zk (wherec = c )arerelated k=−n k k −k to ordinary polynomials f(x) in x= 1(z+z−1) by f(1(z+z−1)) = f[z]. P2 2 2 The general form of the structure relation Suppose we have a family of orthogonal polynomials p (x) with respect to n an orthogonality measure µ on R: p (x) = k xn+··· , hp ,p i= h δ , (2.1) n n n m n n,m where hf,gi := f(x)g(x)dµ(x). (2.2) R Z Write the three-term recurrence relation as xp (x) = A p (x)+B p (x)+C p (x). (2.3) n n n+1 n n n n−1 Then k k h h n n−1 n n A = , C = = A . (2.4) n n n−1 k k h h n+1 n n−1 n−1 The proof of the following proposition is straightforward. Proposition 2.1. Let L be a linear operator acting on the space R[x] of polynomials in one variable with real coefficients such that L is skew sym- metric with respect to the inner product (2.2) (i.e., (1.7) holds) and such that L(xn) = γ xn+1+terms of lower degree, (2.5) n 4 where γ 6= 0. Then the following structure relation and lowering and raising n relations hold: (Lp )(x) =γ A p (x)−γ C p (x), (2.6) n n n n+1 n−1 n n−1 −γ (x−B )p (x)+(Lp )(x) = −(γ +γ )C p (x), (2.7) n n n n n n−1 n n−1 γ (x−B )p (x)+(Lp )(x) =(γ +γ )A p (x). (2.8) n−1 n n n n n−1 n n+1 Skew symmetric operators L as above can be produced from symmetric operators D which have the p as eigenfunctions. First note that n (Xf)(x) := xf(x) (2.9) defines a symmetric operator X on R[x] with respect to the inner product (2.2), i.e., hXf,gi = hf,Xgi. Now the following proposition can be shown immediately. Proposition 2.2. Let D be a linear operator acting on R[x] which is sym- metric with respect to the inner product (2.2), i.e., hDf,gi = hf,Dgi, and which satisfies D(xn) = λ xn+terms of lower degree, (2.10) n where λ 6= λ , so p is an eigenfunction of D: n n−1 n Dp = λ p . (2.11) n n n Then the commutator L := [D,X] = DX −XD (2.12) is skew symmetric with respect to the inner product (2.2) and satisfies (2.5) with γ = λ −λ 6= 0. (2.13) n n+1 n So L also satisfies the structure relation (2.6). In fact, we can reverse Proposition 2.2: From skew symmetric L as in Proposition 2.1 we can produce symmetric D as in Proposition 2.2, and D is uniquely determined by L up to a term which is constant times identity. Proposition 2.3. Let L be as in Proposition 2.1. Define a linear operator D on R[x] by its action on monomials: n−1 D(1) = 0, D(xn)= XkL(xn−k−1). (2.14) k=0 X Then D satisfies the properties of Proposition 2.2 with λ = 0. Any other 0 operator D satisfying (2.12) and having 1 as eigenfunction differs from D given by (2.14) by a constant times identity. 5 Proof Formula(2.12)actingonxn followsdirectlyfrom(2.14). From(2.5) together with DX = L+XD acting on xn we see by induction that (2.10) holds with λ satisfying (2.13). In order to prove that D is symmetric, n observe that, for n> 0, hDxn,xmi−hxn,Dxmi = hDXxn−1,xmi−hxn−1,XDxmi = h(L+XD)xn−1,xmi+hxn−1,(L−DX)xmi = hDxn−1,xm+1i−hxn−1,Dxm+1i. Hence hDxn+m,1i = hDxn,xmi−hxn,Dxmi = −h1,Dxn+mi. So hDxn+m,1i = 0 and hDxn,xmi =hxn,Dxmi. Finally, for the uniqueness, let D and D satisfy (2.14) and and let 1 2 them have 1 as eigenfunction. Then D − D commutes with X. Hence 1 2 (D −D )(xn)= Xn(D −D )(1) = Xn(c1) = cxn. 1 2 1 2 Remark 2.4. If L satisfies (2.12) and D satisfies (2.11) then the lowering andraisingrelations(2.7)and(2.8)arepreservedifweaddtotheirleft-hand sides a term g(x)(D−λ )p (x), where g(x) is any function. n n Remark 2.5. VanMoerbeke[22,§7](alsojointlywithAdlerin[1,§5])gives explicit skew symmetric first order differential operators L (Q in his nota- tion) satisfying Proposition 2.1 for thecase of Jacobi, Laguerreand Hermite polynomials. More generally he gives these operators if the weight function is perturbed by multiplying the weight function with exp( ∞ t xj) (only j=1 j finitely many t nonzero). Then (2.5) and (2.6) are no longer valid, but the j P right-hand side of (2.6) has to be replaced by some linear combination of orthogonal polynomials p (x) with coefficients depending on the t . In all k j these cases L satsifies the so-called string equation [X,L] = f (X) (2.15) 0 with f a polynomial given by f (X) = 1−X2, X, 1 for Jacobi, Laguerre 0 0 and Hermite polynomials, respectively, and for their deformations. This is inspired by the matrix models of the physicists, where the Hermite case occurs, see Witten [23, §4c], in particular (4.43), (4.53), (4.65), (4.66). Remark 2.6. The structure relation (2.6) can be written in a form which is symmetric with respect to the variables n and x. Write p(n,x) instead of p (x). Define operators J and Λ acting on functions on Z by n ≥0 (Jφ)(n) := A φ(n+1)+B φ(n)+C φ(n−1), (2.16) n n n (Λφ)(n) := λ φ(n). (2.17) n 6 Now use (2.9), (2.12), (2.3) and (2.11) in order to rewrite (2.6) as [D,X]p(n, .) (x) = [J,Λ]p(.,x) (n). (2.18) (cid:0) (cid:1) (cid:0) (cid:1) This equation essentially occurs in Duistermaat & Gru¨nbaum [6, (1.7)] and Gru¨nbaum & Haine [10, (3)], where they study the bispectral problem with D a differential operator and n continuous respectively discrete. For p , D and λ q-dependent, a q-analogue of (2.18) has been con- n n sidered in the literature which involves q-commutators instead of ordinary commutators: (q21DX −q−12XD)p(n, .) (x) = (q21JΛ−q−21ΛJ)p(.,x) (n). (2.19) (cid:16) (cid:17) (cid:16) (cid:17) In fact, in Zhedanov [24] formulas (1.4) and (1.8a) may be interpreted as formulas (2.11) and (2.3) in the present paper, respectively. Then formula (1.8b) together with (1.1a) in [24] can be interpreted as (2.19) above. See also Gru¨nbaum & Haine [11] on the bispectral problem in the q-case, where some formulas in section 3 may be close to (2.19) above. 1 Formula (1.1c) in [24] may be interpreted as the q-commutator q2XL− q−12LX being equal to a linear combination of X, D and L. A similar formula is observed in [11, (2.4)], where it is called the q-string equation as a q-analogue of the string equation (2.15). 3 Jacobi polynomials (α,β) Jacobi polynomials P (x) = P (x) (see [7, §10.8]) are orthogonal with n n respect to the inner product 1 hf,gi = f(x)g(x)(1−x)α(1+x)βdx (α,β > −1). Z−1 The coefficients in the three-term recurrence relation (2.3) are 2(n+1)(n+α+β+1) A = , (3.1) n (2n+α+β+1)(2n+α+β +2) β2−α2 B = , (3.2) n (2n+α+β)(2n+α+β+2) 2(n+α)(n+β) C = . (3.3) n (2n+α+β)(2n+α+β+1) For the operator L given by (1.9) we see immediately that (2.5) holds with γ = −1(2n+α+β+2). (3.4) n 2 7 and that the skew symmetry (1.7) holds. Hence, by Proposition 2.1, the structure relation (2.6) is valid. Explicitly it reads as follows: d (1−x2) − 1 α−β +(α+β +2)x P(α,β)(x) dx 2 n (cid:16) (n+1)(n+(cid:0)α+β+1) (α,β) (cid:1)(cid:17)(n+α)(n+β) (α,β) = − P (x)+ P (x). (3.5) 2n+α+β+1 n+1 2n+α+β +1 n−1 We can also make explicit formulas (1.11) and (2.13) with d2 d D = 1(1−x2) +1 β−α−(α+β+2)x , λ = −1n(n+α+β+1). 2 dx2 2 dx n 2 (3.6) (cid:0) (cid:1) Formula (3.5) can be combined with the three-term recurrence relation in order to obtain the structure relation of the form (1.1): d 2n(n+1)(n+α+β +1) (1−x2) P(α,β)(x) = − P(α,β)(x) dx n (2n+α+β +1)(2n+α+β +2) n+1 2n(n+α+β +1)(α−β) + P(α,β)(x) (2n+α+β)(2n+α+β +2) n 2(n+α)(n+β)(n+α+β +1) (α,β) + P (x). (3.7) (2n+α+β)(2n+α+β +1) n−1 4 Askey-Wilson polynomials Askey-Wilson polynomials (see [4], [9, §7.5], [14, §3.1]) are defined by p [z] = p 1(z+z−1) = p 1(z+z−1);a,b,c,d |q n n 2 n 2 (ab,ac,ad;q) q−n,qn−1abcd,az,az−1 (cid:0) (cid:1) n (cid:0) (cid:1) := φ ;q,q . (4.1) an 4 3 ab,ac,ad (cid:18) (cid:19) If a,b,c,d ∈ C satisfy a2,b2,c2,d2,ab,ac,ad,bc,bd,cd ∈/ {q−k |k = 0,1,2,...} (4.2) then these polynomials satisfy the orthogonality relation 1 dz hf,gi := f[z]g[z]w(z) , (4.3) 4πi z IC (z2,z−2;q) ∞ w(z) := , (4.4) (az,az−1,bz,bz−1,cz,cz−1,dz,dz−1;q) ∞ where C is the unit circle traversed in positive direction with suitable de- formations to separate the sequences of poles converging to zero from the sequences of poles diverging to ∞. If a,b,c,d are four reals, or two reals 8 and one pair of complex conjugates, or two pairs of complex conjugates such that |ab|,|ac|,|ad|,|bc|,|bd|,|cd| < 1, then the Askey-Wilson polynomi- als are real-valued and their orthogonality can be rewritten as an integral over x = 1(z +z−1) ∈ [−1,1] plus a finite sum over real x-values outside 2 [−1,1]. This finite sum does not occur if |a|,|b|,|c|,|d| < 1. Now k , h , B , D and λ in §2 can be specified for the Askey-Wilson n n n n case as follows. h 1−abcdq−1 (q,ab,ac,ad,bc,bd,cd;q) k = 2n(abcdqn−1;q) , n = n , n n h 1−abcdq2n−1 (abcdq−1;q) 0 n B = (a+b+c+d)(q−abcdqn−1−abcdqn+abcdq2n) n (cid:16) +(bcd+abd+acd+abc)(1−qn−qn+1+abcdq2n−1) qn−1 (cid:17) × , (4.5) 2(1−abcdq2n−2)(1−abcdq2n) Dp = λ p , where n n n 1(1−q−1)(Df)[z] = v(z)f[qz]− v(z)+v(z−1) f[z]+v(z−1)f[q−1z], 2 (1−az)(1−bz)(1−cz)(1−dz) v(z) = (cid:0) (cid:1) , (1−z2)(1−qz2) 1(1−q−1)λ = (q−n−1)(1−abcdqn−1). (4.6) 2 n Define an operator L acting on symmetric Laurent polynomials: (Lf)[z] := (1−az)(1−bz)(1−cz)(1−dz)z−2f[qz] (cid:16) −(1−a/z)(1−b/z)(1−c/z)(1−d/z)z2f[q−1z] (z−z−1)−1 (4.7) (az,az−1,bz,bz−1,cz,cz−1,dz,dz−1;q2) (cid:17) = 1(1−q2) ∞ 2 (qz2,qz−2;q2) ∞ δq2 (qz2,qz−2;q2)∞f[z] × . (4.8) δ x (qaz,qaz−1,qbz,qbz−1,qcz,qcz−1,qdz,qdz−1;q2) q2 ∞! Here δq 2(g[q12z]−g[q−12z]) g[z] := (4.9) δqx (q21 −q−12)(z −z−1) is a divided q-difference operator (see [4, §5]). It tends to d g(x) as q ↑ 1. dx Then L sends symmetric Laurent polynomials of degree n to symmetric Laurent polynomials of degree n+1, γ = 2(abcdqn −q−n), (4.10) n 9 and L is skew symmetric with respect to the inner product (4.3), so (1.7) holds. For the proof of (1.7) note that (1−az)(1−bz)(1−cz)(1−dz)f[qz]g[z] dz w(z) z2(z−z−1) z IC f[qz]g[z] (q2z2,q−2z−2;q) dz ∞ = (qz)−2(qz−(qz)−1) (qaz,az−1,qbz,bz−1,qcz,cz−1,qdz,dz−1;q) z IC ∞ f[z]g[q−1z] (z2,z−2;q) dz ∞ = z−2(z−z−1) (az,qaz−1,bz,qbz−1,cz,qcz−1,dz,qdz−1;q) z IC ∞ f[z](1−az)(1−bz)(1−cz)(1−dz)g[q−1z] dz = w(z) . z−2(z−z−1) z IC (Actually, the contour deformation above can be done with avoidance of poles in the generic case of complex a,b,c,d such that thefourline segments connectinga,b,c,dwith0avoidthefourhalflines{ta−1,tb−1,tc−1,td−1 |t ≥ 1.) Alternatively, we can observe that L = [D,X], γ = λ − λ with n n+1 n D,λ given by (4.6), and apply Proposition 2.2. n By Proposition 2.1 we have the structure relation (2.6), which can be more explicitly written (with usage of (4.7)) as (1−abcdqn−1)p [z] (Lp )[z] = − n+1 +(1−abqn−1)(1−acqn−1)(1−adqn−1) n qn(1−abcdq2n−1) (1−qn)p [z] ×(1−bcqn−1)(1−bdqn−1)(1−cdqn−1) n−1 (4.11) qn−1(1−abcdq2n−1) We can also write the lowering and raising relations (2.7), (2.8) more explic- itly (with usage of (4.7) and (4.5)): (Lp )[z]−(abcdqn−q−n)(z+z−1−2B )p [z] = (1−abqn−1)(1−acqn−1) n n n (1+q)(1−qn)p [z] ×(1−adqn−1)(1−bcqn−1)(1−bdqn−1)(1−cdqn−1) n−1 , qn(1−abcdq2n−2) (4.12) (Lp )[z]+(abcdqn−1−q1−n)(z+z−1−2B )p [z] n n n (1+q)(1−abcdqn−1) = − p [z]. (4.13) qn(1−abcdq2n) n+1 LoweringandraisingrelationsforAskey-Wilson polynomialswereearlier obtained by Bangerezako [5, (41)]. He obtained his lowering and raising operator by the factorization method, see Proposition 1 in [5]. His lowering and raising relation can be obtained from (4.12) and (4.13), respectively, by adding 1(1−q−1)(z−qz−1)(Dp )[z] to the left-hand sides of these relations 2 n (cf. Remark 2.4). 10