On the Askey-Wilson polynomials

Constr. Approx. (1992) 8:363-369 CONSTRUCTIVE APPROXIMATION 9 1992 Springer-VerlagNew York Inc. On the Askey-Wilson Polynomials N. M. Atakishiyev and S. K. Suslov Abstract. Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach. In [3], [4], and [16] a number of new systems of orthogonal polynomials (i.e., 4F3 and 4q~3 polynomials) were introduced. The main properties of these polynomials have been established on the basis of the theory of hypergeometric and basic hypergeometric series [9], [11]. As is clear now, the Askey-Wilson [3], [4] and Wilson [16] polynomials are the most general extensions of the Jacobi, Laguerre, and Hermite polynomials known at present. The families of polynomials, considered in [1], [3], [4], and [16], together with their special and limiting cases, form a mathematical entity--the classical orthogonal polynomials of a discrete variable on nonuniform lattices [1], [15]. The introduction of such an object made it possible to generalize all the fundamental properties characteristic of Hahn's approach [12], [10]. We shall disc/ass these properties for the most general system--the Askey-Wilson polynomials, in the framework of an approach developed in [15], [7]. 1. Hypergeometrie-Type Difference Equation Consider the hypergeometric-type difference equation in the self-adjoint form (for details, see [15]) (1) (2) A Vxl(z) [cr(z)p(z)~l+2p(z)y(z)=O ' A[cr(z)p(z)] = p(z)~(z)Vx~(z), on the lattice x(z) = cosh 2o9z = ~q" + q-Z), q = e2~,, where xl(z) = x(z + 89and AT(z) = Vf(z + 1) = f(z + 1) - f(z). Date received: March 14, 1991. Date revised: September 30, 1991. Communicated by Mourad Ismail. AMS classification: 33D45. Key words and phrases: Hahn's approach, Askey-Wilson polynomials, Difference operators and equations, Sturm-Liouville problem, Orthogonality, Moments. 363 364 N.M. Atakishiyev and S. K. Sustov In the very general form, we have [7] (3) a(z) = q-2~(q~ _ a)(qZ _ b)(q~ _ c)(q~ _ d), (4) p(z) = p(z, a, b, c, d) = fq(z) H 9(z, v). vma, b,c,d Here f~-l(z ) = i.q(2z)rq(_2z)(q~ _ q-Z), Iql < 1; 0-a(z, V) ---- f i [1 -- v(q z + q-Z)qk d-- o2q2k'], II)1 < 1. k=0 For the definition and properties of Fq(z) we refer the reader to [11]. Since p1(z) = tT(z + 1)p(z + 1) = -ql/2p(z + 89aq x/2, bq~/2, cql/2, dqX/2), by using the operator 6f(z)= A f ( z - 1/2), equation (1) for the Askey-Wilson polynomials may be rewritten in the form [4-] 6x(z)6 [ p(z, aq 1/2, bq 1/2, cq 1/2, dq 1/2) 6Y(z) 6x(z)_J1 = 2"q-1/2P(Z' a, b, c, d)Y(z), where 2,q-1/z = 4q(1 - q)-:(1 - q-")(1 - abcdq"-1) (cf. [4-]). 2. Rodrigues-Type Formula The polynomial solutions Y.(z) = Y.[x(z)] of (I), corresponding to definite values of 2 = 2., can be .found by the Rodrigues-type formula [15] Bn (5) Y.(z) = p ~ v~"l[p.(z),], where p,(z) = p(z + n) 1-~,=1 a(z + k), V V V , xk = x(z + k/2). Since [7] V~")p,(z) = p, z - and for the Askey-Wilson polynomials we have (n , ) pn(z) = (--1)nq-n("-2)/2p Z + ~, aq n/2, bq n/2, cq n/2, dq n/2 , then (5) takes the form [4] (6) Y,(z)=(--1)"B"q-"t"-2)/2Ib~z)l" p(z, a, b, c, d) p(z, aq "/z, bq n/2, cq n/2, dq"/2). On the Askey-Wilson Polynomials 365 3. Series Expansion The general series expansion for polynomials (5) has the form [8] n--1 Y.(z) = B. I-I z~(z~) ~, k-1 (2. -- 2,,)[x(z v + m) -- x(z)] II k=O ,.=o zm(zv)y(m+ 1) 7(n) = (q~/2 _ q-,/Z)(ql/2 _ q-1/2)-1, p=o ' where a(zo) = 0, for other notations, see [15]. F r o m this formula it is easy to obtain the representation for the A s k e y - W i l s o n polynomials through the basic hypergeometric series (7) Y,(z) = 2"Bnq -.(3.- 5)/,* a"(1 - q)" (ab, q),(ac, q).(ad, q). ,,q~3(q-", 9 abcdq"- 1, aqZ, aq-Z ab, ac, ad I q' q)" \ Here (v, q)o = 1, (v, q), = (1 - v)(1 - vq)-..(1 - vq"- 1), n = 1, 2 . . . . . W h e n B, = 2-"(1 --q)nqn(3n-5)/4 the polynomials (7) coincide with those introduced in [4]. They have the p r o p e r t y of self-duality [4], [14]. 4. Formula of Difference Differentiation F o r the polynomials Y,(z) = p.(x, a, b, c, d[q) from (1)-(2), (6), or (7) we can derive the formula of difference differentiation [4] (8) ~ 6 ox~z) p.(x, a, b, c, dlq) 2qCn+1)/2 - - 1-q (1 - q-")(1 -- abcdq"- l)p,_ l(x, aq 1/2, bq 1/z, cq ~/2, dql/2lq), i.e., difference differentiation on a n o n u n i f o r m lattice results in polynomials of the same type (7). 5. Orthogonality Property The orthogonality p r o p e r t y for the A s k e y - W i l s o n polynomials with respect to a continuous measure was established in [4] by direct evaluation of an integral. Another p r o o f is given in [13]. Here we shall briefly discuss this p r o p e r t y by analogy with the p r o o f of an orthogonality of eigenfunctions for the S t u r m Liouville p r o b l e m [7]. Under conditions (9) ;c V[pl(z)x](z)] dz = 0 (k=0,1,2 . . . . ), 366 N . M . A t a k i s h i y e v a n d S. K. Suslov Imz -1 - z~- log -I q C I / -iTr log -1 q 0 Rcz irr log - l q - 1 + irr log -~ q rig. 1 the polynomial solutions o f ( l ) are orthogonal on a contour C in the complex z-plane: I (10) .f,- Ym[X(Z)]~.Ex(z)]p(z)Vxl(z) dz = 0 (m # n). In the case under consideration, it is necessary to choose the line segment z c = iO log-1 q, _ rc < 0 < rq as a contour C (see Fig. 1), and to require that the points z~ = log v/log q, v = a, b, c, d (a(%) = 0), which define four sequences of the poles (see formula (4)) of the function p(z) = - p ( - z ) , going to the right (left), be located on the right (left) of the contour C, i.e., 1 (11) Im z, = Im Zo Re zv > Re z c. Owing to (2), pl(z) = a ( - z ) p ( z ) , and the integrand in (9) does not have the poles inside the parallelogram in the figure. Therefore the integral over the whole of the boundary of this parallelogram is equal to. zero, ensuring the fulfillment of condition (9) due to the 2rci log-Xq periodicity of the integrand. From this it follows that the orthogonality property (10) is the case. Now, using the symmetry of the integrand in (10) with respect to the change z ~ - z , we finally come to the orthogonality relation for the Askey-Wilson polynomials [4] J~ pm(x)pn(x)p(x)dx = 0 (m # n), 1 Since Re(z o - Zc) = l o g l v l / l o g l q l , then from (11) it follows t h a t [ql < 1 a n d m a x . . . . b,c,dtVl < 1. 367 On the Askey-WilsonPolynomials where p(x) f +2(1 i l--2x2)qkwq l 2k] = (1 - x 2 ) - 1/2 k=O I~ f i [1 -- 2vxq k + v2q 2k'] v = a , b , c , d k=O 6. Property of Moments The property of moments for the polynomial solutions of equation (1), generalizing the corresponding property in Hahn's theory [12], was considered in [17] and [6]. For the moments of the form C. = fc[X(Z) - x(z a + n - 1)]~")p(z)Vxl(z) dz, (12) with the generalized power n--1 [x(z) - x(za + n -- 1)] ~") = ~I [x(z) - x(z. + k)] k=O = _ "1] "q-nt,,-1)/2(aq-Z ' q).(aqZ, q)n, 2a] ~(Za) = O, the relation [-6] C.+ 1 = - a(z. + n) + z(z. + n)Vxl(z. + n) C. x.+ lVX. + l(z.) (13) is valid. Here we have used the notation x. = -2.~-1(n). For the Askey-Wilson polynomials from (13) we find (14) . - 1 a ( - z~ - k) (ab, q).(ac, q).(ad, q). Co. Cn = ( - 1)" 17I Co = ( - 1)" k=O Xk+ 1VXk+l(Za) (2a)"q "("- 1)~2(abed, q). We also note that the moments of the form (15) M . = fc(aq-~, q).(aq z, q).p(z)Vxx(z ) dz are equal to (16) Mn= (ab, q).(ac, q),,(ad, q),, Mo. (abcd, q),, Now it remains to evaluate the integral (17) Mo = Mo(a, b, c, at) = ;c p(z, a, b, c, d)Vxl(z ) dz. 368 N.M. Atakishiyev and S. K. Suslov According to (4) M 1= fc(aq-, q)l(aq z, q)lp(z, a, b, c, d)Vxl(z) dz = fc p(z, aq, b, c, d)Vxl(z) dz = Mo(aq, b, c, d). Therefore, in view of (16) 1 - abcd Mo(a, b, c, d) = (1 - ab)(1 - ac)(1 - ad) Mo(a q, b, c, d). From this formula, by using the symmetry of (17) under permutations of parameters a, b, c, d, we can find (18) Mo(a, b, c, d) = (abcd, q)o~ Mo(0, 0, 0, 0), (ab, q)~(ac, q)oo(ad, q)oo(bc, q)~(bd, q)o~(cd, q)~o o0 1 - vqk), Iql < 1. To evaluate the constant Mo(0, 0, 0, where (v, q)oo = l-[k=O( O) = Mo(q) we use the Jacobi triple product identity [11], which gives Mo(q) = 2n(q, q)L 1. As a result we obtain the Askey-Wilson q-beta integral [4], [2], [13]. The representation for the Askey-Wilson polynomials in terms of moments has the form (19, Y~(z, = C det(jl/xitpkd#)i=o.1 ...... -1 \ q)k / k=O, 1 ..... n Mo,o 9 Mo, Mn-l,o 9"" M.-1,. n = const q~o ''' q)n with q~k(Z) = (aq-~, q)k(aq ~, q)k, Mik = f c ~kk(Z) = qgk(Z)I,~b; ~tI(Z)tPk(Z)P(Z'a, b, c, d)Vxl(z ) dz = Mo(aq k, bq i, c, d). Wilson [17] has suggested a method for the direct evaluation of this determinant and pointed out that it can be applied to the case of the 4~o3-polynomials. This proof can be easily given in terms of the CHARM polynomials [5]. The authors are grateful to M. E. H. Ismail and M. Rahman for discussions. On the Askey-Wilson Polynomials 369 References 1. G. ANDREWS, R. ASKEY (1985): Classical Orthogonal Polynomials. In: Lecture Notes in Mathematics, Vol. 1171, Berlin: Springer-Vedag, pp. 36-62. 2. R. ASKEY (1988): Beta integrals and q-extensions. Proceedings of the Ramanujan Centennial International Conference, Ramanujan Mathematical Society, pp.. 85-102. 3. R. ASKEY, J. A. WILSON (1979): A set of orthogonal polynomials that generalize the Racah coefficients or 6j-symbols. SIAM J. Math. Anal., 10(5): 1008-1016. 4. R. ASKEY, J. A. WILSON (1985): Some basic hypergeometric orthogonalpolynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 319:55. 5. N.M. ATAKISHIYEV,G. 1. 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