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304 W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327 rðxÞDqD1=qyðxÞþsðxÞDqyðxÞþkq;nyðxÞ ¼0; where f ðqxÞ f ðxÞ Dqf ðxÞ ¼ ; q 6¼ 1; ðq 1Þx denotes the q-difference operator. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Computer algebra; Maple; Differential equation; Q-difference equation; Structure formula 1. Introduction Families of orthogonal polynomials pnðxÞ (corresponding to a positive-definite measure) satisfy a three-term recurrence equation of the form pnþ1ðxÞ ¼ðAnx þ BnÞpnðxÞ Cnpn 1ðxÞ ðn 2 N0; p 1 0Þ ð1Þ with CnAnAn 1 > 0, see e.g. [5, p. 20]. Moreover, Favard’s theorem states that the converse is also true. On the other hand, in practice one is often interested in an explicit solution of a given recurrence equation. Therefore it is an interesting question to ask whether a given recurrence equation has classical orthogonal polynomial solutions. In this paper an algorithm is developed which answers this question for a large class of classical orthogonal polynomial systems. Furthermore, we present results of our corresponding Maple implementation retode and compare these with the Maple implementation rec2ortho of Koornwinder and Swarttouw [12]. These programs overlap, but rec2ortho does not cover Bessel, Hahn and q-polynomials, whereas retode does not include the Meixner–Pollaczek case. 2. Classical orthogonal polynomials A family yðxÞ ¼pnðxÞ ¼knx n þ k 0 n xn 1 þ ðn 2 N0 :¼ f0; 1; 2; ...g; kn 6¼ 0Þ ð2Þ of polynomials of degree exactly n is a family of classical continuous orthogonal polynomials if it is the solution of a differential equation of the type rðxÞy 00 ðxÞþsðxÞy 0 ðxÞþknyðxÞ ¼0; ð3Þ
W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327 305 where rðxÞ ¼ax2 þ bx þ c is a polynomial of at most second-order and sðxÞ ¼dx þ e ðd 6¼ 0Þ is a polynomial of first-order ([3,13]). Since one demands that pnðxÞ has exact degree n, by equating the coefficients of xn in (3) one gets kn ¼ ðanðn 1ÞþdnÞ: ð4Þ Similarly, a family pnðxÞ of polynomials of degree exactly n, given by (2), is a family of classical discrete orthogonal polynomials if it is the solution of a difference equation of the type rðxÞDryðxÞþsðxÞDyðxÞþknyðxÞ ¼0; ð5Þ where DyðxÞ ¼yðx þ 1Þ yðxÞ and ryðxÞ ¼yðxÞ yðx 1Þ denote the forward and backward difference operators, respectively, and rðxÞ ¼ax2 þ bx þ c and sðxÞ ¼dx þ e are again polynomials of at most secondand of first-order, respectively, see e.g. [18]. Again, (4) follows. Finally, a family pnðxÞ of polynomials of degree exactly n, given by (2), is a family of classical q-orthogonal polynomials if it is the solution of a q-difference equation of the type rðxÞDqD1=qyðxÞþsðxÞDqyðxÞþkq;nyðxÞ ¼0; ð6Þ where f ðqxÞ f ðxÞ Dqf ðxÞ ¼ ; q 6¼ 1; ðq 1Þx denotes the q-difference operator [6], and rðxÞ ¼ax2 þ bx þ c and sðxÞ ¼dx þ e are again polynomials of at most second- and of first-order, respectively. By equating the coefficients of xn in (6) one gets kq;n ¼ a½nŠ 1=q ½n 1Šq d½nŠq; ð7Þ where the abbreviation 1 qn ½nŠq ¼ 1 q denotes the so-called q-brackets. Note that limq!1½nŠq ¼ n. It can be shown (see e.g. [14]) that any solution pnðxÞ of either (3), (5) or (6) satisfies a recurrence equation (1). The following is a general procedure to find the coefficients of the recurrence equation (as well as of similar structural formulas for classical orthogonal polynomials, see [10]) in terms of the coefficients a; b; c; d and e of rðxÞ and sðxÞ: 1. Substitute pnðxÞ ¼knx n þ k 0 n xn 1 þ k 00 n xn 2 þ in the differential equation (3), in the difference equation (5) or in the q-difference equation (6), respectively.
Page 1: Recurrence equations and their clas
Page 5 and 6: and kn 1 Cn ¼ knþ1 ðd ðan þ d
Page 7 and 8: ðqnðxÞ; rnðxÞ; snðxÞ 2Q½n;
Page 9 and 10: If we apply this algorithm to the r
Page 11 and 12: hence An ¼ knþ1 kn and therefore
Page 13 and 14: W. Koepf, D. Schmersau / Appl. Math
Page 15 and 16: Table 2 Normal forms of discrete po
Page 17 and 18: knþ1 kn :¼ An ¼ vn wn according
Page 19 and 20: With Koornwinder-Swarttouw’s rec2
Page 21 and 22: Table 3 Normal forms of q-polynomia
Page 23 and 24: according to (14), in terms of the
Page 25: W. Koepf, D. Schmersau / Appl. Math

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