Recurrence equations and their classical orthogonal polynomial ...

More documents

Recommendations

Info

326 W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327 Using our implementation, these results are obtained by > RE :¼ pðnþ2Þ x pðnþ1Þþalpha q^n ðq^ðnþ1Þ 1Þ pðnÞ¼0; RE :¼ pðn þ 2Þ xpðn þ 1Þþaq n ðq ðnþ1Þ 1ÞpðnÞ ¼0 > REtoqDE(RE,p(n),q,x); Warning: parameters have the values fe ¼ 0; a ¼ dq þ d; c ¼ aqd þ ad; d ¼ d; b ¼ 0g ðx 2 " þ aÞDq Dq pðn; xÞ; 1 ; x ; q; x q xDqðpðn; xÞ; q; xÞ q 1 þ qð 1 þ qnÞpðn; xÞ ðq 1Þ 2 qn ¼ 0; qðqxÞ qðxÞ ¼ a q2x2 # knþ1 ; ¼ 1 þ a kn : Note that q-polynomials are not accessible with Koornwinder–Swarttouw’s rec2ortho. Note: The Maple implementation retode, and a worksheet retode.mws with the examples of this paper can be obtained from http://www.mathematik. uni-kassel.de/ koepf/Publikationen. Acknowledgements The first named author thanks Tom Koornwinder and Rene Swarttouw for helpful discussions on their implementation rec2ortho [12]. Examples 2 and 4 given by recurrence equation (31) were provided by them. Thanks to the support of their institutions I had a very pleasant and interesting visit at the Amsterdam universities in August 1996. References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964. [2] W.A. Al-Salam, The Bessel polynomials, Duke Math. J. 24 (1957) 529–545. [3] S. Bochner, €Uber Sturm–Liouvillesche Polynomsysteme, Math. Z. 29 (1929) 730–736. [4] W.C. Brenke, On polynomial solutions of a class of linear differential equations of the second order, Bull. Amer. Math. Soc. 36 (1930) 77–84. [5] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, NewYork, 1978. [6] W. Hahn, € Uber Orthogonalpolynome, die q-Differenzengleichungen, Math. Nachr. 2 (1949) 4– 34. [7] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Delft; electronic version available at http://aw.twi.tudelft.nl/ koekoek/research.html, 1998.
W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327 327 [8] W. Koepf, Hypergeometric Summation. An Algorithmic Approach to Summation and Special Function Identities, Vieweg, Braunschweig, 1998. [9] W. Koepf, D. Schmersau, Algorithms for classical orthogonal polynomials, Konrad-Zuse- Zentrum Berlin (ZIB), 1996 (preprint SC 96-23). [10] W. Koepf, D. Schmersau, Representations of orthogonal polynomials, J. Comput. Appl. Math. 90 (1998) 57–94. [11] T.H. Koornwinder, Compact quantum groups and q-special functions, in: Baldoni, Picardello (Eds.), Representations of Lie groups and Quantum Groups, Pitman Research Notes in Mathematics Series, vol. 311, Longman, Essex, 1994, pp. 46–128. [12] T.H. Koornwinder, Swarttouw, Rene: rec2ortho: an algorithm for identifying orthogonal polynomials given by their three-term recurrence relation as special functions. http:// turing.wins.uva.nl/ thk/recentpapers/rec2ortho.html, 1996–1998. [13] P. Lesky, Die Charakterisierung der klassischen orthogonalen Polynome durch Sturm– Liouvillesche Differentialgleichungen, Arch. Rat. Mech. Anal. 10 (1962) 341–351. [14] P. Lesky, € Uber Polynoml€osungen von Differentialgleichungen und Differenzengleichungen zweiter Ordnung, Anzeiger der € Osterreichischen Akademie der Wissenschaften, Math.- Naturwiss. Klasse 121 (1985) 29–33. [15] H. Melenk, Solving polynomial equation systems by Groebner type methods, CWI Quart. 3 (2) (1990) 121–136. [16] H. Melenk, Algebraic solution of nonlinear equation systems in REDUCE, Konrad-Zuse- Zentrum f€ur Informationstechnik, Technical Report TR-93-02, 1993. [17] H.M. M€oller, On decomposing systems of polynomial equations with finitely many solutions, Applicable Algebra in Engineering, Communication and Computing (AAECC) 4 (1993) 217– 230. [18] A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991.
Page 1 and 2: Recurrence equations and their clas
Page 3 and 4: W. Koepf, D. Schmersau / Appl. Math
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: according to (14), in terms of the

Recurrence equations and their classical orthogonal polynomial ...

Create successful ePaper yourself

Delete template?

Save as template?