Abstract
We evaluate different Hankel determinants of Rogers–Szegö polynomials, and deduce from it continued fraction expansions for the generating function of RS polynomials. We also give an explicit expression of the orthogonal polynomials associated to moments equal to RS polynomials, and a decomposition of the Hankel form with RS polynomials as coefficients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G.E. Andrews, The Theory of Partitions (Addison-Wesley, Reading, MA, 1976).
R. Askey and M.E.H. Ismail, A generalization of ultraspherical polynomials, in: Studies in Pure Mathematics, ed. P. Erdös (Birkhäuser, Boston, MA, 1983) pp. 55–78.
B. Baillaud and H. Bourget, Correspondance d'Hermite et de Stieltjes (Gauthier-Villars, Paris, 1905).
A. Berkovich and O. Warnaar, Positivity preserving transformations for q-binomial coefficients, math.CO/0302320 (2003).
C. Brezinski, History of Continued Fractions and Padé Approximants (Springer, Berlin, 1991).
K.A. Driver and D. Lubinsky, Convergence of Padé approximants for a q-hypergeometric series, Aequationes Math. 42 (1991) 85–106.
N.J. Fine, Basic Hypergeometric Series and Application, Mathematical Surveys and Monographs, Vol. 27 (Amer. Math. Soc., Providence, RI, 1988).
J. Goldman and G.-C. Rota, The number of subspaces of a vector space, in: Recent Progress in Combinatorics, ed. W.T. Tutte (Academic Press, New York, 1969) pp. 75–83.
K. Hikami and B. Basu-Mallick, Supersymmetric Polychronakos spin chain: Motifs, distribution functions and characters, Nuclear Phys. B 566 (2000) 511–528.
M. Ismail and D. Stanton, More orthogonal polynomials as moments, in: Mathematical Essays in Honor of Gian-Carlo-Rota (Birkhäuser, Bäsel, 1998) pp. 377–396.
V. Kac and P. Cheung, Quantum Calculus (Springer, New York, 2002) pp. 25–26.
A. Lascoux, Symmetric functions and combinatorial operators on polynomials, in: CBMS-AMS Conference Lecture Notes (to appear).
D. Lubinsky and E.B. Saff, Convergence of Padé approximants of partial theta functions and the Rogers-Szego polynomials, Construct. Approx. 3 (1987) 331–361.
I.G. Macdonald, Symmetric Functions and Hall Polynomials (Clarendon Press, Oxford, 1979).
A. Nijenhuis, A.E. Solow and H.S. Wilf, Bijective methods in the theory of finite vector spaces, J. Combin. Theory A 37 (1984) 80–84.
G. Szegö, Ein Beitrag zur Theorie der Thetafunktionen, Sitz. Preuss. Akad.Wiss., Phys.-Math. Klasse 19 (1926) 249–252 (reprinted in: Collected Papers, Vol. 1, ed. R. Askey (Birkhaüser, Boston, MA) pp. 795–805).
T.J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sc. Toulouse 8 (1894) 1–122.
H.S. Wall, Analytic Theory of Continued Fractions (Chelsea, New York, 1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hou, QH., Lascoux, A. & Mu, YP. Continued Fractions for Rogers–Szegö Polynomials. Numerical Algorithms 35, 81–90 (2004). https://doi.org/10.1023/B:NUMA.0000016604.14688.21
Issue Date:
DOI: https://doi.org/10.1023/B:NUMA.0000016604.14688.21