Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Symmetrized Chebyshev polynomials

Author: Igor Rivin
Journal: Proc. Amer. Math. Soc. 133 (2005), 1299-1305
MSC (2000): Primary 05C25, 05C20, 05C38, 41A10, 60F05
Published electronically: November 19, 2004
MathSciNet review: 2111935
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that $T_n(c \cos \theta)$ and $U_n(c \cos \theta)$ are positive definite functions. We further show that a Central Limit Theorem holds for the coefficients of our polynomials.

References [Enhancements On Off] (What's this?)

  • [FellerII] William Feller. An Introduction to Probability Theory and its Applications, Volume II, second edition, John Wiley, New York, 1971. MR 0270403 (42:5292)
  • [Rivin99] Igor Rivin. Growth in Free Groups (and other stories), preprint math.CO/9911076
  • [Rivlin90] Theodore J. Rivlin. Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, second edition, Wiley Interscience, 1990. MR 1060735 (92a:41016)
  • [Sharp01] Richard Sharp. Local limit theorems for free groups, Math. Ann. 321(4), 2001, pp. 889-904. MR 1872533 (2002k:20039)
  • [Schur73] Issai Schur. Arithmetischen über die Tschebyscheffschen Polynome, in Gesammelte Abhandlungen, III, Springer, Berlin, 1973, pp. 422-453. MR 0462891 (57:2858a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 05C25, 05C20, 05C38, 41A10, 60F05

Retrieve articles in all journals with MSC (2000): 05C25, 05C20, 05C38, 41A10, 60F05

Additional Information

Igor Rivin
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122

Keywords: Chebyshev polynomials, positivity, central limit theorem
Received by editor(s): February 7, 2003
Received by editor(s) in revised form: January 9, 2004
Published electronically: November 19, 2004
Additional Notes: These results first appeared in the author’s 1998 preprint “Growth in free groups (and other stories)”, but seems to be of independent interest. The positivity result was preprint math.CA/0301210, but there appears to be no reason to separate it from the limiting distribution result, and many reasons to keep them together. The author would like to thank the Princeton University Mathematics Department for its hospitality, and the NSF DMS for its support. He would also like to thank the anonymous referee for useful comments on an earlier version of this paper
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society