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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Symmetrized Chebyshev polynomials
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by Igor Rivin PDF
Proc. Amer. Math. Soc. 133 (2005), 1299-1305 Request permission

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.
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Additional Information
  • Igor Rivin
  • Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
  • MR Author ID: 295064
  • Email: rivin@math.temple.edu
  • 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
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 1299-1305
  • MSC (2000): Primary 05C25, 05C20, 05C38, 41A10, 60F05
  • DOI: https://doi.org/10.1090/S0002-9939-04-07664-6
  • MathSciNet review: 2111935