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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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$q$-Deformation of the Kac-Sylvester tridiagonal matrix
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by J. F. van Diejen PDF
Proc. Amer. Math. Soc. 149 (2021), 2291-2304 Request permission


Upon replacing integers by $q$-integers, one arrives at a natural $q$-deformation of a remarkably ubiquitous tridiagonal integer matrix whose eigenvalues and eigenvectors were first computed by J.J. Sylvester and M. Kac, respectively. The present note computes the eigenvalues and eigenvectors of this novel $q$-deformed Kac-Sylvester matrix through a connection with Macdonald’s hyperoctahedral Hall-Littlewood polynomials. This yields the spectrum as a minimizer of an associated Morse function reminiscent of Stieltjes’ electrostatic potential for the roots of classical orthogonal polynomials.
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Additional Information
  • J. F. van Diejen
  • Affiliation: Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile
  • MR Author ID: 306808
  • ORCID: 0000-0002-5410-8717
  • Email:
  • Received by editor(s): July 10, 2020
  • Received by editor(s) in revised form: July 17, 2020
  • Published electronically: March 18, 2021
  • Additional Notes: Work was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant # 1210015.
  • Communicated by: Mourad Ismail
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 2291-2304
  • MSC (2020): Primary 15A18; Secondary 05E05, 15B36, 33D52, 47B36, 65F15
  • DOI:
  • MathSciNet review: 4246783