The Ramanujan Journal

, Volume 41, Issue 1–3, pp 147–169 | Cite as

Nonlinear recurrences related to Chebyshev polynomials



We use two nonlinear recurrence relations to define the same sequence of polynomials, a sequence resembling the Chebyshev polynomials of the first kind. Among other properties, we obtain results on their irreducibility and zero distribution. We then study the \(2\times 2\) Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval \((-1,1)\), while those of the other half lie on the unit circle. Some further extensions and generalizations of these results are indicated.


Recurrence relations Polynomial sequences Chebyshev polynomials Zeros Irreducibility 

Mathematics Subject Classification

Primary: 30C15 Secondary: 33C45 12E05 


  1. 1.
    Dilcher, K., Stolarsky, K.B.: Sequences of polynomials whose zeros lie on fixed lemniscates. Period. Math. Hung. 25(2), 179–190 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Based, in part, on notes left by Harry Bateman, vol. II. McGraw-Hill Book Company Inc, New York (1953)Google Scholar
  3. 3.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010). Online companion: NIST Digital Library of Mathematical Functions., Release 1.0.5 of 2012–10-01
  4. 4.
    Rivlin, T.J.: Chebyshev Polynomials, 2nd edn. Wiley, New York (1990)MATHGoogle Scholar
  5. 5.
    Sved, M.: Divisibility—with visibility. Math. Intell. 10(2), 56–64 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    The On-Line Encyclopedia of Integer Sequences, published electronically at (2013)
  7. 7.
    Weisstein, E.W.: Chebyshev Polynomial of the First Kind. From MathWorld-A Wolfram Web Resource.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations