Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

Roman Wituła; Damian Słota

Open Mathematics (2006)

  • Volume: 4, Issue: 3, page 531-546
  • ISSN: 2391-5455

Abstract

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In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x 2n + y 2n , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.

How to cite

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Roman Wituła, and Damian Słota. "Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences." Open Mathematics 4.3 (2006): 531-546. <http://eudml.org/doc/269281>.

@article{RomanWituła2006,
abstract = {In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x 2n + y 2n , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.},
author = {Roman Wituła, Damian Słota},
journal = {Open Mathematics},
keywords = {11B83; 26C99; 11B39},
language = {eng},
number = {3},
pages = {531-546},
title = {Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences},
url = {http://eudml.org/doc/269281},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Roman Wituła
AU - Damian Słota
TI - Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences
JO - Open Mathematics
PY - 2006
VL - 4
IS - 3
SP - 531
EP - 546
AB - In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x 2n + y 2n , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.
LA - eng
KW - 11B83; 26C99; 11B39
UR - http://eudml.org/doc/269281
ER -

References

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  1. [1] L. Carlitz and J.A.H. Hunter: “Sums of Powers of Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 7, (1969), pp. 467–473. Zbl0194.07301
  2. [2] A.F. Horadam: “Basic properties of a certain generalized sequence of numbers”, Fibonacci Quart., Vol. 3, (1965), pp. 161–176. Zbl0131.04103
  3. [3] A.F. Horadam: “Generating functions for powers of a certain generalised sequence of numbers”, Duke Math. J., Vol. 32, (1965), pp. 437–446. http://dx.doi.org/10.1215/S0012-7094-65-03244-8 
  4. [4] T. Koshy: Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001. 
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  6. [6] R.S. Melham: “Sums of Certain Products of Fibonacci and Lucas Numbers - Part I”, Fibonacci Quart., Vol. 37, (1999), pp. 248–251. Zbl0936.11012
  7. [7] R.S. Melham: “Families of Identities Involving Sums of Powers of the Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 37, (1999), pp. 315–319. Zbl0939.11010
  8. [8] R.S. Melham: “Sums of Certain Products of Fibonacci and Lucas Numbers - Part II”, Fibonacci Quart., Vol. 38, (2000), pp. 3–7. Zbl0943.11007
  9. [9] R.S. Melham: “Alternating Sums of Fourth Powers of Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 38, (2000), pp. 254–259. Zbl0943.11008
  10. [10] J. Morgado: “Note on some results of A.F. Horadam and A.G. Shannon concerning a Catalans’s identity on Fibonacci Numbers”, Portugal. Math., Vol. 44, (1987), pp. 243–252. 
  11. [11] J. Morgado: “Note on the Chebyshev polynomials and applications to the Fibonacci numbers”, Portugal. Math., Vol. 52, (1995), pp. 363–378. Zbl0844.11012
  12. [12] S. Paszkowski: Numerical Applications of Chebyshev Polynomials and Series, PWN, Warsaw, 1975 (in Polish). Zbl0423.65012
  13. [13] P. Ribenboim: Fermats’s Last Theorem For Amateurs, Springer, New York 1999. 
  14. [14] J. Riordan: “Generating functions for powers of Fibonacci numbers”, Duke Math. J., Vol. 29, (1962), pp. 5–12. http://dx.doi.org/10.1215/S0012-7094-62-02902-2 Zbl0101.28801
  15. [15] T. Rivlin: Chebyshev Polynomials from Approximation Theory to Algebra and Number Theory, 2nd ed., Wiley, New York, 1990. Zbl0734.41029
  16. [16] R. Wituła and D. Słota: “On Modified Chebyshev Polynomials”, J. Math. Anal. Appl., (2006), in print. Zbl1124.33012

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