# On some properties of Chebyshev polynomials

Hacène Belbachir; Farid Bencherif

Discussiones Mathematicae - General Algebra and Applications (2008)

- Volume: 28, Issue: 1, page 121-133
- ISSN: 1509-9415

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topHacène Belbachir, and Farid Bencherif. "On some properties of Chebyshev polynomials." Discussiones Mathematicae - General Algebra and Applications 28.1 (2008): 121-133. <http://eudml.org/doc/276945>.

@article{HacèneBelbachir2008,

abstract = {Letting $T_\{n\}$ (resp. $U_\{n\}$) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences $(X^\{k\}T_\{n-k\})_\{k\}$ and $(X^\{k\}U_\{n-k\})_\{k\}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space $_\{n\}[X]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_\{n\}$ and $U_\{n\}$ admit remarkableness integer coordinates on each of the two basis.},

author = {Hacène Belbachir, Farid Bencherif},

journal = {Discussiones Mathematicae - General Algebra and Applications},

keywords = {Chebyshev polynomials; integer coordinates},

language = {eng},

number = {1},

pages = {121-133},

title = {On some properties of Chebyshev polynomials},

url = {http://eudml.org/doc/276945},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Hacène Belbachir

AU - Farid Bencherif

TI - On some properties of Chebyshev polynomials

JO - Discussiones Mathematicae - General Algebra and Applications

PY - 2008

VL - 28

IS - 1

SP - 121

EP - 133

AB - Letting $T_{n}$ (resp. $U_{n}$) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences $(X^{k}T_{n-k})_{k}$ and $(X^{k}U_{n-k})_{k}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space $_{n}[X]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_{n}$ and $U_{n}$ admit remarkableness integer coordinates on each of the two basis.

LA - eng

KW - Chebyshev polynomials; integer coordinates

UR - http://eudml.org/doc/276945

ER -

## References

top- [1] H. Belbachir and F. Bencherif, Linear recurrent sequences and powers of a square matrix, Integers 6 (A12) (2006), 1-17. Zbl1122.11008
- [2] E. Lucas, Théorie des Nombres, Ghautier-Villars, Paris 1891.
- [3] T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, second edition, Wiley Interscience 1990. Zbl0734.41029

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