A geometric property of the roots of Chebyshev polynomials

• Autorzy: Wituła R. , Hetmaniok E. , Słota D.
• Języki publikacji: EN

Abstrakty

EN

In this text a new property of geometric nature of the Chebyshev polynomials is given. It is proven that the lengths of diagonals of a regular n-gon with the side of length equal to one are the sums of positive roots of the respective renormalized Chebyshev polynomials of one from among four types. Some new special decompositions of differences of values of the Chebyshev polynomials are also presented.

Słowa kluczowe

EN

Chebyshev polynomials; Vieta-Lucas polynomials

PL

wielomiany Czebyszewa; wielomiany Vieta-Lucasa

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2014
• Tom: Vol. 13, nr 1
• Strony: 143--148
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Wituła R.

• Institute of Mathematics, Silesian University of Technology Gliwice, Poland

autor

Hetmaniok E.

• Institute of Mathematics, Silesian University of Technology Gliwice, Poland

autor

Słota D.

• Institute of Mathematics, Silesian University of Technology Gliwice, Poland

Bibliografia

• [1] Bialostocki A., Bialostocki D., Fluster M., Holman W., McAleer M., A geometric property of renormalized Chebyshev polynomials, College Math. J. (in review).
• [2] Mason J.C., Handscomb D.C., Chebyshev Polynomials, Chapman & Hall/CRC, New York 2003.
• [3] Paszkowski S., Numerical Applications of Chebyshev Polynomials and Series, PWN, Warsaw 1975 (in Polish).
• [4] Rivlin T., Chebyshev Polynomials from Approximation Theory to Algebra and Number Theory, 2nd ed., Wiley, New York 1990.
• [5] Robbins N., Vieta’s triangular array and a related family of polynomials, Internat. J. Math.Math. Sci. 1991, 14, 239-244.
• [6] Wituła R., Słota D., On modified Chebyshev polynomials, J. Math. Anal. Appl. 2006, 324,321-343.
• [7] Bayad A., Cangul I.N., The minimal polynomial of 2 cos ( pi/q) and Dickson polynomials,Appl. Math. Comput. 2012, 218, 7014-7022.
• [8] Knopfmacher A., Mansour T., Munagi A., Prodinger H., Staircase words and Chebyshev polynomials, Appl. Anal. Discrete Math. 2010, 4, 81-95.
• [9] Wituła R., Hetmaniok E., Słota D., Prodinger’s algebraic identities and their applications,Int. J. Pure Appl. Math. 2010, 64, 225-237.
• [10] Wituła R., Słota D., Decompositions of certain symmetric functions of powers of cosine and sine functions, Int. J. Pure Appl. Math. 2009, 50, 1-12.
• [11] Zhang Z., Wang J., On some identities involving the Chebyshev polynomials, Fibonacci Quart. 2004, 42, 245-249.
• [12] Castillo K., Lamblem R.L., Sri Ranga A., On a moment problem associated with Chebyshev polynomials, Appl. Math. Comput. 2012, 218, 9571-9574.