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

• Autorzy: Roman Wituła , Damian Słota
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Cauchy polynomials; Ferrers-Jackson polynomials; Chebyshev polynomials; Fibonacci and Lucas numbers; recurrence sequences; 11B83; 26C99; 11B39

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 3
• Strony: 531-546

Daty

wydano

2006-09-01

online

2006-09-01

Twórcy

autor

Roman Wituła

• Silesian University of Technology

autor

Damian Słota

• Silesian University of Technology

Bibliografia

• [1] L. Carlitz and J.A.H. Hunter: “Sums of Powers of Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 7, (1969), pp. 467–473.
• [2] A.F. Horadam: “Basic properties of a certain generalized sequence of numbers”, Fibonacci Quart., Vol. 3, (1965), pp. 161–176.
• [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] T. Koshy: Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001.
• [5] T. Mansour: “A formula for the generating functions of powers of Horadam’s sequence”, Australas. J. Combin., Vol. 30, (2004), pp. 207–212.
• [6] R.S. Melham: “Sums of Certain Products of Fibonacci and Lucas Numbers - Part I”, Fibonacci Quart., Vol. 37, (1999), pp. 248–251.
• [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.
• [8] R.S. Melham: “Sums of Certain Products of Fibonacci and Lucas Numbers - Part II”, Fibonacci Quart., Vol. 38, (2000), pp. 3–7.
• [9] R.S. Melham: “Alternating Sums of Fourth Powers of Fibonacci and Lucas Numbers”, Fibonacci Quart., Vol. 38, (2000), pp. 254–259.
• [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] J. Morgado: “Note on the Chebyshev polynomials and applications to the Fibonacci numbers”, Portugal. Math., Vol. 52, (1995), pp. 363–378.
• [12] S. Paszkowski: Numerical Applications of Chebyshev Polynomials and Series, PWN, Warsaw, 1975 (in Polish).
• [13] P. Ribenboim: Fermats’s Last Theorem For Amateurs, Springer, New York 1999.
• [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
• [15] T. Rivlin: Chebyshev Polynomials from Approximation Theory to Algebra and Number Theory, 2nd ed., Wiley, New York, 1990.
• [16] R. Wituła and D. Słota: “On Modified Chebyshev Polynomials”, J. Math. Anal. Appl., (2006), in print.

Identyfikatory

DOI

10.2478/s11533-006-0022-9