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2017•Mathematica SlovacaRequires access

δ-Fibonacci and δ-lucas numbers, δ-fibonacci and δ-lucas polynomials

Roman Wituła, Edyta� Hetmaniok, Damian Słota, Mariusz Pleszczyński

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Abstract

Abstract In this paper, with reference to the previous work [WITUŁA, R.—SŁOTA, D.: δ-Fibonacci numbers, Appl. Anal. Discrete Math. 3 (2009), 310–329] concerning the, so called, δ-Fibonacci numbers, the concepts of δ-Lucas numbers, δ-Fibonacci and δ-Lucas polynomials are introduced. There are discussed the basic properties of such objects, as well as their applications, especially for description of certain polynomials and identities of algebraic and trigonometric type. Many from among these identities describe the binomial transformations of the respective integer sequences and polynomials. Similarly as for δ-Fibonacci numbers, also for δ-Lucas numbers some attractive identities–bridges are obtained, connecting these numbers in practice with every sequence of integer numbers.

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Abstract In this paper, with reference to the previous work [WITUŁA, R.—SŁOTA, D.: δ-Fibonacci numbers, Appl. Anal. Discrete Math. 3 (2009), 310–329] concerning the, so called, δ-Fibonacci numbers, the concepts of δ-Lucas numbers, δ-Fibonacci and δ-Lucas polynomials are introduced. There are discussed the basic properties of such objects, as well as their applications, especially for description of certain polynomials and identities of algebraic and trigonometric type. Many from among these identities describe the binomial transformations of the respective integer sequences and polynomials. Similarly as for δ-Fibonacci numbers, also for δ-Lucas numbers some attractive identities–bridges are obtained, connecting these numbers in practice with every sequence of integer numbers.

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Available abstract

Abstract In this paper, with reference to the previous work [WITUŁA, R.—SŁOTA, D.: δ-Fibonacci numbers, Appl. Anal. Discrete Math. 3 (2009), 310–329] concerning the, so called, δ-Fibonacci numbers, the concepts of δ-Lucas numbers, δ-Fibonacci and δ-Lucas polynomials are introduced. There are discussed the basic properties of such objects, as well as their applications, especially for description of certain polynomials and identities of algebraic and trigonometric type. Many from among these identities describe the binomial transformations of the respective integer sequences and polynomials. Similarly as for δ-Fibonacci numbers, also for δ-Lucas numbers some attractive identities–bridges are obtained, connecting these numbers in practice with every sequence of integer numbers.

Key concepts: Fibonacci polynomials, Fibonacci number, Lucas number, Lucas sequence, Mathematics, Pisano period, Combinatorics, Integer (computer science)

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