Open AccessSome Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j
Author(s) -
Yüksel Soykan
Publication year - 2021
Publication title -
archives of current research international
Language(s) - English
Resource type - Journals
ISSN - 2454-7077
DOI - 10.9734/acri/2021/v21i330235
Subject(s) - fibonacci number , lucas number , pisano period , fibonacci polynomials , recurrence relation , mathematics , lucas sequence , mathematical proof , combinatorics , sequence (biology) , mathematical induction , algebra over a field , discrete mathematics , pure mathematics , orthogonal polynomials , classical orthogonal polynomials , geometry , biology , genetics
In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and recurrence properties of generalized Fibonacci sequence.