On Generalized Fibonacci Numbers

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Communications in Advanced Mathematical Sciences
Volume:3 Issue:4
On Generalized Fibonacci Numbers

On Generalized Fibonacci Numbers

Authors : Fidel ODUOL, Isaac Owino OKOTH

Pages : 186-202

Doi:10.33434/cams.771023

View : 28 | Download : 14

Publication Date : 2020-12-22

Article Type : Research Paper

Abstract :Fibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of $r$-sum Fibonacci numbers. The recurrence relation is maintained but initial conditions are varied. Among results obtained are Binet`s formula, generating function, explicit sum formula, sum of first $n$ terms, sum of first $n$ terms with even indices, sum of first $n$ terms with odd indices, alternating sum of $n$ terms of $r-$sum Fibonacci sequence, Honsberger`s identity, determinant identities and a generalized identity from which Cassini`s identity, Catalan`s identity and d`Ocagne`s identity follow immediately.
Keywords : Binet`s formula, Fibonacci sequence, generating function, r shifted Fibonacci sequence

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