On Generalized Fibonacci Numbers

Year 2020, Volume: 3 Issue: 4, 186 - 202, 22.12.2020

Fidel Oduol Isaac Owino Okoth

https://doi.org/10.33434/cams.771023

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

Supporting Institution

Maseno University

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