Generalized Multiplicative Coupled Fibonacci Sequence and its Properties

Sikhwal, Omprakash; Vyas, Yashwant; Bhatnagar, Shikha

doi:10.5120/ijca2017912838

Cited by 1 publication

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“…Various generalizations of the aforementioned sequence have been derived since it was first discovered by Fibonacci in the 13 th century. Fibonacci sequence has been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions [1,3,4,5,7,9,10], and by varying the recurrence relation and maintaining the initial conditions [2,4,8,9,11,13,12,15]. Some of the properties that have been obtained by various researchers are not limited to finding a closed form for the n th term of the sequence, sum of the first n terms of the sequence, sum of the first n terms with odd (or even) indices of the sequence, explicit sum formula, Catalan's identity, Cassini's identity, d'Ocagne's identity, Honsberger's identity, determinant identities, and generating function among many others.…”

Section: Introductionmentioning

confidence: 99%

On Generalized Fibonacci Numbers

Oduol

Okoth

2020

Communications in Advanced Mathematical Sciences

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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.

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