What question did this study set out to answer?

The research focuses on deriving identities and binomial sums related to the generalized Fibonacci sequence.

March 1, 2026

Some equalities and binomial sums about the generalized Fibonacci number u(n)

Key Points

The research focuses on deriving identities and binomial sums related to the generalized Fibonacci sequence.
Defined the generalized Fibonacci sequence with a recurrence relation.
Derived three interrelated functions using the sequence terms.
Introduced a new approach to obtain identities and binomial sums.
Identified numerous identities involving generalized Fibonacci numbers.
Established relationships between binomial sums and the Fibonacci sequence.
Derived alternate binomial sums showcasing new mathematical properties.

Abstract

In this study, we take the generalized Fibonacci sequence u (n) as u (0) = 0; u (1) = 1 and u (n) = ru (n-1) + u (n-2) for n > 1, where r is a non-zero integer. Based on Halton's paper in 4, we derive three interrelated functions involving the terms of generalized Fibonacci sequence u (n). Using these three functions we introduce a simple approach to obtain a lot of identities, binomial sums and alternate binomial sums involving the terms of generalized Fibonacci sequence u (n).

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