Weighted generalized Fibonacci sums

TL;DR

This paper derives formulas for weighted sums, such as binomial and double binomial sums, related to the generalized Fibonacci sequence, expanding the understanding of Fibonacci-like sequences with arbitrary initial conditions.

Contribution

It introduces new formulas for weighted sums of generalized Fibonacci sequences, including binomial and double binomial sums, for arbitrary initial values.

Findings

Derived formulas for binomial sums of generalized Fibonacci sequences

Extended identities to double binomial sums

Applicable to sequences with arbitrary initial conditions

Abstract

We derive weighted sums, including binomial and double binomial sums, for the generalized Fibonacci sequence $\{G_m\}$ where for $m\ge 2$, $G_m=G_{m-1}+G_{m-2}$ with initial values $G_0$ and $G_1$.