# On the self-convolution of generalized Fibonacci numbers

## TL;DR

This paper unifies various known identities involving self-convolution of generalized Fibonacci numbers into a single framework using bivariate generating functions, providing explicit formulas for the coefficients and generalizing previous results.

## Contribution

It introduces a unified equation for these identities and derives explicit coefficient formulas, extending prior work by Zhang, Zao-Wang, and Mansour.

## Key findings

- Unified equation for self-convolution identities
- Explicit formulas for convolution coefficients
- Generalized forms of previous Fibonacci identities

## Abstract

We focus on a family of equalities pioneered by Zhang and generalized by Zao and Wang and hence by Mansour which involves self convolution of generalized Fibonacci numbers. We show that all these formulas are nicely stated in only one equation involving a bivariate ordinary generating function and we give also a formula for the coefficients appearing in that context. As a consequence, we give the general forms for the equalities of Zhang, Zao-Wang and Mansour.