Abstract
There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.
Highlights
The purpose of this note is to establish the following elementary summation formulas involving the famous Fibonacci numbers: Theorem 1
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One of the most significant properties of these bundles is their recursive structure reflecting the boundary stratification of M g,n. This property, known as factorization, was cleverly used by Mukhopadhyay to compute the rank of the g2, level 1, case of these bundles and the formula he obtained [12] is the middle expression in Equation (1) involving the golden ratio; by using factorization in two different ways, we obtain the two Fibonacci summation expressions in that equation
Summary
The purpose of this note is to establish the following elementary summation formulas involving the famous Fibonacci numbers: Theorem 1. One of the most significant properties of these bundles is their recursive structure reflecting the boundary stratification of M g,n This property, known as factorization, was cleverly used by Mukhopadhyay to compute the rank of the g2 , level 1, case of these bundles and the formula he obtained [12] is the middle expression in Equation (1) involving the golden ratio; by using factorization in two different ways, we obtain the two Fibonacci summation expressions in that equation. The details of this assertion are the content of this paper.
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