# A recurrence relation for the odd order moments of the Fabius function

## TL;DR

This paper derives recurrence relations for the odd order moments of the Fabius function, linking them to even moments and providing a closed-form for the coefficients, advancing understanding of its moment structure.

## Contribution

It introduces a recurrence relation for the odd order moments of the Fabius function and a closed-form expression for the coefficients involved.

## Key findings

- Recurrence relation for odd moments in terms of even moments
- Matrix formulation of the recurrence relations
- Closed-form expression for the coefficients

## Abstract

A simple recurrence relation for the even order moments of the Fabius function is proven. Also, a very similar formula for the odd order moments in terms of the even order moments is proved. The matrices corresponding to these formulas (and their inverses) are multiplied so as to obtain a matrix that correspond to a recurrence relation for the odd order moments in terms of themselves. The theorem at the end gives a closed-form for the coefficients.