# Orthorecursive expansion of unity

## TL;DR

This paper investigates a recursively defined sequence related to orthorecursive expansion, establishing its properties, growth estimates, series identities, and connections to harmonic numbers, with conjectures supported by numerical evidence.

## Contribution

It introduces and analyzes a new sequence defined by a recursive relation, linking it to harmonic numbers and providing growth estimates and identities.

## Key findings

- Derived growth estimates for the sequence c_n
- Established series identities involving c_n and harmonic numbers
- Formulated conjectures based on numerical computations

## Abstract

We study the properties of a sequence cn defined by the recursive relation \[\frac{c_0}{n + 1}+\frac{c_1}{n + 2}+\ldots+\frac{c_n}{2n + 1}=0\] for $n>1$ and $c_0=1$. This sequence also has an alternative definition in terms of certain norm minimization in the space $L^2([0, 1])$. We prove estimates on growth order of $c_n$ and the sequence of its partial sums, infinite series identities, connecting $c_n$ with harmonic numbers $H_n$ and also formulate some conjectures based on numerical computations.