# On Polynomial Recursive Sequences

## TL;DR

This paper explores the limits of polynomial recursive sequences, showing that some sequences, like un = nn, cannot be expressed in this form.

## Contribution

The paper proves that the sequence un = nn is not polynomial recursive, a novel result in the field of recursive sequences.

## Key findings

- Polynomial recursive sequences are a nonlinear extension of linear recursive sequences.
- The sequence un = nn is not polynomial recursive, demonstrating a boundary of this class.

## Abstract

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is bn = n!. Our main result is that the sequence un = nn is not polynomial recursive.