Hadamard grade of power series

TL;DR

This paper introduces the concept of Hadamard grade for power series, measuring the minimal number of algebraic series needed to express a given series through Hadamard products, and explores its properties.

Contribution

It defines and investigates the Hadamard grade of power series, a novel measure related to algebraic and transcendental series.

Findings

The Hadamard grade can be finite or infinite.

Connections between Hadamard grade and algebraic properties are discussed.

The notion provides new insights into the structure of power series.

Abstract

The Hadamard product of two power series $\sum a_n z^n$ and $\sum b_n z^n$ is the power series $\sum a_n b_n z^n$. We define the (Hadamard) grade of a power series $A$ to be the least number (finite or infinite) of algebraic power series, the Hadamard product of which equals $A$. We study and discuss this notion.