Bracket notation for the `coefficient of' operator

TL;DR

This paper explores a generalized bracket notation for coefficients in power series, aiming to improve algebraic manipulation and clarity compared to existing notations.

Contribution

It introduces a new linear bracket notation for coefficients that enhances algebraic manipulation and explores its properties.

Findings

The new notation simplifies algebraic operations on power series.

It generalizes existing coefficient notations for better flexibility.

The paper discusses properties that make the notation more useful in mathematical contexts.

Abstract

When $G(z)$ is a power series in $z$, many authors now write `$[z^n] G(z)$' for the coefficient of $z^n$ in $G(z)$, using a notation introduced by Goulden and Jackson in [\GJ, p. 1]. More controversial, however, is the proposal of the same authors [\GJ, p. 160] to let `$[z^n/n!] G(z)$' denote the coefficient of $z^n/n!$, i.e., $n!$ times the coefficient of $z^n$. An alternative generalization of $[z^n] G(z)$, in which we define $[F(z)] G(z)$ to be a linear function of both $F$ and $G$, seems to be more useful because it facilitates algebraic manipulations. The purpose of this paper is to explore some of the properties of such a definition. The remarks are dedicated to Tony Hoare because of his lifelong interest in the improvement of notations that facilitate manipulation.