1.3. Some notation#
There are a few pieces of notation that we will try to use consistently. There is potential for confusion as in other branches of mathematics some of this notation would have a different interpretation. We briefly summarize it in a table and give more detailed descriptions below:
Notation |
Interpretation |
|---|---|
\(f(x), g(x), \ldots\) |
Ordinary Generating Functions |
\(\mu(x), \psi(x), \Omega(x), \ldots\) |
Probability Generating Functions |
\([x^n]f(x)\), \([x^n]\mu(x)\) |
The coefficient of \(x^n\) in the series expansion of \(f(x)\) or \(\mu(x)\) |
\(f^{(n)}(x)\), \(\mu^{(n)}(x)\) |
function composition: \(f(f(f(\cdots f(x) \cdots)))\) or \(\mu(\mu(\mu(\cdots \mu(x)\cdots)))\) |
\(f(x)^n\), \(\mu(x)^n\) |
function multiplication: \(f(x)f(x)\cdots f(x)\) or \(\mu(x)\mu(x)\cdots \mu(x)\) |
\((x \partial_x)^n f(x)\), \((x \partial_x)^n \mu(x)\) |
\(x\frac{d}{dx}\left(x \frac{d}{dx}\left(\cdots x\frac{d}{dx} f(x)\cdots\right)\right)\) or \(x\frac{d}{dx}\left(x \frac{d}{dx}\left(\cdots x\frac{d}{dx} \mu(x)\cdots\right)\right)\) |
We will use the Latin alphabet for Ordinary Generating Functions and the Greek alphabet for Probability Generating Functions.
We will frequently need to reference the coefficient of a particular power of \(x\) in the series expansion of \(f(x)\). For this, we will use
\[ \big[x^n\big]f(x) = \text{ the coefficient of } x^n \text{ in the Taylor Series expansion of } f(x) \]
For example if \(f(x)=(1+x)^2=1+2x+x^2\), then \([x]f(x) = 2\).
Function composition will occur frequently. Given the function \(f(x)\), the function \(f^{(2)}(x)\) is \(f(f(x))\). More generally
This notation is often used in other contexts to represent \(n\) derivatives of a function \(f(x)\) but we will only use it to denote function composition.
Multiplication of functions will also occur frequently. We use \(f(x)^2\) to represent \(f(x)f(x)\). More generally
Derivatives occur often. Usually we will use the standard notation \(\frac{d^n}{dx^n}f(x)\) to denote the \(n\)th derivative, and if \(n\) is small we will often write \(f''(x)\), but never \(f^{(n)}(x)\) as this is reserved for function composition.
We are often interested in \(xf'(x)\). We will use the notation \(x \partial_x f(x)\) to denote \(x f'(x)\) and