Lagmental Vicfred

Exponential Generating Functions Remember Labels by Vicfred

tags: mathematics, generating functions, recurrences, Lagrange inversion

The factorial denominator makes products distribute labels between independent components. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

A sequence \((a_n)_{n\ge0}\) becomes a formal series \(A(x)=\sum_{n\ge0}a_nx^n\). Algebra on \(A(x)\) translates recurrences, convolution, and recursive constructions into coefficient identities.

$$ A(x)=\sum_{n\ge0}a_n\frac{x^n}{n!} $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ [x^n/n!]A(x)B(x)=\sum_{k=0}^{n}\binom nka_kb_{n-k} $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \exp(x)=\sum_{n\ge0}\frac{x^n}{n!},\qquad\exp(\exp x-1)=\sum_{n\ge0}B_n\frac{x^n}{n!} $$

The reusable statement

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad A(x)=\sum_{n\ge0}a_n\frac{x^n}{n!},\\[5pt] \mathsf{C}\;&:\quad [x^n/n!]A(x)B(x)=\sum_{k=0}^{n}\binom nka_kb_{n-k}. \end{aligned} $$

A nearby false statement

Formal power series permit algebra without analytic convergence, but substitution and inversion still require the correct constant terms.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [x^n/n!]A(x)B(x)=\sum_{k=0}^{n}\binom nka_kb_{n-k} \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Sat 17 November 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.