TITLE: Umbral calculus with negative indices (and powers)
QUESTION [6 upvotes]: Can we do umbral calculus with negative indices (and powers)? Can we write $a_{-n} \equiv a^{-n}$ or $L[a_{-n}] = a^{-n}$ where $L$ is a linear functional and $n$ need not be negative?
The common convention is to use $\mathbb N$ or $\mathbb N \cup \{0\}$ to index sequences, but we can use any countable set, say $\mathbb Z$ and redefine our sequence: $a_{-n} \equiv {b_m}$ with
$$
m =
\begin{cases}
2n - 1&
\text{if } n > 0\\
-2n&
\text{if } n \leq 0\\
\end{cases}.
$$
However, does this mean we can use negative indices, and writing $a_{-n} \equiv a^{-n}$ or $L[a_{-n}] = a^{-n}$ is justified?
Please provide references.

REPLY [2 votes]: Never seen such, but why not? In the worst case, you can define $b_n = a_{-n}$ and forge ahead. Negative binomial coefficients make very good sense, in a weird way Stirling numbers of the first and second kind are "negative indices" of one another.