Umbral calculus with negative indices (and powers)

Question

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.

Answer 1

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.

Answer 2

I'm not sure if this is what you're looking for, but you can probably extend a Sheffer sequence to negative indices using the closed forms found on Rota's Finite Operator Calculus.

Specifically if $s_n$ is a sheffer set with associated delta operator $Q = DP$ ($D$ is the derivative operator and $P$ is invertible) whose set of basic polynomials is $p_n$, and $S$ is the operator that satisfies $Ss_n = p_n$, then one has the following identities (Q' denotes the Pincherle derivative of the operator $Q$ defined through $Qx - xQ$): $$p_n(x) = Q’P^{-n-1}x^n$$ $$s_n = S^{-1}p_n$$

By extending the notation of these operators from a ring of polynomials to a field of rational functions, one can define negative index Sheffer polynomials.

For example for the falling factorials $x_{(n)}$ one has (if I'm not mistaken) that: $$x_{(-n)} = \frac{1}{(x+1)(x+2)...(x+n)}$$