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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.

YuiTo Cheng
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glebovg
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2 Answers2

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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.

vonbrand
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  • I also think that if $a_{−1}, a_{−2}, a_{-3}, \ldots$ is our infinite sequence, then if we insist on using nonnegative integers we let ${a_{-n}}{n \in \mathbb N} = {b{n - 1}}_{n \in \mathbb N}$. But it seems unconventional and I could not find any articles or books that discuss this. Can you justify it rigorously so that I can give you 50 rep? – glebovg May 02 '13 at 03:16
  • Can you elaborate? – glebovg May 02 '13 at 23:37
  • @glebovg If you are in umbral calculus, how does this answer your question? – Did May 12 '13 at 12:16
  • @Did It does not, hence my comments. Half the bounty value was automatically awarded to the top voted answer. – glebovg May 13 '13 at 00:14
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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: $$x_{(-n)} = \frac{1}{(x+1)(x+2)...(x+n)}$$

Rafael Hipólito
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