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Let $a = (a_0,a_1,a_2,\ldots)$ be an infinite sequence of real numbers.

What is the name of the sequence

\begin{equation*} Da = (a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots) \qquad ? \end{equation*}

Also, is the operator $D$ called the "forward difference operator"?

eternalGoldenBraid
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  • Do you need a fancier or shorter name than "sequence of consecutive differences"? Defining "forward difference operator" $D$ by $(Da)i = a{i+1} - a_i$ seems like a very clear approach. – hardmath Mar 21 '15 at 21:32
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    Alternatively, you word it as $D=S_l-I$ where $S_l$ is the canonical left-shift operator. I.e. $S_l: (a_0,a_1,a_2,\dots)\mapsto (a_1,a_2,a_3,\dots)$ – JMoravitz Mar 21 '15 at 21:36

1 Answers1

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The sequence $Da$ is often called the first difference of sequence $a$. Search for "first difference of" sequence will bring up many examples of usage.

Naturally, $D^2a=D(D(a))$ is the second difference of $a$, and so on. Same terminology as with derivatives, except the word "derivative" is replaced with "sequence".

"Forward difference operator" is a standard term for $D$.