The Add, Differentiate, Map, Multiply, and Nops commands perform operations on the three kinds of symbolic sequences implemented in the Maple system, namely:

1.  A sequence of numbers - say from n to m - frequently displayed as

2.  A sequence of one object, say a, repeated say p times, frequently displayed as

3.  A more general sequence, as in 1., but of different objects and not necessarily numbers, frequently displayed as

or likewise a sequence of functions $f\left(n\right),\mathrm{...},f\left(m\right)$. In all these cases, none of n, m, or p are known: they are just symbols, or algebraic expressions, representing integer values. These symbolic sequences are represented in Maple using the $ operator. To cases 1., 2. and 3. correspond, respectively, the input $(n .. m), a $ p and a[i] $ (i = n .. m).

Add and Multiply can either add or multiply the elements of a symbolic sequence (that is, receive only one operand), as well as add or multiply many elements, where possibly more than one is a symbolic sequence.

Nops generalizes nops in that if the single argument is a symbolic sequence, it returns the number of elements it contains expressed in terms of the symbols representing integers that define the sequence.

Differentiate and Map respectively generalize diff and map in that they work the same way as the lowercase commands, but when the first argumen,t in the case of Differentiate, or the second argument in the case of Map, is a symbolic sequence, the operation is applied in such a way that the result represents the operation applied to all the elements of the sequence.

$\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Sequences}\right)$

$\left[\mathrm{Add}\,\mathrm{Differentiate}\,\mathrm{Map}\,\mathrm{Multiply}\,\mathrm{Nops}\right]$

$\mathrm{interface}\left(\mathrm{typesetting}=\mathrm{extended}\right)\:$

$\mathrm{S\_\_1}≔\mathrm{`\$`}\left(n..m\right)$

$\mathrm{S\_\_1}≔n,\mathrm{...},m$

$\mathrm{S\_\_2}≔\mathrm{`\$`}\left(a\,p\right)$

$\mathrm{S\_\_2}≔\underset{\text{p times}}{\underbrace{a,\mathrm{...},a}}$

$\mathrm{S\_\_3}≔\mathrm{`\$`}\left(a\left[i\right]\,i=n..m\right)$

$\mathrm{S\_\_3}≔a_n,\mathrm{...},a_m$

$\mathrm{Nops}\left(\mathrm{S\_\_1}\right)$

$m-n+1$

$\mathrm{Nops}\left(\mathrm{S\_\_2}\right)$

$p$

$\mathrm{Nops}\left(\mathrm{S\_\_3}\right)$

$m-n+1$

$\mathrm{Add}\left(\mathrm{S\_\_1}\right)$

$\frac{\left(m-n+1\right)\left(n+m\right)}{2}$

$\mathrm{Add}\left(\mathrm{S\_\_2}\right)$

$ap$

$\mathrm{Add}\left(\mathrm{S\_\_3}\right)$

$\sum _{i=n}^m{a}_i$

$\mathrm{Multiply}\left(\mathrm{S\_\_1}\right)$

$\frac{m!}{\left(n-1\right)!}$

$\mathrm{Multiply}\left(\mathrm{S\_\_2}\right)$

$a^p$

$\mathrm{Multiply}\left(\mathrm{S\_\_3}\right)$

$\prod _{i=n}^m{a}_i$

$\mathrm{Differentiate}\left(\mathrm{S\_\_1}\,k\right)$

$\underset{\text{m-n+1 times}}{\underbrace{0,\mathrm{...},0}}$

$\mathrm{Differentiate}\left(\mathrm{S\_\_2}\,a\right)$

$\underset{\text{p times}}{\underbrace{1,\mathrm{...},1}}$

$\mathrm{Differentiate}\left(\mathrm{S\_\_3}\,a\left[k\right]\right)$

$\left\{\begin{array}{cc}1& k=n\\ 0& \mathrm{otherwise}\end{array}\right.,\mathrm{...},\left\{\begin{array}{cc}1& k=m\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{Map}\left(f\,\mathrm{S\_\_1}\right)$

$f\left(n\right),\mathrm{...},f\left(m\right)$

$\mathrm{Map}\left(\mathrm{Int}\,\mathrm{S\_\_1}\,x\right)$

$\int n\ⅆx,\mathrm{...},\int m\ⅆx$

$\mathrm{Map}\left(\mathrm{`+`}\,\mathrm{S\_\_3}\,r\right)$

$a_n+r,\mathrm{...},a_m+r$

$\mathrm{Map}\left(\mathrm{`*`}\,\mathrm{S\_\_2}\,r\right)$

$\underset{\text{p times}}{\underbrace{ar,\mathrm{...},ar}}$

The MathematicalFunctions[Sequences][Add], MathematicalFunctions[Sequences][Differentiate], MathematicalFunctions[Sequences][Map], MathematicalFunctions[Sequences][Multiply] and MathematicalFunctions[Sequences][Nops] commands were introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.