The product command is for computing symbolic products. It is used to compute a formula for an indefinite or definite product. If a formula cannot be computed, Maple returns the product unevaluated. A typical example would be product(x+k, k=0..n-1) which returns the formula GAMMA(x+n)/GAMMA(x). If you want to multiply a finite sequence of values, rather than compute a formula, use the mul command.  For example mul(x+k, k=0..2) returns x*(x+1)*(x+2). Although the product command can be used to compute explicit products, the mul command should be used in programs for explicit products.

You can enter the product command using either the 1-D or 2-D calling sequence.  For example, product(x+k, k=0..n-1) is equivalent to $\prod _{k=0}^{n-1}\left(x+k\right)$.

The call product(f, k) computes the indefinite product of f(k) with respect to k. That is, it computes a formula g such that g(k+1)/g(k) = f(k) for all k.

The call product(f, k = m..n) computes the definite product of f(k) over the given range m..n, that is, it computes f(m) f(m+1) ... f(n). The definite product is equivalent to g(n+1)/g(m) where g is the indefinite product. For example, product(n,n) = product(k, k=1..n-1) = GAMMA(n).

If m = n+1 then the value returned is 1. If m > n+1 then 1/product(f, k=n+1..m-1) is the value returned.

The call product(f, k= alpha) computes the definite product of f(k) over the roots of a polynomial where alpha must be a RootOf.

The call product(f, k= expr) substitutes the value of expr for k in f.

If Maple cannot find a closed form for the product, the function call itself is returned. (The prettyprinter displays the product function using a stylized product sign.)

The capitalized function name Product is the inert product function, which simply returns unevaluated.  It appears gray so that it is easily distinguished from a returned product calling sequence.

$\mathrm{product}\left(k^2\,k=1..4\right)$

$576$

$\mathrm{mul}\left(k^2\,k=1..4\right)$

$576$

$\mathrm{product}\left(k^2\,k=1..n\right)$

${\mathrm{\Gamma }\left(n+1\right)}^2$

$\mathrm{product}\left(k^2\,k\right)$

${\mathrm{\Gamma }\left(k\right)}^2$

$\mathrm{product}\left(a\left[k\right]\,k=0..4\right)$

$a_0{a}_1{a}_2{a}_3{a}_4$

$\mathrm{product}\left(a\left[k\right]\,k=0..n\right)$

$\prod _{k=0}^n{a}_k$

$\mathrm{Product}\left(n+k\,k=0..m\right)=\mathrm{product}\left(n+k\,k=0..m\right)$

$\prod _{k=0}^m\left(n+k\right)=\frac{\mathrm{\Gamma }\left(n+m+1\right)}{\mathrm{\Gamma }\left(n\right)}$

$\mathrm{product}\left(kx\,k=1..5\right)$

$120x^5$

$\mathrm{product}\left(k\,k=x..5x\right)$

$\frac{\mathrm{\Gamma }\left(5x+1\right)}{\mathrm{\Gamma }\left(x\right)}$

$\mathrm{product}\left(\frac{1}{k}\,k=1..\mathrm{\infty }\right)$

$0$

The product command was updated in Maple 2016; see Advanced Math.