The GreatestFactorialFactorization command computes a greatest factorial factorization $\[c,\[\[\mathrm{g1},\mathrm{e1}\],\[\mathrm{g2},\mathrm{e2}\],...\]\]$ of f w.r.t. x. It satisfies the following properties.

 $f=c\mathrm{g1}\left(x\right)\mathrm{g1}\left(x-1\right)...\mathrm{g1}\left(x-\mathrm{e1}+1\right)\mathrm{g2}\left(x\right)\mathrm{g2}\left(x-1\right)...\mathrm{g2}\left(x-\mathrm{e2}+1\right)...$ 

 $\gcd \left(\mathrm{gi}\left(x\right)\mathrm{gi}\left(x-1\right)...\mathrm{gi}\left(x-\mathrm{e1}+1\right),\mathrm{gj}\left(x+1\right)\mathrm{gj}\left(x+\mathrm{ej}\right)\right)=1$ for $1<=i<=j$ 

$c$ is constant w.r.t. x, and $\mathrm{g1},...$ are nonconstant primitive polynomials w.r.t. x, and $0<\mathrm{e1}<\mathrm{e2}<\mathrm{...}$ are integers.

The greatest factorial factorization is unique up to multiplication by units.

GreatestFactorialFactorization can handle the same types of coefficients as the Maple function gcd.

If f is constant w.r.t. x, then the return value is $\left[f\&comma;\left[\right]\right]$.

Partial factorizations of the input are not taken into account.

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\&colon;$

$\mathrm{GreatestFactorialFactorization}\left(-x^8+x^2\&comma;x\right)$

$\left[\mathrm{-1}\&comma;\left[\left[x\&comma;1\right]\&comma;\left[x^2+x+1\&comma;2\right]\&comma;\left[x+1\&comma;3\right]\right]\right]$

$\mathrm{GreatestFactorialFactorization}\left(\mathrm{expand}\left(\mathrm{pochhammer}\left(x\&comma;3\right)\mathrm{pochhammer}\left(x\&comma;5\right)\right)\&comma;x\right)$

$\left[1\&comma;\left[\left[x+2\&comma;3\right]\&comma;\left[x+4\&comma;5\right]\right]\right]$

Paule, Peter. "Greatest factorial factorization and symbolic summation." Journal of Symbolic Computation Vol. 20, (1995): 235-268.

Gerhard, Juergen. "Modular algorithms for polynomial basis conversion and greatest factorial factorization." Proceedings of the Seventh Rhine Workshop on Computer Algebra, RWCA pp. 125-141 ed. T. Mulders, 2000.