Let F be a rational function of n over a field K of characteristic 0. The PolynomialNormalForm(F,n) command constructs the polynomial normal form for F.

The output is a sequence of 4 elements $z,a,b,c$ where z is an element of K, and $a,b,c$ are monic polynomials over K such that: $F\=\frac{zaE\left(c\right)}{bc}\.$  $\gcd \left(a,E^{k\left(b\right)}\right)=1\mathrm{for\; all}\mathrm{non}-\mathrm{negative\; integers}k\.$ $\gcd \left(a\,c\right)=1,\gcd \left(b\,E\left(c\right)\right)=1.$

Note: E is the automorphism of K(n) defined by {E(F(n)) = F(n+1)}.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\:$

$F≔\frac{\frac{3}{2}n\left(n+2\right)\left(3n+2\right)\left(3n+4\right)}{\left(n-1\right)\left(2n+9\right){\left(n+4\right)}^2}$

$F≔\frac{3n\left(n+2\right)\left(3n+2\right)\left(3n+4\right)}{2\left(n-1\right)\left(2n+9\right){\left(n+4\right)}^2}$

$z,a,b,c≔\mathrm{PolynomialNormalForm}\left(F\,n\right)$

$z,a,b,c≔\frac{27}{4},\left(n+2\right)\left(n+\frac{2}{3}\right)\left(n+\frac{4}{3}\right),\left(n+\frac{9}{2}\right){\left(n+4\right)}^2,n-1$

$\mathrm{evalb}\left(F=\mathrm{normal}\left(\frac{z\left(\frac{a}{b}\right)\mathrm{subs}\left(n=n+1\,c\right)}{c}\right)\right)$

$\mathrm{true}$

$\mathrm{LREtools}\left[\mathrm{dispersion}\right]\left(b\,a\,n\right)$

$\mathrm{FAIL}$

$\gcd \left(a\,c\right),\gcd \left(b\,\mathrm{subs}\left(n=n+1\,c\right)\right)$

$1,1$

Gosper, R.W., Jr. "Decision procedure for indefinite hypergeometric summation." Proc. Natl. Acad. Sci. USA. Vol. 75. (1977): 40-42.

Petkovsek, M. "Hypergeometric solutions of linear recurrences with polynomial coefficients." J. Symb. Comput. Vol. 14. (1992): 243-264.