PolynomialNormalForm - Maple Help

SumTools[Hypergeometric]

 PolynomialNormalForm
 construct the polynomial normal form of a rational function

 Calling Sequence PolynomialNormalForm(F, n)

Parameters

 F - rational function of n n - variable

Description

 • 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: $F=\frac{zaE\left(c\right)}{bc}.$  $\mathrm{gcd}\left(a,{E}^{k\left(b\right)}\right)=1\mathrm{for all}\mathrm{non}-\mathrm{negative integers}k.$ $\mathrm{gcd}\left(a,c\right)=1,\mathrm{gcd}\left(b,E\left(c\right)\right)=1.$
 Note: E is the automorphism of K(n) defined by {E(F(n)) = F(n+1)}.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $F≔\frac{\frac{3}{2}n\left(n+2\right)\left(3n+2\right)\left(3n+4\right)}{\left(n-1\right)\left(2n+9\right){\left(n+4\right)}^{2}}$
 ${F}{≔}\frac{{3}{}{n}{}\left({n}{+}{2}\right){}\left({3}{}{n}{+}{2}\right){}\left({3}{}{n}{+}{4}\right)}{{2}{}\left({n}{-}{1}\right){}\left({2}{}{n}{+}{9}\right){}{\left({n}{+}{4}\right)}^{{2}}}$ (1)
 > $z,a,b,c≔\mathrm{PolynomialNormalForm}\left(F,n\right)$
 ${z}{,}{a}{,}{b}{,}{c}{≔}\frac{{27}}{{4}}{,}\left({n}{+}{2}\right){}\left({n}{+}\frac{{2}}{{3}}\right){}\left({n}{+}\frac{{4}}{{3}}\right){,}\left({n}{+}\frac{{9}}{{2}}\right){}{\left({n}{+}{4}\right)}^{{2}}{,}{n}{-}{1}$ (2)

Check the results.

Condition 1 is satisfied.

 > $\mathrm{evalb}\left(F=\mathrm{normal}\left(\frac{z\left(\frac{a}{b}\right)\mathrm{subs}\left(n=n+1,c\right)}{c}\right)\right)$
 ${\mathrm{true}}$ (3)

Condition 2 is satisfied.

 > $\mathrm{LREtools}\left[\mathrm{dispersion}\right]\left(b,a,n\right)$
 ${\mathrm{FAIL}}$ (4)

Condition 3 is satisfied.

 > $\mathrm{gcd}\left(a,c\right),\mathrm{gcd}\left(b,\mathrm{subs}\left(n=n+1,c\right)\right)$
 ${1}{,}{1}$ (5)

References

 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.