The HGDispersion(p, q, x, r) command returns the hypergeometric dispersion of p and q, that is,

The polynomials can contain hypergeometric terms in their coefficients. These terms are defined in the formal parameter r. Each hypergeometric term in the list is specified by a name, for example, t. It can be specified directly in the form of an equation, for example, $t=n!$, or specified as a list consisting of the name of the term variable and the consecutive term ratio, for example, $\left[t\,n+1\right]$.

The computation of hypergeometric dispersions is reduced to solving the $\mathrm{\sigma }$-orbit problem (see OrbitProblemSolution) in the shortened tower of hypergeometric extensions.

$\mathrm{with}\left(\mathrm{LREtools}\left[\mathrm{HypergeometricTerm}\right]\right)\:$

$\mathrm{alias}\left(\mathrm{\phi }=3+\frac{4\mathrm{RootOf}\left(x^2+1\right)}{5}\right)\:$

$p≔{\mathrm{\phi }}^4{s}^2+{\mathrm{\phi }}^{2}s+1$

$p≔{\mathrm{\phi }}^4{s}^2+{\mathrm{\phi }}^{2}s+1$

$q≔s^2+s+1$

$q≔s^2+s+1$

$\mathrm{ext}≔\left[s={\mathrm{\phi }}^x\right]$

$\mathrm{ext}≔\left[s={\mathrm{\phi }}^x\right]$

$\mathrm{HGDispersion}\left(p\,q\,x\,\mathrm{ext}\right)$

$2$

$\mathrm{alias}\left(\mathrm{\phi }=\mathrm{RootOf}\left(x^3-5\right)\right)\:$

$p≔{\mathrm{\phi }}^4{s}^2+{\mathrm{\phi }}^{2}s+1$

$p≔{\mathrm{\phi }}^4{s}^2+{\mathrm{\phi }}^{2}s+1$

$q≔s^2+s+1$

$q≔s^2+s+1$

$\mathrm{ext}≔\left[s={\mathrm{\phi }}^x\right]$

$\mathrm{ext}≔\left[s={\mathrm{\phi }}^x\right]$

$\mathrm{HGDispersion}\left(p\,q\,x\,\mathrm{ext}\right)$

$\mathrm{-1}$

$p≔2^4{s}^2+2^{2}s+1+v$

$p≔16s^2+4s+v+1$

$q≔16{\left(x+3\right)}^2{\left(x+2\right)}^2{\left(x+1\right)}^2{s}^2+4\left(x+3\right)\left(x+2\right)\left(x+1\right)s+1+8v$

$q≔16{\left(x+3\right)}^2{\left(x+2\right)}^2{\left(x+1\right)}^2{s}^2+4\left(x+3\right)\left(x+2\right)\left(x+1\right)s+1+8v$

$\mathrm{ext}≔\left[v=2^x\,s=x!\right]$

$\mathrm{ext}≔\left[v=2^x\,s=x!\right]$

$\mathrm{HGDispersion}\left(sq\,pv+s\,x\,\mathrm{ext}\right)$

$3$

Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.