The OrbitProblemSolution($\mathrm{\alpha }$, $\mathrm{\beta }$, x, r) command returns the solution of a $\mathrm{\sigma }$-orbit problem, that is, a positive integer n such that $E^{n-1}\mathrm{\alpha }\cdot \dots \cdot E\mathrm{\alpha }\cdot \mathrm{\alpha }=\mathrm{\beta }$. $\mathrm{\alpha }$ and $\mathrm{\beta }$ can be algebraic numbers or polynomials in K(r), where K is the ground field and r is the tower of hypergeometric extensions. Each $r_i$ is specified by a hypergeometric term, that is, $\frac{{\mathrm{Er}}_i}{r_i}$ is a rational function over K. E is the shift operator.

If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are algebraic numbers then the procedure solves the classic orbit problem (${\mathrm{\alpha }}^n=\mathrm{\beta }$). Otherwise, it solves the $\mathrm{\sigma }$-orbit problem for polynomials in the tower of hypergeometric extensions. This means that the polynomials can contain hypergeometric terms in their coefficients. These terms are defined in the 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 OrbitProblemSolution function returns $\mathrm{-1}$ if there is no solution.

If the arguments of the $\mathrm{\sigma }$-orbit problem are algebraic numbers, then the routine directly computes the solution. Otherwise, a hypergeometric dispersion is calculated. For an empty tower of hypergeometric extensions, a simple dispersion is calculated.

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

$\mathrm{OrbitProblemSolution}\left(s+1\,\left(s+1\right)\left(2s+1\right)\,x\,\left[s=2^x\right]\right)$

$2$

$\mathrm{OrbitProblemSolution}\left(t+s\,\left(t+s\right)\left(\left(x+1\right)t+2s\right)\,x\,\left[t=x!\,s=2^x\right]\right)$

$2$

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