Overview of the LREtools[HypergeometricTerm] Subpackage - Maple Programming Help

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Overview of the LREtools[HypergeometricTerm] Subpackage

 Calling Sequence LREtools[HypergeometricTerm][command](arguments) command(arguments)

Description

 • The commands in the LREtools[HypergeometricTerm] subpackage solve the following problems.
 1 Find polynomial solutions of a linear difference equation with polynomial coefficients depending on hypergeometric terms.
 2 Find the hypergeometric dispersion of two polynomials depending on hypergeometric terms.
 3 Find a solution of an orbit problem.
 4 Find a universal denominator of the rational solutions of a linear difference equation with polynomial coefficients depending on hypergeometric terms.
 5 Find rational solutions of a linear difference equation with polynomial coefficients depending on hypergeometric terms.
 • Each command in the LREtools[HypergeometricTerm] subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.
 As the underlying implementation of the LREtools[HypergeometricTerm] subpackage is a module, it is also possible to use the form LREtools[HypergeometricTerm]:-command to access a command. For more information, see Module Members.

List of LREtools[HypergeometricTerm] Commands

 • The following is a list of available commands.

 • To display the help page for a particular LREtools[HypergeometricTerm] command, see Getting Help with a Command in a Package.

Examples

 > $\mathrm{with}\left({\mathrm{LREtools}}_{\mathrm{HypergeometricTerm}}\right):$
 > $\mathrm{eq}≔y\left(n+2\right)-\left(t+n\right)y\left(n+1\right)+n\left(t-1\right)y\left(n\right)$
 ${\mathrm{eq}}{:=}{y}{}\left({n}{+}{2}\right){-}\left({t}{+}{n}\right){}{y}{}\left({n}{+}{1}\right){+}{n}{}\left({t}{-}{1}\right){}{y}{}\left({n}\right)$ (1)
 > $\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(n\right),t=n!\right)$
 $\frac{{t}{}{{\mathrm{_C}}}_{{1}}}{{n}}{,}\left[{t}{,}{n}{+}{1}\right]$ (2)
 > $\mathrm{eq}≔y\left(n+1\right)\left(1+\left(n+1\right)t\right)-y\left(n\right)\left(1+t\right)$
 ${\mathrm{eq}}{:=}{y}{}\left({n}{+}{1}\right){}\left({1}{+}\left({n}{+}{1}\right){}{t}\right){-}{y}{}\left({n}\right){}\left({1}{+}{t}\right)$ (3)
 > $\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(n\right),t=n!\right)$
 ${0}{,}\left[{t}{,}{n}{+}{1}\right]$ (4)
 > $\mathrm{RationalSolution}\left(\mathrm{eq},y\left(n\right),t=n!\right)$
 $\frac{{{\mathrm{_C}}}_{{1}}}{{1}{+}{t}}{,}\left[{t}{,}{n}{+}{1}\right]$ (5)
 > $\mathrm{eq}≔45y\left(x\right)-9y\left(x\right)x-18y\left(x+3\right)+9y\left(x+3\right)x$
 ${\mathrm{eq}}{:=}{45}{}{y}{}\left({x}\right){-}{9}{}{y}{}\left({x}\right){}{x}{-}{18}{}{y}{}\left({x}{+}{3}\right){+}{9}{}{y}{}\left({x}{+}{3}\right){}{x}$ (6)
 > $\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(x\right),\left[t,\frac{9\cdot 1}{10-7x-8{x}^{2}}\right]\right)$
 $\frac{{{\mathrm{_C}}}_{{1}}}{{x}{-}{5}}{,}\left[{t}{,}\frac{{9}}{{-}{8}{}{{x}}^{{2}}{-}{7}{}{x}{+}{10}}\right]$ (7)
 > $p≔\left(x+2+{2}^{2}s\right)\left(x+100+{e}^{100}v\right);$$q≔s\left(x+s\right)\left(v+x\right);$$\mathrm{ext}≔\left[s={2}^{x},v={e}^{x}\right]$
 ${p}{:=}\left({x}{+}{2}{+}{4}{}{s}\right){}\left({{e}}^{{100}}{}{v}{+}{x}{+}{100}\right)$
 ${q}{:=}{s}{}\left({x}{+}{s}\right){}\left({v}{+}{x}\right)$
 ${\mathrm{ext}}{:=}\left[{s}{=}{{2}}^{{x}}{,}{v}{=}{{e}}^{{x}}\right]$ (8)
 > $\mathrm{HGDispersion}\left(p,q,x,\mathrm{ext}\right)$
 ${100}$ (9)

References

 Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.
 Bronstein, M. "On solutions of Linear Ordinary Difference Equations in their Coefficients Field." INRIA Research Report. No. 3797. November 1999.