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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.
 The long form, LREtools:-HypergeometricTerm:-command, is always available. The short form can be used after loading the package.

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.