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LREtools[HypergeometricTerm]

 PolynomialSolution
 return the polynomial solution of linear difference equation depending on a hypergeometric term

 Calling Sequence PolynomialSolution(eq, var, term)

Parameters

 eq - linear difference equation depending on a hypergeometric term var - function variable for which to solve, for example, z(n) term - hypergeometric term

Description

 • The PolynomialSolution(eq, var, term) command returns the polynomial solution of the linear difference equation eq. If such a solution does not exist, the function returns NULL.
 • The hypergeometric term in the linear difference equation is specified by a name, for example, t. The meaning of the term is defined by the parameter term. 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 term variable and the consecutive term ratio, for example, $\left[t,n+1\right]$.
 • If the third parameter is omitted, then the input equation can contain a hypergeometric term directly (not a name). In this case, the procedure extracts the term from the equation, transforms the equation to the form with a name representing a hypergeometric term, and then solves the transformed equation.
 • The term "polynomial solution" means a solution $y\left(x\right)$ in $Q\left(x\right)\left[t,{t}^{-1}\right]$ , that is, in the form $y={y}_{d}{t}^{d}+...+{y}_{g}{t}^{g}$ where $d\le g$ and ${y}_{d},...,{y}_{g}$ are in $Q\left(x\right)$.
 • The solution is the function, corresponding to var. The solution involves arbitrary constants of the form, for example, _c1 and _c2.

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)
 > $\mathrm{eq}≔y\left(n+2\right)-\left(n!+n\right)y\left(n+1\right)+n\left(n!-1\right)y\left(n\right)$
 ${\mathrm{eq}}{≔}{y}{}\left({n}{+}{2}\right){-}\left({n}{!}{+}{n}\right){}{y}{}\left({n}{+}{1}\right){+}{n}{}\left({n}{!}{-}{1}\right){}{y}{}\left({n}\right)$ (3)
 > $\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(n\right)\right)$
 $\frac{{t}{}{{\mathrm{_C}}}_{{1}}}{{n}}{,}\left[{t}{,}{n}{+}{1}\right]$ (4)
 > $\mathrm{eq}≔\left(t+{n}^{2}\right)z\left(n+1\right)-\left(2nt+2t+{n}^{2}+2n+1\right)z\left(n\right)$
 ${\mathrm{eq}}{≔}\left({{n}}^{{2}}{+}{t}\right){}{z}{}\left({n}{+}{1}\right){-}\left({{n}}^{{2}}{+}{2}{}{t}{}{n}{+}{2}{}{n}{+}{2}{}{t}{+}{1}\right){}{z}{}\left({n}\right)$ (5)
 > $\mathrm{PolynomialSolution}\left(\mathrm{eq},z\left(n\right),t={2}^{n}n!\right)$
 ${{\mathrm{_C}}}_{{1}}{}{{n}}^{{2}}{+}{t}{}{{\mathrm{_C}}}_{{1}}{,}\left[{t}{,}{2}{}{n}{+}{2}\right]$ (6)
 > $\mathrm{eq}≔45y\left(x\right)-9y\left(x\right)x-18y\left(x+3\right)+9y\left(x+3\right)x$
 ${\mathrm{eq}}{≔}{45}{}{y}{}\left({x}\right){-}{9}{}{y}{}\left({x}\right){}{x}{-}{18}{}{y}{}\left({x}{+}{3}\right){+}{9}{}{y}{}\left({x}{+}{3}\right){}{x}$ (7)
 > $\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(x\right),\left[t,\frac{9\cdot 1}{10-7x-8{x}^{2}}\right]\right)$
 $\frac{{{\mathrm{_C}}}_{{1}}}{{x}{-}{5}}{,}\left[{t}{,}\frac{{9}}{{-}{8}{}{{x}}^{{2}}{-}{7}{}{x}{+}{10}}\right]$ (8)

References

 Bronstein, M. "On solutions of Linear Ordinary Difference Equations in their Coefficients Field." INRIA Research Report. No. 3797. November 1999.