LREtools[HypergeometricTerm] - Maple Programming Help

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LREtools[HypergeometricTerm]

 SubstituteTerm
 return solution with a directly substituted hypergeometric term instead of a name representing it in the solution

 Calling Sequence SubstituteTerm(sol, term, n)

Parameters

 sol - rational function in n and a name representing a hypergeometric term term - hypergeometric term n - independent variable

Description

 • The SubstituteTerm(sol, term, n) returns the solution with a directly substituted hypergeometric term instead of a name representing it in the solution.
 • 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. The term is 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 solution is the function, corresponding to n. The solution involves arbitrary constants of the form, for example, _c1 and _c2.

Examples

 > $\mathrm{with}\left({\mathrm{LREtools}}_{\mathrm{HypergeometricTerm}}\right):$
 > $\mathrm{eq}≔y\left(n+2\right)-\left(n!+n\right)y\left(n+1\right)+n\left(n!-1\right)y\left(n\right)$
 ${\mathrm{eq}}{:=}{y}{}\left({n}{+}{2}\right){-}\left({n}{!}{+}{n}\right){}{y}{}\left({n}{+}{1}\right){+}{n}{}\left({n}{!}{-}{1}\right){}{y}{}\left({n}\right)$ (1)
 > $\mathrm{sol},r≔\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(n\right)\right)$
 ${\mathrm{sol}}{,}{r}{:=}\frac{{t}{}{{\mathrm{_C}}}_{{1}}}{{n}}{,}\left[{t}{,}{n}{+}{1}\right]$ (2)
 > $\mathrm{SubstituteTerm}\left(\mathrm{sol},r,n\right)$
 ${\mathrm{Γ}}{}\left({n}\right){}{{\mathrm{_C}}}_{{1}}$ (3)
 > $\mathrm{eq}≔\left(t+{n}^{2}\right)z\left(n+1\right)-\left(2nt+2t+{n}^{2}+2n+1\right)z\left(n\right)$
 ${\mathrm{eq}}{:=}\left({{n}}^{{2}}{+}{t}\right){}{z}{}\left({n}{+}{1}\right){-}\left({{n}}^{{2}}{+}{2}{}{n}{}{t}{+}{2}{}{n}{+}{2}{}{t}{+}{1}\right){}{z}{}\left({n}\right)$ (4)
 > $\mathrm{sol},r≔\mathrm{PolynomialSolution}\left(\mathrm{eq},z\left(n\right),t={2}^{n}n!\right)$
 ${\mathrm{sol}}{,}{r}{:=}{{n}}^{{2}}{}{{\mathrm{_C}}}_{{1}}{+}{t}{}{{\mathrm{_C}}}_{{1}}{,}\left[{t}{,}{2}{}{n}{+}{2}\right]$ (5)
 > $\mathrm{SubstituteTerm}\left(\mathrm{sol},r,n\right)$
 ${{\mathrm{_C}}}_{{1}}{}\left({{2}}^{{n}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right){+}{{n}}^{{2}}\right)$ (6)

References

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