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 diffeqtorec
 convert a linear differential equation into a recurrence

 Calling Sequence diffeqtorec(deq, y(z), u(n))

Parameters

 deq - linear differential equation in y(z) with polynomial coefficients y - name; function name z - name; variable of the function y u - name; recurrence name n - name; index of the recurrence u

Description

 • The diffeqtorec(deq, y(z), u(n)) command converts a linear differential equation, deq, into a recurrence.
 • Let $f$ be a power series solution of the differential equation.
 If u(n) is the nth Taylor coefficient of $f$ around zero, the diffeqtorec function returns a linear recurrence for the numbers u(n), with rational coefficients in n.
 • The syntax is the same as that of dsolve.  Combined with algeqtodiffeq, this function produces a linear recurrence for the Taylor coefficients of an algebraic function.

Examples

 > $\mathrm{with}\left(\mathrm{gfun}\right):$
 > $\mathrm{diffeqtorec}\left(y\left(z\right)=a\left(\frac{ⅆ}{ⅆz}y\left(z\right)\right),y\left(z\right),v\left(n\right)\right)$
 ${v}{}\left({n}\right){+}\left({-}{a}{}{n}{-}{a}\right){}{v}{}\left({n}{+}{1}\right)$ (1)
 > $\mathrm{deq}≔\mathrm{algeqtodiffeq}\left(y=1+z\left({y}^{2}+{y}^{3}\right),y\left(z\right),\left\{\right\}\right):$
 > $\mathrm{diffeqtorec}\left(\mathrm{deq},y\left(z\right),u\left(m\right)\right)$
 $\left\{\left({-}{2}{}{{m}}^{{2}}{-}{m}\right){}{u}{}\left({m}\right){+}\left({-}{18}{}{{m}}^{{2}}{-}{30}{}{m}{-}{9}\right){}{u}{}\left({m}{+}{1}\right){+}\left({46}{}{{m}}^{{2}}{+}{227}{}{m}{+}{279}\right){}{u}{}\left({m}{+}{2}\right){+}\left({-}{4}{}{{m}}^{{2}}{-}{26}{}{m}{-}{42}\right){}{u}{}\left({m}{+}{3}\right){,}{u}{}\left({0}\right){=}{1}{,}{u}{}\left({1}\right){=}{2}{,}{u}{}\left({2}\right){=}{10}\right\}$ (2)