gfun[diffeqtolist] - compute coefficients from a differential equation

gfun[diffeqtoseries] - compute a truncated series from a differential equation

gfun[rectolist] - compute coefficients from a recurrence

gfun[rectoseries] - compute a truncated series from a recurrence

Calling Sequence

diffeqtolist(deq, y(x), N)

diffeqtoseries(deq, y(x), N)

rectolist(rec, u(n), N)

rectoseries(rec, u(n), x, N)

Parameters

deq - a linear differential equation, optionally together with initial conditions

y(x) - the unknown function and independent variable of the differential equation

rec - a linear recurrence, optionally together with initial conditions

u(n) - the unknown sequence and recurrence index

x - the variable of the output series

N - a nonnegative integer giving the last coefficient index

Description

The rectolist command returns the list [u(0), ..., u(N)] determined by rec. It uses rectoproc with its list option.
The rectoseries command converts this list to the ordinary generating series in x. Its result contains terms through degree N and has order term O(x^(N+1)).
The diffeqtolist and diffeqtoseries commands first convert deq to a recurrence by using diffeqtorec. The output coefficients are the Taylor coefficients at the origin.
Initial conditions may be included in rec or deq. If they do not determine a unique solution, arbitrary constants may occur in the output.
These commands are part of the gfun package. Their short forms are available after executing with(gfun); their long forms, such as gfun[diffeqtolist](...), are always available.

Examples

> with(gfun):
rec := {u(n+2)=u(n+1)+u(n),u(0)=0,u(1)=1};

\[\{u \! \left(0\right) = 0, u \! \left(1\right) = 1, u \! \left(n +2\right) = u \! \left(n +1\right)+u \! \left(n \right)\}\]

> rectolist(rec,u(n),10);

\[[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]\]

> rectoseries(rec,u(n),x,10);

\[x +x^{2}+2 x^{3}+3 x^{4}+5 x^{5}+8 x^{6}+13 x^{7}+21 x^{8}+34 x^{9}+55 x^{10}+\mathrm{O}\! \left(x^{11}\right)\]

> deq := {diff(y(x),x)-y(x),y(0)=1};

\[\{\frac{d}{d x}y \! \left(x \right)-y \! \left(x \right), y \! \left(0\right) = 1\}\]

> diffeqtolist(deq,y(x),7);

\[\left[1, 1, {\frac{1}{2}}, {\frac{1}{6}}, {\frac{1}{24}}, {\frac{1}{120}}, {\frac{1}{720}}, {\frac{1}{5040}}\right]\]

> diffeqtoseries(deq,y(x),7);

\[1+x +\frac{1}{2} x^{2}+\frac{1}{6} x^{3}+\frac{1}{24} x^{4}+\frac{1}{120} x^{5}+\frac{1}{720} x^{6}+\frac{1}{5040} x^{7}+\mathrm{O}\! \left(x^{8}\right)\]