Re: Maple gfun

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, Dec 22, 2021 at 04:57:54PM +0100, Bill Allombert wrote:
> On Wed, Dec 22, 2021 at 02:21:57PM +0100, Ruud H.G. van Tol wrote:
> > 
> > On 2021-12-22 14:09, Bill Allombert wrote:
> > > On Wed, Dec 22, 2021 at 01:48:03PM +0100, Ruud H.G. van Tol wrote:
> > 
> > > > Would something like Maple's gfun be useful for PARI?
> > > 
> > > What does gfun ? Could you give some example ?
> > 
> > https://dl.acm.org/doi/10.1145/178365.178368
> 
> What I need is an example with some GP input and some expected GP output.
> 
> I will give you one:
> 
> ? S=sum(i=0,20,binomial(2*i,i)*T^i)+O(T^21);
> ? seralgdep(S,2,2)
> %8 = (4*T-1)*x^2+1
> 
> So S = sqrt(1/(1-4*T)) and indeed:
> 
> ? S==sqrt(1/(1-4*T))
> %10 = 1

A new function (available in the git branch bill-serdiffdep).

? S=sum(i=0,20,binomial(3*i,i)*T^i)+O(T^21);
? serdiffdep(S,3,3)
%3 = [(27*T^2-4*T)*x^2+(54*T-2)*x+6,0]

So S satisfies the linear equation 

(27*T^2-4*T)*S'' + (54*T-2)*S' +6*S = 0

? S=exp(T^2)+T^2
%4 =
%1+2*T^2+1/2*T^4+1/6*T^6+1/24*T^8+1/120*T^10+1/720*T^12+1/5040*T^14+1/40320*T^16+O(T^18)
? serdiffdep(S,3,3)
%5 = [x-2*T,-2*T^3+2*T]

So S satisfies the linear equation:
S'-2*T*S = -2*T^3+2*T

Is it what you are looking for ?

Cheers,
Bill