Re: [Axiom-developer] Patches

[Martin Rubey]

Waldek Hebisch <address@hidden> writes:

> Martin Rubey wrote:

> Of course, currently Axiom has no support for holonomic functions.

Yes. :-(

> But in few places we use derivatives: changing variables in integrals,
> computing Laplace transforms.  It is quite possible that Axiom never uses
> derivative of bessel function with respect to parameter.  But checking this
> would be a substantial ongoing effort.
> 
> > > Once we get better support for special functions this may be very serious
> > > problem.
> > 
> > Probably. By the way: most (probably all) special functions would be
> > covered by my favourite would-be category/domain hierarchy of
> > differentially algebraic functions. Then we could say something like
> > 
> > polygamma(a, x)$HOLO(???)

ups, that should read

besselJ(a, x)$HOLO(???)

or some such. (And, of course, I have no idea whether it is holonomic in both
variables, although I doubt it)

> > and get the corresponding differential equation.
> > 
> 
> Hmm, gamma and consequently also polygamma(a, x) as a function of x is
> differential transcendental.  Also handling of non-holonomic differentially
> algebraic functions seem to be a research problem -- do you have some
> interesting results here?

Not me, but Joris van der Hoeven told me that there is indeed a zero test for
differentially algebraic functions (in the sense: differentially algebraic
function = function satisfying ADE = exists a polynomial p with p(f, f',
f'',...) = 0)

Furthermore, ADE's are closed under addition, multiplication, etc., and there
exist algorithms to do all that.  They are probably not extremely efficient,
but since the range of functions covered is vast, this would nevertheless be
extremely good to have.

What I have implemented in my guessing package is a reasonably fast way to
guess the differential equation for such functions.  Try 

(7) -> guessADE [1,1,2,9/2,32/3,625/24,324/5]

   (7)
   [
     [
       function =
         BRACKET
              n
            [x ]f(x):
                    ,          3       2             ,        ,,
                - xf (x) + f(x)  - f(x) = 0,f(0)= 1,f (0)= 1,f  (0)= 4

             ,
                 ,,,         (iv)
                f   (0)= 27,f    (0)= 256

       ,
      order= 0]
     ]
    Type: List Record(function: Expression Integer,order: NonNegativeInteger)


Thanks for the polygamma reference.

Martin