Re: [Axiom-developer] Patches

[Waldek Hebisch]

Martin Rubey wrote:
> Waldek Hebisch <address@hidden> writes:
> 
> > > But the behaviour is consistent for Gamma, Bessel and Polygamma.  It is 
> > > not
> > > difficult to change this behaviour to leaving the derivative unevaluated,
> > > but I'm not sure whether that would really be better.  If you are
> > > absolutely sure, please let me know as soon as possible.
> > 
> > Yes, currently we produce mathematically incorrect result.  In principle 
> > user
> > may get wrong results even if input does not contain explicit derivative.
> 
> Oh? How is that?
> 

Consider something like naive test for holonomic functions: compute
derivatives up to some fixed order and check for linear dependence.
Such test would immediatly conclude that besselK(a, x) is holonomic
as a function of a.  I did not check this but I think that
besselK(a, x) is not holonomic as a function of a, and certainly
we would get wrong differential equation.  Once you have differential
equation in hand you can do a lot of transformations.

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

> > > How about polygamma?  should D(polygamma(x, x), x) throw an error?  I 
> > > guess so.
> > > But if we follow you, Bessel* should leave the derivative with respect to 
> > > the
> > > first argument - i.e., leave it unevaluated.
> 
> > polygamma(a, x) has sensible definition also for non-integral a, so just
> > leaving D(polygamma(x, x), x) unevaluated is reasonable.
> 
> I could not find such a definition.  Could you please send me such a 
> definition
> or a reference?
>


>From http://mathworld.wolfram.com/PolygammaFunction.html:

   A special function which is given by the (n+1) st derivative of the
   logarithm of the gamma function Gamma(z)
   ....
   ....
   psi_n(z) is implemented in Mathematica as PolyGamma[n, z] for positive
   integer n . In fact, PolyGamma[nu, z] is supported for all complex nu
   (Grossman 1976; Espinosa and Moll 2004).

I do not know which definition the references use, but a derivatives
may be defined for fractional orders via convolution:

{d \over dx}^n f = f*mu_{-n-1}

where 

mu_l(x) = x^l/\Gamma(l+1)  for x > 0

and mu_l(x) = 0 for x < 0.

This definition of derivative is for non-integral n, for integral
n you get normal derivative as a limit.

The definition above will get function which is analytic in n.  Because
analytic functions have strong restictions on possible zeros other
definitions are likely to give the same value.
 

-- 
                              Waldek Hebisch
address@hidden