Re: [Reduce-algebra-developers] expansion of gamma function

[Alan B. <ala...@gm...>]

On 15/10/15 04:58, Andrey G. Grozin wrote:
> On Wed, 14 Oct 2015, Rainer Schöpf wrote:
>> However, it means that gamma, psi and friends must be treated specially.
With the current set of rules for differentiating gamma, psi and 
polygamma, tps will be unable to build a suitable recurrence relation to 
successfully expand these functions as power series. With the current 
state of my knowledge of the gamma function I can't say whether it is 
feasible to recast the differentiation rules in a more suitable form for 
tps.

Rainer is probably better placed to judge whether taylor can be adapted 
to handle the expansion of these functions by adding special rules for 
gamma & friends.

> Yes. But other systems (maxima, mathematica, ...) can expand gamma, 
> while reduce cannot :-(
>
Does anyone in the Reduce community know what the algorithms these 
systems use?

tps can find the series for gamma(1+x) about x=0 as described in my 
earlier post (This works if one uses an up-to-date version of Reduce!!):

     s := pssum(n=2, (-1)^n*zeta(n)/n, x, 0, n);
     s1 := ps(s-euler_gamma*x,x,0);
     s2 := ps(exp(s1), x, 0);

Perhaps not ideal, but it works!
>> In addition, the above formula doesn't help for the expandsion of 
>> gamma(f(x))
>> where f(x) is some non-trivial, but well known function, like 
>> gamma(x^2-x^3) .
> For any f(x) such that f(0) is integer it does. First reduce 
> gamma(f(x)) to gamma(f(x)+n) where n is integer and f(0)+n=1, then use 
> the above formula. In particular, for your example x^2-x^3 this works 
> fine.
>
> It does not help if f(0) is not integer.
>
> Andrey
>

Using pscompose will help in some cases:

s3:=ps(sin x, x, 0);
pscompose(s2, s3);

yields the series for gamma(1+sin x), but the series s3 must have an 
order greater than 0 (in order that the power series composition 
algorithm works).  So this doesn't help if f(0) is non-integral.

Alan