Gamma of complex numbers incorrectly simplifies to factorial

[sagetrac-JoalHeagney]

If I type the following into sage-5.5 built from source,

(gamma(-i-3/2)).simplify_full()

I get the following result:

factorial(-I - 5/2)

This:

n(factorial(-I - 5/2))

gives the obvious error:

TypeError: cannot evaluate symbolic expression numerically

while:

n(gamma(-I-5/2))

gives no problems.

-0.0417366258078936 + 0.0863691073697635*I

I've only noticed this happening in 5.5 - no such problems in 5.4 (that I remember).

CC: @benjaminfjones @sagetrac-dsm @eviatarbach @paulmasson

Component: symbolics

Keywords: gamma, simplify_full, simplify_factorial, maxima

Issue created by migration from https://trac.sagemath.org/ticket/13869

[burcin]

comment:1

This seems to be our doing. In simplify_factorial() we have:

return self.parent()(self._maxima_().makefact().factcomb().minfactorial())

Here is what the Maxima manual says about makefact():

transforms occurrences of binomial,gamma, and beta functions in exp to factorials. 

There is also a corresponding makegamma() function, but this will change factorials to gamma as well.

[kcrisman]

comment:2

In Sage 5.2:

sage: (gamma(-i-3/2)).simplify_full()
factorial(-I - 5/2)
sage: a = gamma(-i-3/2)
sage: a._maxima_().makefact()
(-%i-5/2)!

This seems to be our doing. In simplify_factorial() we have:

So I agree with Burcin that this comes from us.

The problem is that we do rely on this for some examples of the (mostly improved) simplification. E.g. from #6636

sage: var('k')
k
sage: f = binomial(n,k)*factorial(k)*factorial(n-k)
sage: f._maxima_()
k!*binomial(n,k)*(n-k)!
sage: f._maxima_().makefact()
n!

On the other hand, does Maxima really want these occurrences with non-integer inputs to be transformed thusly?

[sagetrac-JoalHeagney]

comment:3

So, should we report upstream?

[kcrisman]

comment:4

Replying to @sagetrac-JoalHeagney:

So, should we report upstream?

I'm not a special functions expert, so maybe it would be nice if someone who had work interrupted reported.

[sagetrac-JoalHeagney]

comment:5

Hmm, maybe not. From what I can see from the maxima documentation, it's kinda hinted that if you call makefact, it's assumed that you know what you're asking for.

Would it be possible to change the simplification code to use makegamma instead?

In the sage -maxima interface, I was able to do the following (these are all out of order cut-and-paste from my maxima session.)

(%i18) makegamma(binomial(n,k)*factorial(k)*factorial(n-k));
(%o18)                           gamma(n + 1)

(%i12) gamma_expand:true;
(%o12)                               true
(%i13) gamma(2+i);
(%o13)                        i (i + 1) gamma(i)
(%i16) makegamma(binomial(n,k)*factorial(k)*factorial(n-k));
(%o16)                            n gamma(n)

The benefits here, are that makegamma and gamma_expand functions seem to provide the same simplifications (if not more) that minfactorial and makefact do, but gamma functions can handle (by definition) more fields (RR, CC).

I don't know if we want gamma_expand to be true all the time, but it does seem to give another level of simplification, pulling out i(i+1) in my second example.

Then we could document the .makefact function so that users could convert sagemath (new default) gamma answers to factorials for themselves?

[sagetrac-JoalHeagney]

comment:6

In the documentation of maxima http://maxima.sourceforge.net/docs/manual/en/maxima_15.html there are also a lot of other expand options we may want to investigate as well:

E.g. besselexpand, beta_expand, expintexpand, hypergeometric and expand_hypergeometric,