#17123: Extending binomial(n,k) to negative integers n, k.
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Reporter: pluschny | Owner:
Type: enhancement | Status: new
Priority: minor | Milestone: sage-6.4
Component: combinatorics | Resolution:
Keywords: binomial | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by pluschny):
John, this is why I offered to discuss things first on sage-devel and hear
more arguments.
"Apparently Maple and Mathematica use the proposed version. Is that
sufficient justification to change our version?"
Of course not, although for me this is a strong hint that something is
missing in the current Sage implementation.
But this is a reason:
{{{
sage: binomial(-1, -1)
0
}}}
More generally, binomial(z, z) != 1 is absurd, a bug IMO, and definitely a
reason to change things. binomial(z, z) = 1 for all z is the first thing
to be preserved.
Next the relation of the diagonals to the Fibonacci numbers is another
important thing which should be preserved (and the proposed extension does
this although apparently Kronenburg misses to mention this in his paper).
The additive formula is nice in the region where the formula applies, but
the multiplicative formula is the more important one, a viewpoint taken by
all major modern authors as far as I understand.
--
Ticket URL: <http://trac.sagemath.org/ticket/17123#comment:12>
Sage <http://www.sagemath.org>
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