#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 rws):
The pseudocode in the ticket description does not seem to fit the paper
given as reference, quote:
{{{
For nonnegative integer n and integer k this reduces to [1]:
(1.2) binomial(n,k) = n!/(k!(n-k)!) if 0<=k<=n, 0 otherwise
... This results in the following binomial coefficient identity,
which with identity (1.2) allows computation of the binomial coefficient
for all integer
arguments.
(2.1) Theorem 2.1. For negative integer n and integer k:
binomial(n,k) = (-1)^k * binomial(-n+k-1,k) if k>0,
(-1)^(n-k) * binomial(-k-1,n-k) if k<=n,
0 otherwise
...
}}}
so in my opinion the definition should rather be
{{{
def BINOMIAL(n, k):
if n in ZZ and k in ZZ:
if n >= 0:
return binomial(n, k)
if k >= 0:
return (-1)^k*binomial(-n+k-1, k)
if k <= n:
return (-1)^(n-k)*binomial(-k-1, n-k)
return 0
else:
return binomial(n, k)
}}}
This leaves a discrepancy between `binomial(*,k<0)` and `BINOMIAL(*,k<0)`
in the area `n<0, k<n` and to get to the point I would ask darij to
provide a Knuth reference that supports the behaviour of `binomial` for
these values.
--
Ticket URL: <http://trac.sagemath.org/ticket/17123#comment:16>
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