#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 darij):
Peter:
(1) If it is an extension, then call it `binomial_extension` or whatever;
do not usurp `binomial`.
(2) Maple and Mathematica do not set standards in combinatorics.
The 0 definition does enlarge a relation to its largest possible domain of
validity -- the recurrence relation, to the whole integer lattice. Unlike
the symmetry, the recurrence actually is used all the time in proofs. The
0 definition also matches with the idea that (-1)^k (-n choose k) is the
number of k-element multisets of elements of {1, 2, ..., n}. It is most
definitely guided by mathematical consideration; check, e.g., the
identities (5.22) to (5.26) in Graham-Knuth-Patashnik, specifically (5.22)
(Vandermonde's convolution, an extremely important identity). It might not
be the most interesting definition, but we should not be aiming for that;
I don't think we want 1 + 2 + 3 + ... to yield -1/12 just because the most
interesting definition for summing powers of positive integers involves
the zeta function. We should be going for the definition which is the most
reliable and standard in *mathematical* literature. And here it is the 0
one.
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Ticket URL: <http://trac.sagemath.org/ticket/17123#comment:19>
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