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an algebraic identity leading to Wilson's theorem
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For any positive integer $n$ and any real or complex number $x$
$$ \sum_{k=0}^{n} (-1)^k {n \choose k}(x-k)^n = n! $$
Furthermore, if $n>m$ then
$$ \sum_{k=0}^{n} (-1)^k {n \choose k}(x-k)^m = 0 $$
- 1
- S. M Ruiz.
An algebraic identity leading to Wilson's theorem.
The Mathemtical Gazette, 80(489):579-582, November 1996.
math.GM/0406086.
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"an algebraic identity leading to Wilson's theorem" is owned by GeraW. [ full author list (2) ]
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Cross-references: complex number, real, integer, positive
This is version 13 of an algebraic identity leading to Wilson's theorem, born on 2004-08-04, modified 2005-04-18.
Object id is 6069, canonical name is RuizIdentity.
Accessed 3011 times total.
Classification:
| AMS MSC: | 05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions) | | | 11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities) |
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Pending Errata and Addenda
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