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Zbl 0559.05008
Gessel, Ira; Stanton, Dennis
Short proofs of Saalschütz's and Dixon's theorems.
(English)
[J] J. Comb. Theory, Ser. A 38, 87-90 (1985). ISSN 0097-3165

Short proofs are given of the identities: $$\sum\sb{k\ge 0}\left( \matrix a\\ l-k\endmatrix \right)\left( \matrix b\\ m-k\endmatrix \right)\left( \matrix a+b+k\\ k\endmatrix \right)=\left( \matrix a+m\\ l\endmatrix \right)\left( \matrix b+l\\ m\endmatrix \right),$$ $$\sum\sb{k\ge 0}(-1)\sp k\left( \matrix n\\ k\endmatrix \right)\left( \matrix m\\ l-n+k\endmatrix \right)\left( \matrix l\\ l-m+k\endmatrix \right)=\cases 0&\quad\text{ if l+m-n odd},\\ (-1)\sp{m-r}{n+r\choose n}{n\choose l-r} &\quad\text{ if l+m-n=2r}.\endcases$$ The proofs depend upon the fact that, for any Laurent series $f(x,y)$, the constant term of $f(x/1+y$, $y/1+x)$ equals that of $1/(1-xyf(x,y))$.
[I.Anderson]
MSC 2000:
*05A19 Combinatorial identities
05A10 Combinatorial functions

Keywords: binomial coefficients

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