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Short proofs of Saalschütz’s and Dixon’s theorems. (English) Zbl 0559.05008
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 $1/(1-xyf(x,y))$.
Reviewer: I.Anderson

05A19Combinatorial identities, bijective combinatorics
05A10Combinatorial functions
Full Text: DOI
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[2] Cartier, P.; Foata, D.: Problèmes combinatoire de commutation et réarrangements. Lecture notes in mathematics no. 85 (1969) · Zbl 0186.30101
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[4] Jacobi, C. G. J: Originally published in J. Reine angew. Math.. J. reine angew. Math. 6, 257-286 (1830)
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