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Durfee rectangles and the Jacobi triple product identity. (English) Zbl 0782.05008

Das Jacobische dreifache Produkt kann folgendermaßen formuliert werden: \[ (t;q)_ \infty\cdot (t^{-1} q;q)_ \infty\cdot (q;q)_ \infty= \sum^ \infty_{n=-\infty} (-1)^ n q^{{n\choose 2}} t^ n, \] wobei \((x; q)_ 0=1\) und \((x;q)_ n= (1-x)(1- xq)\cdots (1- xq^{n-1})\) ist.
In diesem Artikel wird mittels der Durfee-Rechtecke der Partitionen ein sehr elementarer Beweis für die Identität des Jacobischen dreifachen Produkts sowie der endlichen Analoga vorgelegt. Schließlich wird ein Beispiel angeführt.

MSC:

05A19 Combinatorial identities, bijective combinatorics
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References:

[1] Andrews, G.E.,Generalizations of the Durfee square, J. London Math. Soc.,3 (1971), 563–570. · Zbl 0223.05005 · doi:10.1112/jlms/s2-3.3.563
[2] Andrews, G.E.,Two theorems of Gauss and allied identities proved arithmetically, Pacific J. Math.,41 (1972), 563–578. · Zbl 0219.10021 · doi:10.2140/pjm.1972.41.563
[3] Andrews, G.E., The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol.2 (addison-Wesley, Reading, MA, 1976).
[4] Andrews, G.E.,Partitions and Durfee dissection, Amer. J. of Math.,101 (1979), 735–742. · Zbl 0409.10006 · doi:10.2307/2373804
[5] Gessel, I.M.,Some generalized Durfee square identities, Discrete Math.,49 (1984), 41–44. · Zbl 0547.05009 · doi:10.1016/0012-365X(84)90149-3
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