Multi-dimensional q-summations and multi-colored partitions
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Motivated by Alladi’s recent multi-dimensional generalization of Sylvester’s classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy’s identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.
KeywordsSylvester’s identity Cauchy’s identity Multiple summations Multi-colored partitions Combinatorial proof
Mathematics Subject Classification05A17 11P84
We would like to acknowledge our gratitude to Ae Ja Yee for her helpful suggestions, which strengthen our original version of Theorem 1.2. We also want to thank the referee for the careful reading and useful comments.
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