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On Combinatorial Extensions of Some Ramanujan’s Mock Theta Functions
Ukrainian Mathematical Journal ( IF 0.5 ) Pub Date : 2020-06-01 , DOI: 10.1007/s11253-020-01763-4
M. Goyal

Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions of Ramanujan. Using a bijection between the associated lattice path functions and the $(n+t)$-color partitions and then between the associated lattice path functions and the weighted lattice path functions, we extend the works by Agarwal and Agarwal and Rana to five new 3-way combinatorial identities. These results are further extended to 4-way combinatorial identities by using bijection between the $(n+t)$-color partitions and the partitions with certain antihook differences. These interesting results present elegant combinatorial links between Ramanujan's mock theta functions, $(n+t)$-color partitions, weighted lattice paths, associated lattice paths, and antihook differences.

中文翻译:

一些拉马努金模拟 Theta 函数的组合扩展

S. Ramanujan 的五个模拟 theta 函数通过某些关联的晶格路径函数和反钩子差异进行组合解释。这些结果为 Ramanujan 的五个模拟 theta 函数提供了新的组合解释。使用关联格子路径函数和 $(n+t)$-color 分区之间的双射,然后在关联格子路径函数和加权格子路径函数之间,我们将 Agarwal、Agarwal 和 Rana 的工作扩展到五个新的 3 -way 组合恒等式。通过使用 $(n+t)$-color 分区和具有某些 antihook 差异的分区之间的双射,将这些结果进一步扩展到 4-way 组合恒等式。这些有趣的结果呈现了 Ramanujan 的模拟 theta 函数、$(n+t)$-color 分区之间的优雅组合链接,
更新日期:2020-06-01
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