Split \((n+t)\)-color partitions and Gordon-McIntosh eight order mock theta functions. (English) Zbl 1300.05025
Summary: The first author [ibid. 11, No. 1, Research Paper N14, 6 p. (2004; Zbl 1063.05006)] gave the combinatorial interpretations of four mock theta functions of Srinivasa Ramanujan using \(n\)-color partitions which were introduced by A. K. Agarwal and G. E. Andrews [J. Comb. Theory, Ser. A 45, 40–49 (1987; Zbl 0618.05003)]. In this paper we introduce a new class of partitions and call them ”split \((n+t)\)-color partitions”. These new partitions generalize Agarwal-Andrews \((n+t)\)-color partitions. We use these new combinatorial objects and give combinatorial meaning to two basic functions of Gordon-McIntosh found in [B. Gordon and R. J. McIntosh, J. Lond. Math. Soc., II. Ser. 62, No. 2, 321–335 (2000; Zbl 1031.11007)]. They used these functions to establish the modular transformation formulas for certain eight order mock theta functions. The work done here has a great potential for future research.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A17 | Combinatorial aspects of partitions of integers |
| 11P81 | Elementary theory of partitions |
Keywords:
mock theta functions; \((n+t)\)-color partitions; split \((n+t)\)-color partitions; combinatorial interpretationsReferences:
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