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2010•The Quarterly Journal of MathematicsRequires access

PARTITION IDENTITIES FOR RAMANUJAN'S THIRD-ORDER MOCK THETA FUNCTIONS

W. Y. C. Chen, Kathy Q. Ji, Eric H. Liu

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Abstract

We find two involutions on partitions that lead to partition identities for Ramanujan's third-order mock theta functions ϕ(−q) and ψ(−q). We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third-order mock theta functions are consequences of these identities. Our combinatorial constructions also apply to Andrews’ generalizations of Ramanujan's identities.

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What this paper is about

We find two involutions on partitions that lead to partition identities for Ramanujan's third-order mock theta functions ϕ(−q) and ψ(−q). We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third-order mock theta functions are consequences of these identities. Our combinatorial constructions also apply to Andrews’ generalizations of Ramanujan's identities.

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Available abstract

We find two involutions on partitions that lead to partition identities for Ramanujan's third-order mock theta functions ϕ(−q) and ψ(−q). We also give an involution for Fine's partition identity on the mock theta function f(q). The two classical identities of Ramanujan on third-order mock theta functions are consequences of these identities. Our combinatorial constructions also apply to Andrews’ generalizations of Ramanujan's identities.

Key concepts: Ramanujan's sum, Ramanujan theta function, Mathematics, Partition (number theory), Theta function, Ramanujan tau function, Third order, Combinatorics

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