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Product identities in two variables $x, q$ expand infinite products asinfinite sums, which are linear combinations of theta functions; famousexamples include Jacobi's triple product identity, Watson's quintuple identity,and Hirschhorn's septuple identity. We view these series expansions asrepresentations in canonical bases of certain vector spaces of quasiperiodicmeromorphic functions (related to sections of line and vector bundles), andfind new identities for two nonuple products, an undecuple product, and severaltwo-variable Rogers-Ramanujan type sums. Our main theorem explains acorrespondence between the septuple product identity and the two originalRogers-Ramanujan identities, involving two-variable analogues of fifth-ordermock theta functions. We also prove a similar correspondence between an octupleproduct identity of Ewell and two simpler variations of the Rogers-Ramanujanidentities, which is related to third-order mock theta functions, andconjecture other occurrences of this phenomenon. As applications, we specializeour results to obtain identities for quotients of generalized Dedekind etafunctions and mock theta functions.

Several new product identities in relation to two-variable Rogers-Ramanujan type sums and mock theta functions