Several new product identities in relation to Rogers–Ramanujan type sums and mock theta functions

Abstract

Product identities in two variables xq expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi’s triple product identity, Watson’s quintuple identity, and Hirschhorn’s septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers–Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers–Ramanujan identities; this amounts to an unexpected proportionality of canonical basis vectors, two of which can be viewed as two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers–Ramanujan identities, related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized eta functions and mock theta functions.

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Acknowledgements

The author wishes to thank Professor Terence Tao for his helpful guidance, suggestions and support for this project. The author is also deeply grateful to Professors S. Ole Warnaar and William Duke for insightful discussions and feedback.

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Correspondence to Alexandru Pascadi.

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Pascadi, A. Several new product identities in relation to Rogers–Ramanujan type sums and mock theta functions. Res Math Sci 8, 16 (2021). https://doi.org/10.1007/s40687-021-00245-8

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Keywords

  • Product identities
  • Rogers–Ramanujan identities
  • Generalized eta functions
  • Mock theta functions

Mathematics Subject Classification

  • 11P84
  • 33D15
  • 11B65