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

Several new product identities in relation to Rogers–Ramanujan type sums and mock theta functions Product identities in two variables x, q 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature 2021
eISSN
2197-9847
DOI
10.1007/s40687-021-00245-8
Publisher site
See Article on Publisher Site

Abstract

Product identities in two variables x, q 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.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Feb 22, 2021

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