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

Alexandru Pascadi

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, involving 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, which is 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 Dedekind eta functions and mock theta functions.

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

Abstract

Product identities in two variables 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, involving 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, which is 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 Dedekind eta functions and mock theta functions.

Paper Structure

This paper contains 16 sections, 54 theorems, 207 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Overview and Notation
  3. Methods
  4. Layout of paper
  5. Notation
  6. Structure of the relevant function spaces and first applications
  7. Dimensionality bounds
  8. Canonical basis vectors
  9. How to identify two functions in the same space
  10. Higher-order product identities and generalized eta functions
  11. M-coefficients and higher-order identities
  12. Identities for quotients of generalized eta functions
  13. Rogers--Ramanujan type identities and mock theta functions
  14. Two-variable generalizations of Rogers--Ramanujan-type identities
  15. W-coefficients, second proofs of main theorems and change-of-basis identities
  16. ...and 1 more sections

Key Result

Proposition \oldthetheorem

As an identity of functions in $\mathcal{T}_\mathbb{C}\left(q^2x^7\right)$,

Figures (3)

  • Figure 1: Relationships between results (arrows show logical implications or equivalences; green marks new results, to the best of the author's knowledge; '2-var.' $=$ 'two-variable').
  • Figure 2: Plots of $[{-qx^5};{1}] + [{-qx^5};{2}]$ (left) and $(q; q)\left(x^{-1}; q\right)[{qx^2 - x};{0}]$ (right) on $\{\textnormal{Re}\ z, \textnormal{Im}\ z \in [0, 1)\}$, for $\tau = i$ (using domain coloring; plot made in Wolfram Mathematica).
  • Figure 3: Relations between infinite products; '$A \xrightarrow{F} B$' means that $B = A \cdot \alpha F$ for some entire function $F(z) = \left\langle \pm q^b x^a; q^a \right\rangle$ and some $\alpha \in \mathbb{C}^\times$ (which may depend on $q$).

Theorems & Definitions (155)

  • Proposition \oldthetheorem: First nonuple product identity
  • Proposition \oldthetheorem: Two-variable statement of \ref{['eq:rog-ram']}
  • Remark
  • Theorem \oldthetheorem: Septuple identity vs. Rogers--Ramanujan
  • Theorem \oldthetheorem: Octuple identity vs. Rogers--Ramanujan variation
  • Corollary \oldthetheorem: Eta quotient polynomial
  • Corollary \oldthetheorem: Fifth-order mock theta sums
  • Remark
  • Proposition \oldthetheorem: Two-variable $2 \times 2$ determinant identity
  • Remark
  • ...and 145 more