A finite-theta certificate for sum-side modularity

Definitions and mathematical statement

Let $\tau\in\mathbb H$, $q=e^{2\pi i\tau}$, and $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$, with the convergent infinite-product convention for $|q|<1$. Define from their unilateral sums \[ G(q)=\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n},\qquad H(q)=\sum_{n\geq0}\frac{q^{n^2+n}}{(q;q)_n}. \] The original motivation is the Rogers--Ramanujan identities \[ G(q)=\frac1{(q;q^5)_\infty(q^4;q^5)_\infty},\qquad H(q)=\frac1{(q^2;q^5)_\infty(q^3;q^5)_\infty}. \] It asks for a method proving modularity directly from the sum side, rather than taking these product evaluations as input. Normalize $g(\tau)=q^{-1/60}G(q)$, $h(\tau)=q^{11/60}H(q)$, with fractional powers interpreted through $\tau$.

The aid's reported product-free route uses Bressoud's terminating Gaussian-polynomial identities, recurrence certification, and compact-open convergence to unary theta differences, followed by Poisson summation. The exact terminating identities and recurrences are not supplied in the record. A precise theta target for the standard normalization is \[ \Theta_r(\tau)=\sum_{n\equiv r\ (\mathrm{mod}\ 20)}q^{n^2/40},\qquad g=\frac{\Theta_1-\Theta_9}{\eta},\qquad h=\frac{\Theta_3-\Theta_7}{\eta}, \] where $\eta(\tau)=q^{1/24}\prod_{j\geq1}(1-q^j)$. The supplied route does not invoke either Rogers--Ramanujan product identity or Jacobi's triple product as an input. In this normalization its reported exact transformation laws are \[ \begin{pmatrix}g(-1/\tau)\\h(-1/\tau)\end{pmatrix} =\frac2{\sqrt5}\begin{pmatrix} \sin(2\pi/5)&\sin(\pi/5)\\ \sin(\pi/5)&-\sin(2\pi/5) \end{pmatrix}\begin{pmatrix}g(\tau)\\h(\tau)\end{pmatrix}, \] \[ \begin{pmatrix}g(\tau+1)\\h(\tau+1)\end{pmatrix} =\begin{pmatrix}e^{-2\pi i/60}&0\\0&e^{22\pi i/60}\end{pmatrix} \begin{pmatrix}g(\tau)\\h(\tau)\end{pmatrix}. \] This specifies weight-zero vector-valued modularity, allowing the cusp pole indicated by $q^{-1/60}$, rather than claiming both entries are holomorphic modular forms at every cusp.

The more general sufficient certificate in the aid has the following explicit finite form. For an integer $M\geq1$, put \[ \Theta_{M,r}(\tau)=\sum_{n\equiv r\ (\mathrm{mod}\ 2M)}q^{n^2/(4M)}. \] After division by $\eta$, the residue vector has finite transformation matrices \[ \mathsf S_{r,s}=(2M)^{-1/2}e^{-2\pi i rs/(2M)},\qquad \mathsf T_{r,s}=1_{r=s}e^{2\pi i(r^2/(4M)-1/24)}. \] If a finite coefficient space of linear combinations is preserved under the induced row actions of $\mathsf S,\mathsf T$, and a recurrence-certified sum-side finitization converges compactly to those eta-denominated combinations, then it yields the corresponding vector-valued modular transformation laws. Relations such as $\Theta_{M,r}=\Theta_{M,-r}$ must be respected, either by symmetric coefficient vectors or by taking the appropriate quotient. This is a sufficient certificate for sums admitting the required finite theta realization, not a universal algorithm deciding modularity of arbitrary q-hypergeometric series.

Short English statement

The source asks how to see modularity directly from q-hypergeometric sums. The supplied route passes through terminating Gaussian-polynomial identities and compact convergence to unary theta differences, then uses Poisson summation. For the Rogers--Ramanujan pair it gives explicit transformation matrices; finite theta-space closure is a sufficient certificate, not a general modularity decision procedure.

Sources

• Local registry: permanent UnsolvedMath ID $20000729$, exact public code catalogue entry; aim-analytic-number-theory-notes.json, notes[92].
• AIM workshop Mock modular forms in combinatorics and arithmetic geometry, March 8--12, 2010, Problem 1.2: https://aimath.org/WWN/mockmodular/mockmodular.pdf. This URL is supplied provenance; no PDF access is claimed.
• Original motivation: the Rogers--Ramanujan identities for $G,H$, without full bibliographic references.
• Supplied research summary, "A finite-theta certificate for sum-side modularity," attempt 1, cites Bressoud's terminating Gaussian-polynomial identities, Poisson summation, and finite Weil-matrix closure without full theorem citations. Original remarks and literature fields are empty.