Torsion continuous q-Hermite expansions as Siegel modular units

Definitions and mathematical statement

The source asks for relations between modularity of a series and its expansion in $q$-orthogonal polynomials. It specifies neither a polynomial family, expansion coefficients, modular group nor weight, so it does not assert that every such expansion is modular.

The supplied special case uses $q=e^{2\pi i\tau}$, $\operatorname{Im}\tau>0$, with $q^{1/2}=e^{\pi i\tau}$, and continuous $q$-Hermite polynomials normalized by $H_0=1$, $H_1=2x$, and \[ H_{n+1}(x\mid q)=2xH_n(x\mid q)-(1-q^n)H_{n-1}(x\mid q). \] Write $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$ and use the corresponding infinite product. For $\beta\in{\mathbb{Q}}$, $\zeta=e^{2\pi i\beta}$, define \[ F_\beta(\tau)=q^{1/24}\sum_{n\ge0} \frac{H_n(\cos(2\pi\beta)\mid q)q^{n/2}}{(q;q)_n}. \] The report asserts \[ F_\beta(\tau)=\frac{q^{1/24}} {(\zeta q^{1/2};q)_\infty(\zeta^{-1}q^{1/2};q)_\infty} =-e^{-\pi i\beta/2}\,g_{(1/2,\beta)}(\tau)^{-1}. \] Its Siegel-function normalization is fixed by \[ g_{(1/2,\beta)}=-e^{-\pi i\beta/2}q^{-1/24} (\zeta q^{1/2};q)_\infty(\zeta^{-1}q^{1/2};q)_\infty. \] If $M$ is the least positive integer with $M/2,M\beta\in{\mathbb{Z}}$, the report gives $F_\beta^{12M}$ as a modular unit on $X(M)$. A modular unit is a meromorphic modular function with zeros and poles confined to cusps; this power statement is distinct from claiming $F_\beta$ has trivial multiplier itself.

The reported logarithmic derivative is a weight-two modular form for the corresponding level convention and has expansion \[ \frac1{2\pi i}\frac{d}{d\tau}\log F_\beta =\frac1{24}+\sum_{m\ge1}\left[\frac12 \sum_{\substack{d\mid m\\d\text{ odd}}} d\left(\zeta^{m/d}+\zeta^{-m/d}\right)\right]q^{m/2}. \] The scalar multiplier and logarithm branch do not affect the derivative. The report emphasizes that mere existence of an orthogonal-polynomial expansion cannot certify modularity without extra arithmetic structure; here rational torsion specialization and the explicit Siegel-product identity provide that structure. It supplies no universal criterion for arbitrary $q$-orthogonal families or coefficients.

Short English statement

Seek connections between $q$-orthogonal expansions and modularity. The supplied report gives a rationally specialized continuous $q$-Hermite series equal to an inverse Siegel function, with an explicit modular-unit power and derivative coefficients. It does not turn arbitrary orthogonal expansions into modularity criteria.

Sources

• Local UnsolvedMath registry, permanent ID $20000730$, source code catalogue entry; AIM analytic number theory corpus, aim-analytic-number-theory-notes.json, notes[93]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Mock modular forms in combinatorics and arithmetic geometry, March 8--12, 2010, Problem 1.3: https://aimath.org/WWN/mockmodular/mockmodular.pdf. Original remarks and literature fields are empty.
• Registry dated August 14, 2026: statement exact, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. The torsion $q$-Hermite/Siegel identity and derivative certificate are machine-generated research aids, not peer reviewed, requiring independent verification before citation. No full primary references or proofs are supplied.