Linear Independence Across Tschakaloff Parameters

Definitions and mathematical statement

For a complex parameter $q$ with $|q|>1$, define the Tschakaloff function \[ T_q(z)=\sum_{\nu=0}^{\infty}q^{-\nu(\nu+1)/2}z^\nu. \] The series converges absolutely for every $z\in{\mathbb{C}}$. Fix the rational argument $z$ intended by the source, and let $\mathcal A_z\subseteq\{q\in{\mathbb{C}}:|q|>1\}$ denote its intended set of admissible parameters. The proposed linear independence over ${\mathbb{Q}}$ means \[ \begin{gathered} \text{for every finite list of distinct }q_1,\ldots,q_s\in\mathcal A_z \text{ and every }a_1,\ldots,a_s\in{\mathbb{Q}},\\ \sum_{i=1}^{s}a_iT_{q_i}(z)=0\quad\Longrightarrow\quad a_1=\cdots=a_s=0. \end{gathered} \] The argument $z$ is common to these values; the independence question varies $q$. The number $1$ is not adjoined to the list of values, since the source does not ask for that stronger independence statement.

The qualifier "admissible" is essential but undefined: the record does not say whether $q$ is integral, rational or algebraic, or state restrictions on parameter relations or on $z$. In particular, $T_q(0)=1$ for every $q$, so the unrestricted assertion for rational $z=0$ and two distinct admissible parameters fails. A nonzero-argument condition or another admissibility convention is required, but is not silently supplied. The source's canonical-version claim does not itself recover these missing conditions.

Short English statement

For a fixed rational argument, prove rational linear independence of Tschakaloff values at distinct admissible parameters. The admissible parameter set is undefined, and the unrestricted zero-argument case gives identical values, so the intended qualifications must be recovered.

Sources

• Supplied report: Diophantische Approximationen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/21, pp. 1115--1190; proposed year 2007. https://doi.org/10.4171/owr/2007/21.
• Supplied Vaananen--Zudilin references: Russian Mathematical Surveys, https://doi.org/10.1070/RM2007v062n01ABEH004386; and Journal of Number Theory, https://doi.org/10.1016/j.jnt.2008.03.003.
• Local UnsolvedMath registry, permanent ID 30000752, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note calls this the complete canonical version, including the series, of duplicate OWR-1537-001. That companion is permanent ID 30000749 and is labelled unrecoverable only as the less complete duplicate.
• The supplied 21 August 2026 triage labels the question open, reports restrictive parameter configurations in existing results, and refers to a 2025 Vaananen--Zudilin note still describing the cross-parameter problem as open. No separate bibliographic identification of that 2025 note is provided beyond the two references above; the dated claim is retained without independent verification.