Modularity of Higher-$t$ Generating-Function Coefficients

Definitions and mathematical statement

The supplied record refers to integrals $R_{k,\ell}(X)$, but does not supply their defining integrands, integration domains, normalization, admissible indices, or the domain of $X$. For the depth-one family, the displayed expansion records the indexing \[ \mathcal G(t,X)=\sum_{k=2}^{\infty}R_{k,1}(X)t^{k-2}. \] Here $\mathcal G$ denotes the generating expression mentioned in the source; this notation does not assert convergence. At the level of formal coefficient indexing, its coefficient of $t^m$ is $R_{m+2,1}(X)$ for integers $m\geq 0$. The supplied transcription of the hypergeometric identity labelled Theorem 2 is corrupted and does not determine its complete right-hand side reliably.

Research question, subject to unresolved source definitions. Determine whether the coefficients of higher powers of $t$ in $\mathcal G(t,X)$ admit interpretations as modular forms or higher-order modular antiderivatives. Identify and prove the corresponding structure for the integrals $R_{k,\ell}(X)$ at depths $\ell>1$. The original excerpt also raises the task of writing linear differential equations for the higher-power coefficients.

A precise modularity assertion requires the source definitions of the integrals, the admissible indices and $X$-domain, the relation between $X$ and a modular parameter, the relevant modular group and transformation data, and the intended meaning of a higher-order modular antiderivative. The cutoff meant by "higher powers" is unspecified. The differential-equation variable, permitted coefficients, and desired order are also unspecified. Consequently, the supplied evidence supports this qualified research question but does not determine a fully quantified conjecture.

The supplied assessment says that modular identifications are available only for low powers and that higher powers and the general higher-depth structure remain difficult and unproved. The original excerpt specifically reports partial results for $R_{k,2}(X)$. These are source-reported qualifications, not mathematical or literature conclusions independently verified here.

Short English statement

The question asks whether the higher coefficients of a generating function for $R_{k,1}(X)$ have modular interpretations, and what analogous structure exists for $R_{k,\ell}(X)$ with $\ell>1$. The integral definitions and modularity conventions are missing, and the displayed hypergeometric formula is corrupted, so a fully precise conjecture requires source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30006595$, source problem code catalogue entry, title "Modularity of Higher-$t$ Generating-Function Coefficients." The supplied record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is specified.
• Multiple Zeta Values and Modular Forms (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/7, pp. 433--486. Supplied source and literature-source link, labelled "Oberwolfach Report 7/2026": https://doi.org/10.4171/owr/2026/7. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and reports modular identifications only for low powers, with higher powers and the general higher-depth structure remaining difficult and unproved. This is supplied dated curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and states that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; the report has not been independently consulted for this draft.