Open problems on the Rogers–Ramanujan dissection of a theta function: multi-factor dissections, an $s$-parameter companion family, physical interpretation, and partition-theoretic content

Dixit and Kumar (Math. Ann. 393 (2025), 667–696; arXiv:2411.06412) proved that for every positive integer $s$ the Jacobi theta function $\sum_{n\in\mathbb{Z}}a^{n}q^{n^{2}/(2s)}$ admits a Rogers–Ramanujan dissection: it equals a finite sum of $s$ products of pairs of generalized Rogers–Ramanujan functions, recovering Ramanujan's generalized modular relation from the Lost Notebook at $s=2$. Their concluding section poses four open problems, which share a common objective — to extend and interpret this dissection theorem: (1) whether the identities, including the vanishing family (1.17), reflect an underlying physical phenomenon, inversion-type formula, or homogeneous system; (2) whether Ramanujan's companion identity (6.1) admits an $s$-parameter extension analogous to (1.10); (3) whether dissections exist whose left-hand sides are finite sums of products of three or more generalized Rogers–Ramanujan functions; and (4) whether the series $\sum_{m\ge0}q^{sm(m-1)/2+m(k+s/2)}/(q)_m$ for parameter cases not covered by Bressoud–Santos–Mondek admit partition-theoretic interpretations. Adjacent work applies the $s=2$ identity as a tool and extends (6.1) bilaterally in a different direction (references 2 and 3), but none of the four problems has been resolved.

Background

Fix a complex number $q$ with $|q|<1$ and use the standard $q$-series notation $(A)_0=1$, $(A)_n=(A;q)_n=(1-A)(1-Aq)\cdots(1-Aq^{n-1})$ for $n\ge1$, and $(A)_\infty=(A;q)_\infty=\lim_{n\to\infty}(A;q)_n$. The Rogers–Ramanujan functions are the series $G(q)=\sum_{n\ge0}q^{n^{2}}/(q)_n$ and $H(q)=\sum_{n\ge0}q^{n^{2}+n}/(q)_n$; the celebrated Rogers–Ramanujan identities rewrite them as infinite products, and they permeate number theory, combinatorics, and physics — for instance in Baxter's exactly solved lattice models such as the hard-hexagon model (a two-dimensional lattice gas with a nearest-neighbor exclusion rule, solved exactly by Baxter using the Rogers–Ramanujan identities), and in the framework of Rogers–Ramanujan-type identities and their arithmetic properties developed by Griffin, Ono and Warnaar. Mock theta functions are the $q$-series of Ramanujan's last letter to Hardy, which behave like theta functions asymptotically at roots of unity but lack their modular transformation laws; they come in orders (third, fifth, seventh, ...). By a generalized Rogers–Ramanujan function one means any series of the form $\sum_{n\ge0}a^{n}q^{(cn^{2}+dn)/s}/(bq)_n$ with $a,b\in\mathbb{C}$, $a\ne0$, $c,d\in\mathbb{R}$, $s\in\mathbb{N}$.

On page 27 of Ramanujan's Lost Notebook (the collection of his unpublished results rediscovered by George E. Andrews in 1976), Ramanujan claimed a generalized modular relation: for $a\in\mathbb{C}\setminus\{0\}$ and $b\in\mathbb{C}$,

$$\sum_{m=0}^{\infty}\frac{a^{-2m}q^{m^{2}}}{(bq)_{m}}\sum_{n=0}^{\infty}\frac{a^{n}b^{n}q^{n^{2}/4}}{(q)_{n}}+\sum_{m=0}^{\infty}\frac{a^{-2m-1}q^{m^{2}+m}}{(bq)_{m}}\sum_{n=0}^{\infty}\frac{a^{n}b^{n}q^{(n+1)^{2}/4}}{(q)_{n}}=\frac{1}{(bq)_{\infty}}\sum_{n=-\infty}^{\infty}a^{n}q^{n^{2}/4}-(1-b)\sum_{n=1}^{\infty}a^{n}q^{n^{2}/4}\sum_{\ell=0}^{n-1}\frac{b^{\ell}}{(q)_{\ell}},$$

proved by Andrews. Setting $a=b=1$ and rescaling $q$ yields the modular relation $G(q)G(q^{4})+qH(q)H(q^{4})=(-q;q^{2})_{\infty}^{2}$, and $a=b=-1$ yields a relation between two fifth-order mock theta functions and $G(q)$, $H(q)$. On page 26 of the Lost Notebook (Entry 7.2.4 in Andrews–Berndt's edition, Part II) Ramanujan claimed a companion identity (proved by Andrews), a difference of two products of theta functions and generalized Rogers–Ramanujan series:

$$\sum_{n=-\infty}^{\infty}a^{n}q^{n^{2}/4}\sum_{n=0}^{\infty}\frac{(-a)^{n}q^{n^{2}/4}}{(q)_{n}}-\sum_{n=-\infty}^{\infty}(-a)^{n}q^{n^{2}/4}\sum_{n=0}^{\infty}\frac{a^{n}q^{n^{2}/4}}{(q)_{n}}=2q^{1/4}(q)_{\infty}\sum_{m=0}^{\infty}\frac{a^{-2m-1}q^{m^{2}+m}}{(q)_{m}},$$

whose special cases include two of Watson's identities connecting $G(q)$, $H(q)$ with third-order mock theta functions.

Dixit and Kumar generalized the page-27 identity to a parameter $s\in\mathbb{N}$ (their Theorem 1.1): for $a\in\mathbb{C}\setminus\{0\}$, $b\in\mathbb{C}$,

$$\sum_{k=0}^{s-1}\left\{\sum_{m=0}^{\infty}\frac{a^{-sm-k}q^{(sm+k)^{2}/(2s)}}{(bq)_{m}}\sum_{n=0}^{\infty}\frac{a^{n}b^{n}q^{n(n+2js-2k)/(2s)}}{(q)_{n}}\right\}=\frac{1}{(bq)_{\infty}}\sum_{n=-\infty}^{\infty}a^{n}q^{n^{2}/(2s)}-(1-b)\sum_{n=1}^{\infty}a^{n}q^{n^{2}/(2s)}\sum_{\ell=0}^{n-1}\frac{b^{\ell}}{(q)_{\ell}},$$

where $j=0$ if $k=0$ and $j=1$ otherwise. Since for $b=1$ the right-hand side is (up to $1/(q)_\infty$) the Jacobi theta function $\sum_{n\in\mathbb{Z}}a^{n}q^{n^{2}/(2s)}$, this represents a theta function as a finite sum of $s$ products of pairs of generalized Rogers–Ramanujan functions — a Rogers–Ramanujan dissection. The case $s=2$ recovers Ramanujan's identity; $s=1$ gives a generalization of the Jacobi triple product identity (the classical formula expanding a theta function as an infinite product); and for $s>2$ the authors give asymptotic evidence that the identities transcend the modular world, so they cannot be expected to follow from modular-forms machinery. Letting $a=-q^{-1/(2s)}$ annihilates the theta term and yields the vanishing family (1.17)

$$\sum_{k=0}^{s-1}\left\{\sum_{m=0}^{\infty}\frac{(-1)^{sm-k}q^{(sm+k)(sm+k+1)/(2s)}}{(q)_{m}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n+2js-2k-1)/(2s)}}{(q)_{n}}\right\}=0.$$

Two further threads matter for the open problems. First, sums and differences of products of quadruples of Rogers–Ramanujan functions $G(q)$, $H(q)$ are known — in Robins' thesis, in recent work of Channabasavayya, Keerthana and Ranganatha, and of Gugg — and these sit inside Ramanujan's forty identities for $G(q)$ and $H(q)$ classified by Berndt et al.; since the pair-product relation $G(q)G(q^{4})+qH(q)H(q^{4})=(-q;q^{2})_{\infty}^{2}$ was a special case of Ramanujan's generalized relation, it is natural to ask whether the triple- and quadruple-product relations admit analogous generalizations. Second, Bressoud, Santos and Mondek proved that, for their parameter ranges, series of the shape $\sum_{m\ge0}q^{\ell m(m-1)/2+\text{(linear in }m\text{)}}/(q)_m$ are generating functions of three infinite families of equinumerous restricted partitions; within the $a=b=1$ dissection series

$$\sum_{m=0}^{\infty}\frac{q^{sm(m-1)/2+m(k+s/2)}}{(q)_{m}},\qquad s\in\mathbb{N},\ 0\le k\le s-1,$$

Dixit and Kumar note that the cases with $s$ even and $k\le s/2$ fall under the Bressoud–Santos–Mondek theorem, while the remaining cases have no known partition-theoretic interpretation.

Problem Statement

Solve the four interlinked open problems posed at the end of Section 7 of Dixit and Kumar's paper on the Rogers–Ramanujan dissection of a theta function, which together ask for the completion of the theory of that dissection — its structural meaning, its extension to the companion identity and to multi-factor products, and its partition-theoretic content. The four parts are independent questions of the same program: a complete solution addresses all four, while an answer to any single part is a self-contained contribution to it. For (3), a proof in the triple case already answers the existence question; for (4), an interpretation of a single previously uncovered family of $(s,k)$ suffices.

(1) Structural interpretation. Does the $s$-parameter family (1.10) — in particular its $s=2$ case (1.1) together with the companion identity (6.1), which vaguely resemble a pair of inversion formulas — represent an underlying physical phenomenon or an inversion-type formula? Does the vanishing family (1.17) hint at some homogeneous system? That is, exhibit a well-defined physical or combinatorial model, or an inversion/homogeneous-system framework, whose exact analysis realizes these identities (in the way, for example, that the Rogers–Ramanujan identities are realized in Baxter's exactly solved lattice models).

(2) Companion-identity extension. Ramanujan's identity (1.1) was generalized to the $s$-parameter family (1.10). Extend the companion identity (6.1) in the same manner: find, and prove, an $s$-parameter family of identities (a finite sum of $s$ terms in the spirit of (1.10)) whose $s=2$ case is (6.1), or show that such a family cannot exist.

(3) Multi-factor dissections. Do there exist analogues of (1.1), more generally of (1.10), in which the left-hand side is a finite sum of products of three or more generalized Rogers–Ramanujan functions — i.e., multi-factor (triple, quadruple, ...) Rogers–Ramanujan dissections of theta functions — and if so, what are their right-hand sides? (This is motivated by the known identities for sums and differences of products of quadruples of the Rogers–Ramanujan functions $G(q)$, $H(q)$ in Robins' thesis, Channabasavayya–Keerthana–Ranganatha's Theorem 3.1, and Gugg's Theorem 4.1, which generalize Ramanujan's forty identities; the authors suggest asymptotic analysis as the most promising search method.)

(4) Partition-theoretic content of the remaining cases. For the series $\sum_{m=0}^{\infty}q^{sm(m-1)/2+m(k+s/2)}/(q)_{m}$ with $s\in\mathbb{N}$ and $0\le k\le s-1$ occurring in Theorem 1.1 with $a=b=1$: the cases with $s$ even and $k\le s/2$ are covered by the Bressoud–Santos–Mondek theorem, which exhibits three restricted-partition families with these series as generating functions. Extract partition-theoretic interpretations and combinatorial consequences for the remaining cases ($s$ odd, or $s$ even with $k>s/2$) — e.g., restricted-partition classes, bijections, or arithmetic of coefficients.

References

1. A. Dixit and G. Kumar, The Rogers–Ramanujan dissection of a theta function, Mathematische Annalen 393 (2025), no. 1, 667–696. DOI: 10.1007/s00208-025-03265-9 (preprint arXiv:2411.06412). https://doi.org/10.1007/s00208-025-03265-9
2. N. H. Zhou, Theta functions and transformations of bilateral basic hypergeometric series, preprint (2026), arXiv:2605.05094 (v2). https://arxiv.org/abs/2605.05094
3. A. Dixit, G. Kumar and A. Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers–Ramanujan partitions, preprint (2025), arXiv:2508.04359. https://arxiv.org/abs/2508.04359
4. D. M. Bressoud, J. P. O. Santos and P. Mondek, A family of partition identities proved combinatorially, The Ramanujan Journal 4 (2000), 311–315. DOI: 10.1023/A:1009805221257. https://doi.org/10.1023/A:1009805221257
5. Channabasavayya, G. K. Keerthana and Ranganatha Dasappa, On further modular relations for the Rogers–Ramanujan functions, Indian Journal of Pure and Applied Mathematics 56 (2025), no. 1, 113–124. DOI: 10.1007/s13226-023-00458-3. https://doi.org/10.1007/s13226-023-00458-3
6. C. Gugg, On conjectures of Koike and Somos for modular identities for the Rogers–Ramanujan functions, International Journal of Number Theory 17 (2021), no. 2, 435–472. DOI: 10.1142/S1793042120400138. https://doi.org/10.1142/S1793042120400138
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