Shi–Wang's triple Nahm sum identity

Less than 1 year old · traced to

Let tt be a formal variable. For n≥0n\geq 0, write (t4;t4)n=∏r=1n(1−t4r)(t^4;t^4)_n=\prod_{r=1}^{n}(1-t^{4r}), and use (x,y;Q)∞=(x;Q)∞(y;Q)∞(x,y;Q)_\infty=(x;Q)_\infty(y;Q)_\infty for products of qq-Pochhammer symbols. Then

Shi–Wang's conjecture.

∑a,b,d≥0t4a2+4ab+3b2−2bd+d2+4a+2b(t4;t4)a(t4;t4)b(t4;t4)d=(t6;t6)∞(t8;t8)∞(t2,t10;t12)∞(t4;t4)∞2(t,t11;t12)∞(t5,t7;t12)∞.\sum_{a,b,d\geq 0}\frac{t^{4a^2+4ab+3b^2-2bd+d^2+4a+2b}}{(t^4;t^4)_a(t^4;t^4)_b(t^4;t^4)_d}=\frac{(t^6;t^6)_\infty(t^8;t^8)_\infty(t^2,t^{10};t^{12})_\infty}{(t^4;t^4)_\infty^2(t,t^{11};t^{12})_\infty(t^5,t^7;t^{12})_\infty}.

The paper introduces this identity as the conjecture stated in Shi and Wang and discusses its connection with one of the paper's proved identities. The supplied text does not state that the identity itself has been resolved.

References

Primary source

Haijun Li, “Proofs of five conjectural identities on modular rank four Nahm sums”, arXiv:2606.25866 (2026).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.