MathDB

Hirose's duality conjecture

For every word ww in the class defining Hirose's iterated qq-integrals on the four-punctured projective line, with the prescribed position-dependent qq-shifts of the parameters, one has the duality identity Lq(w)=Lq(τ(w))L_q(w)=L_q(\tau(w)), where τ\tau is Hirose's word-duality operation.

References

Primary source

arXiv

Additional references

iterated-integralsq-seriesspecial-functions
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims a complete proof of Hirose’s duality conjecture, but the claim has not received independent mathematical assessment.

Hirose’s conjecture asserts the duality identity Lq(w)=Lq(τ(w))L_q(w)=L_q(\tau(w)) for iterated qq-integrals with position-dependent shifts, connecting these integrals with basic hypergeometric identities. Hirose introduced it in a preprint published on May 1, 2026, proving only special cases.

Known results

  • The identity holds in the special case AD=BCqm(w)AD=BCq^{m(w)} (Hirose, 2026).
  • A specialization recovers Yamamoto’s already-proved duality for one-variable multiple qq-polylogarithms (Hirose, 2026).

September 2026 claimed proof

A preprint by Shin-ichiro Seki claims the general identity using a symmetric terminating 4ϕ3{}_4\phi_3 connector. No retrieved source independently checks the argument, and the original paper does not contain this proof.

Current status (as of October 2026): The conjecture has established special cases, while the general statement is only claimed proved in an unrefereed preprint and remains unverified.

Sources

Solutions 0

No solutions have been posted yet.