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Coefficient-duality and Möbius-covariance problem for formal power series

Let F(z)∈C[[z]]F(z)\in\mathbb{C}[[z]] and let α∈C\alpha\in\mathbb{C}. The following conditions are equivalent:\n\n1. FF is Möbius-covariant of weight α\alpha, namely\n

F(−z1−z)=(1−z)−αF(z).F\left(-\frac{z}{1-z}\right)=(1-z)^{-\alpha}F(z).

\n2. For every nonnegative integer rr and every h(z)∈C[[z]]h(z)\in\mathbb{C}[[z]], FF satisfies the Möbius coefficient duality of weight α\alpha:\n

[zr] h(z)F(z)=[zr] h(z1+z)(1+z)r−1−αF(−z).[z^r]\,h(z)F(z)=[z^r]\,h\left(\frac{z}{1+z}\right)(1+z)^{r-1-\alpha}F(-z).

\n3. The coefficient identity in condition 2 holds for every nonnegative integer rr when h=1h=1.\n4. There exists a unique formal power series ΦF,α(t)∈C[[t]]\Phi_{F,\alpha}(t)\in\mathbb{C}[[t]] such that\n

F(z)=(1−z)α/2ΦF,α((z2−z)2).F(z)=(1-z)^{\alpha/2}\Phi_{F,\alpha}\left(\left(\frac{z}{2-z}\right)^2\right).
Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Parity normal form

    F(z)=(1−z)α/2ΦF,α ⁣((z2−z)2)F(z)=(1-z)^{\alpha/2}\Phi_{F,\alpha}\!\left(\left(\frac{z}{2-z}\right)^2\right) for some ΦF,α∈C[[u]]\Phi_{F,\alpha}\in\mathbb{C}[[u]].

    source: Möbius Covariance and Coefficient Duality: From Bernoulli Series to Enumerative Applications

References

Primary source

arXiv

Additional references

enumerative-combinatoricsformal-power-series
Posted

Progress summary

Refreshed
Claimed progress

A new paper gives a common structural explanation for several coefficient identities and derives further formulas, but it does not settle one single named conjecture.

The problem concerns when coefficient duality for formal power series is equivalent to covariance under a fractional-linear change of variable. The latest work develops this equivalence and applications rather than proving or disproving one separately named conjecture.

August 2026 structural framework

Max A. Alekseyev’s paper proves that, for F(z)∈C[[z]]F(z)\in\mathbb{C}[[z]], coefficient duality is equivalent to

F ⁣(−z1−z)=(1−z)−αF(z),F\!\left(-\frac{z}{1-z}\right)=(1-z)^{-\alpha}F(z),

and to the parity normal form F(z)=(1−z)α/2ΦF,α ⁣((z2−z)2)F(z)=(1-z)^{\alpha/2}\Phi_{F,\alpha}\!\left(\left(\frac{z}{2-z}\right)^2\right). It derives weighted and convolution variants, a Ramanujan-type summation theorem, and recurrence families for colored matchings and generalized central trinomial coefficients. No refutation, competing proof, or independent verification was found.

Current status (as of August 2026): The structural duality and its stated applications are available in a new preprint, but no complete resolution of a single named conjecture is recorded.

Sources

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