Extension to odd arities and half-integer differential orders

Extend the alternating-composition construction and the definition of \(\consta(N/2)\) to every natural arity \(N\ge2\), including odd \(N\) and the resulting half-integer differential orders, potentially using vector fields on the circle rather than the real line.

Background

The construction in the paper is restricted to even arities N=2pN=2p, because it alternates $2p$ differential operators of integer order pp. The authors contrast this restriction with the fact that Wronskian brackets satisfy the relevant strongly homotopy Lie identities for arbitrary arities.

The proposed extension would require making sense of the construction for odd NN, corresponding formally to half-integer differential orders. The paper mentions the circle S1\mathbb S^1 as a possible domain for such a generalization.

References

Can the construction of alternating compositions in Eq.~eq:opdef and the count of Wronskians in Eq.~eq:main be extended to make \consta(N/2) well defined for all natural numbers N\geqslant 2, including odd integers that yield the differential orders p\in\tfrac{1}{2}Z\,?

eq:opdef:

Ap[w1,…,wN](f)(x):=∑σ∈SN(−1)σ wσ(1)∂xp∘wσ(2)∂xp∘⋯∘wσ(N)∂xp (f(x));\mathcal{A}_p[w_1,\ldots,w_N](f)(x) \mathrel{{:}{=}} \sum_{\sigma \in S_N} (-1)^\sigma\, w_{\sigma(1)}\partial_x^p \circ w_{\sigma(2)}\partial_x^p \circ \cdots \circ w_{\sigma(N)}\partial_x^p\,(f(x));%,

eq:main:

$\mathcal{A}_p[w_1,\ldots,w_N](f)(x) = \consta(p) \cdot \Wronsk \bigl(w_1, \ldots, w_N\bigr)(x) \cdot \partial_x^p(f(x)), $

— New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations  (2610.08636 - Shah et al., 6 Oct 2026) in Section 6, “Open problems and perspectives,” Open problem labeled OpnAnyN