Q-linear independence of the finite sums D_{r,A}(j)

Determine whether, for a fixed integer r ≥ 1, the elements D_{r,A}(j) ∈ A defined by D_{r,A}(n) = (∑_{k=0}^{p-1} k^n/(k!)^r mod p)_p are linearly independent over Q for j = 0, 1, ..., r − 1.

Background

The authors introduce a Bessel-type generalization D_{r,A}(n) of the finite Dobiński sums and prove a structural decomposition analogous to the classical case. In the complex-analytic setting they establish linear independence of D_r(0), ..., D_r(r−1) over algebraic numbers, but the corresponding statement in A is unresolved.

Establishing Q-linear independence of D_{r,A}(j) would mirror the independence results proved for their classical counterparts and clarify the algebraic structure generated by these finite analogues within A.

References

However, it remains unknown whether $e_A$ is irrational in the ring $A$, or whether $D_{r,A}(j)$ for $0 \le j \le r-1$ are $Q$-linearly independent.

— On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials  (2604.01578 - Matsusaka et al., 2 Apr 2026) in Section 1 (Introduction)