Evaluation of ψn at ±1 via alternating harmonic numbers and Bell polynomials

Prove that for every integer n ≥ 1, the evaluation of ψn at ±1 satisfies ψn(±1, …, ±1) = −Bn(∓H1, ∓2! H2, …, ∓n! Hn), where Bn denotes the complete Bell polynomial and Hk are alternating harmonic numbers, and that the generating function satisfies ∑_{n ≥ 1} ψn(±1, …, ±1) x^n/n! = 1 − (x+1)^{∓1/(1−x)}, with corresponding sign choices.

Background

In Section 4, the paper studies sums of coefficients and special evaluations of ψn, observing sequences that appear to relate to known OEIS entries. This motivates a conjectural closed form for ψn evaluated at ±1 in terms of complete Bell polynomials evaluated at sequences involving alternating harmonic numbers.

The conjecture also proposes an explicit exponential generating function, offering a compact analytic description of these special values of ψn.

References

Supported by further computations, we arrive at the following conjecture. For $n \geq 1$, we have $\psi_n(\pm 1, \ldots, \pm 1) = - B_n(\mp H_1, \mp 2! \, H_2, \ldots, \mp n! \, H_n)$, and the generating function is given by $\sum_{n \geq 1} \psi_n(\pm 1, \ldots, \pm 1) \frac{xn}{n!} = 1 - (x+1){\mp 1/(1-x)}$, choosing the corresponding signs, respectively.

— Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients (2509.05235 - Kellner, 5 Sep 2025) in Section 4 (end), Conjecture