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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.

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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