On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials

We study a finite analogue of Dobiński’s formula, which is related to the Napier constant $e$, and its Bessel-type generalizations. Furthermore, using Gregory polynomials, we extend the results of Kaneko–Matsusaka–Seki on finite analogues of Euler’s constant, and compare them with the Wilson-type analogue ${\gamma }_{\mathcal{A}}^{\mathrm{W}}$.