On finite analogues of Dobi\'{n}ski's formula and of Euler's constant via Gregory polynomials

TL;DR

This paper explores finite analogues of classical mathematical formulas related to e and Euler's constant, utilizing Gregory polynomials and comparing different approaches.

Contribution

It introduces finite analogues of Dobiński's formula and Euler's constant using Gregory polynomials, extending prior results and providing new comparisons.

Findings

Finite analogue of Dobiński's formula related to e

Extension of Euler's constant analogues via Gregory polynomials

Comparison with Wilson-type analogue of Euler's constant

Abstract

We study a finite analogue of Dobi\'{n}ski'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 $\gamma_\mathcal{A}^\mathrm{W}$.