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

Toshiki Matsusaka, Taichi Miyazaki, Shunta Yara

Abstract

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 $γ_\mathcal{A}^\mathrm{W}$.

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

Abstract

We study a finite analogue of Dobiński's formula, which is related to the Napier constant , 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 .

Paper Structure

This paper contains 10 sections, 18 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Generalized Dobiński's formula and its $\mathcal{A}$-analogue
  3. $\overline{\mathbb{Q}}$-linear independence
  4. Generalized Dobiński's formula
  5. Finite analogues of Euler's constant
  6. Gregory polynomials
  7. Generalizations of Kaneko--Matsusaka--Seki's results
  8. An analogue of $\gamma$ (Mascheroni), revisited
  9. An interlude (Gregory), revisited
  10. Yet more variations (Kluyver), revisited

Key Result

Theorem 1.1

For any integer $n\ge 0$, we have where the sequence $(g(n))_{n\ge 0} = (0,1,1,3,9,31,121,523,\ldots)$ is given by $g(0) = 0, g(1) = 1$, and by the same recurrence relation as the Bell numbers, that is, $($see also OEIS OEIS$)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4: Refined Siegel--Shidlovskii theorem Beukers2006
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 27 more