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 $e$, 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 $γ_\mathcal{A}^\mathrm{W}$.

• The paper introduces finite A-analogues of Dobiński’s formula and the constant e, deriving novel recurrence relations using Bell numbers and combinatorial corrections.
• It generalizes classical constants to finite settings via refined algebraic methods and proves linear independence results using Siegel–Shidlovskii techniques.
• A new family of Gregory polynomials is constructed to develop finite analogues of Euler’s constant, explicitly connecting them with logarithmic identities and combinatorial structures.

Finite Analogues of Dobiński's Formula and Euler's Constant via Gregory Polynomials

Introduction and Background

The study situates itself in the emerging arithmetic of the ring $A$, a quotient of products of $\mathbb{Z}/p\mathbb{Z}$ indexed by all primes modulo direct sums, which has become the setting for various "finite analogues" of classical constants and functions. Following the foundational work on finite multiple zeta values by Kaneko and Zagier, this paper systematically develops finite $A$-analogues for concepts intricately tied to combinatorics and analysis: Dobiński's formula (and the constant $e$) and Euler's constant $\gamma$, using the combinatorics of Bell numbers and Gregory polynomials.

Finite Dobiński-Type Formulas and Bessel Generalizations

The classical Dobiński formula expresses the sum $D(n) = \sum_{k=0}^\infty k^n / k!$ as $b(n)e$, where $b(n)$ are the Bell numbers. The authors introduce the finite $A$-analogue: $$D_A(n) = \left( \sum_{k=0}^{p-1} \frac{k^n}{k!} \bmod p \right)_p \in A$$ and define $\mathbb{Z}/p\mathbb{Z}$0 as the $\mathbb{Z}/p\mathbb{Z}$1-analogue of $\mathbb{Z}/p\mathbb{Z}$2.

Main structural result: For all $\mathbb{Z}/p\mathbb{Z}$3,

$\mathbb{Z}/p\mathbb{Z}$4

where $\mathbb{Z}/p\mathbb{Z}$5, $\mathbb{Z}/p\mathbb{Z}$6, and $\mathbb{Z}/p\mathbb{Z}$7, the same recurrence as for Bell numbers but with distinct initial values.

The paper further extends these constructions to generalizations $\mathbb{Z}/p\mathbb{Z}$8 for $\mathbb{Z}/p\mathbb{Z}$9 (recovering Bessel-type functions for $A$0), and introduces their $A$1-analogues $A$2. A key theorem establishes that for each $A$3 and all $A$4,

$A$5

with $A$6 defined via a universal recurrence and explicit initial conditions, and $A$7 as a finite-combinatorial correction.

A technically significant contribution is the proof—using refined Siegel–Shidlovskii methods—that $A$8 are linearly independent over $A$9 for any $e$0, strengthening the transcendence-theoretic perspective on these finite sums.

Finite Analogues of Euler's Constant Using Gregory Polynomials

Building on classical formulas for $e$1 involving Gregory coefficients $e$2, the paper introduces a new family of Gregory polynomials $e$3 via generating functions: $e$4 and studies their arithmetic and analytic properties in depth, including integral representations and asymptotics.

Core innovation: Finite $e$5-analogues of Euler's constant, parameterized by $e$6 and $e$7, are defined: $e$8

$e$9

where $\gamma$0, harmonizing the analogy with classical formulas due to Mascheroni and Kluyver.

A pivotal result is the explicit comparison among the various finite analogues: $\gamma$1 where $\gamma$2 is the Wilson-type $\gamma$3-analogue of Euler's constant, showing that all these candidates differ only by explicit $\gamma$4-linear combinations of logarithmic $\gamma$5-special values and constants.

Additional theorems rigorously connect the finite analogues through binomial, Stirling, and recursive combinatorial identities, establishing that the variations in such constants (including $\gamma$6-analogues parametrized by $\gamma$7 and $\gamma$8) are explicitly and universally expressible in terms of each other.

Theoretical and Arithmetic Implications

The results emphasize the deep parallelism between finite and classical cases: the structure constants in the finite ring $\gamma$9 reflect those of the classical analytical constants, but with subtle correction sequences (such as $D(n) = \sum_{k=0}^\infty k^n / k!$0) reflecting the truncation to $D(n) = \sum_{k=0}^\infty k^n / k!$1 and the arithmetic mod $D(n) = \sum_{k=0}^\infty k^n / k!$2.

A significant theoretical implication is the elucidation of the limited vanishing of $D(n) = \sum_{k=0}^\infty k^n / k!$3's $D(n) = \sum_{k=0}^\infty k^n / k!$4-components, paralleling rare phenomena in the sequences of Wolstenholme and Wilson primes, underscoring challenges in the arithmetic of $D(n) = \sum_{k=0}^\infty k^n / k!$5 such as non-vanishing and independence problems.

The explicit generalization using the $D(n) = \sum_{k=0}^\infty k^n / k!$6 parameter in Gregory polynomials allows streamlined proofs and broader structural statements, potentially facilitating new avenues for constructing and understanding finite analogues for a vast array of analytical constants.

Speculations and Future Directions

The paper leaves open the important question of the $D(n) = \sum_{k=0}^\infty k^n / k!$7-irrationality of $D(n) = \sum_{k=0}^\infty k^n / k!$8 and the $D(n) = \sum_{k=0}^\infty k^n / k!$9-independence of $b(n)e$0 for general $b(n)e$1. These problems are deeply connected to major conjectures such as ABC, as well as to computational and combinatorial phenomena concerning primes.

The explicit representation of all finite Euler constants in terms of logarithmic analogues hints at a possible general classification of all "finite" special values in $b(n)e$2 via similarly uniform transformations, with potential bearing on the broader theory of transcendence and algebraic independence in arithmetic contexts.

The combinatorial and algebraic techniques introduced (especially the recurrence and generating function approaches) set a methodological standard for future attempts to construct and interpret further $b(n)e$3-analogues of analytic entities, not only constants but potentially functional identities within special function theory and beyond.

Conclusion

This study presents a comprehensive construction of finite $b(n)e$4-analogues of Dobiński's formula, the constant $b(n)e$5, and Euler's constant $b(n)e$6, anchored in the interplay of combinatorics (Bell numbers, Gregory polynomials) and arithmetic algebra (the ring $b(n)e$7). The paper establishes strong universal identities among the various analogues, leverages fine transcendence-theoretic tools to prove algebraic independence, and systematically connects finite and classical special values. The structural clarity and explicitness of the obtained formulas provide foundational tools and conceptual advances for further exploration in the arithmetic of finite analogues.

Reference:

"On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials" (2604.01578)