A New Expression for the Bernoulli Numbers and its Applications

Technical Summary: A New Expression for the Bernoulli Numbers and Its Applications

Problem Statement
The paper addresses the problem of finding closed-form expressions for finite discrete convolutions involving Stirling numbers of both kinds and harmonic numbers. While explicit formulas for Bernoulli numbers ($B_n$) are well-established (e.g., Worpitzky's identity and its generalizations), the authors investigate the specific sum:
$$\sum_{k=j}^{n} (-1)^{k-j} \left\{ \begin{matrix} n \\ k \end{matrix} \right\} \left[ \begin{matrix} k \\ j \end{matrix} \right] H_k$$
where $\left\{ \begin{matrix} n \\ k \end{matrix} \right\}$ and $\left[ \begin{matrix} k \\ j \end{matrix} \right]$ denote Stirling numbers of the second and first kind, respectively, and $H_k$ is the $k$-th harmonic number. Previous works by Mez˝o and Merca had extended identities involving Stirling numbers to Bernoulli polynomials, and recent papers introduced specific cases involving harmonic numbers. This work seeks to generalize these findings into a unified formula.

Methodology
The authors employ a combination of generating functions, recurrence relations, and combinatorial identities.

1. Combinatorial Derivation: The proof of the main theorem relies on the standard recurrence relations for Stirling numbers of the first and second kinds. By defining the sum as $B_{n,j}$ and applying these recurrences, the authors derive a reduction formula that relates $B_{n+1,j}$ to $B_{n,j-1}$ and a term involving binomial coefficients and Bernoulli numbers.
2. Generating Functions: To establish applications, the authors utilize generating functions for Stirling numbers, harmonic numbers, and poly-Bernoulli polynomials. Specifically, they manipulate the generating function of the Stirling numbers of the second kind to evaluate sums involving $H_k$.
3. Differentiation and Integration: The derivation of the cumulative sum formula involves differentiating the main identity with respect to a variable $x$ and integrating specific forms to relate the sums to poly-Bernoulli polynomials and Cauchy numbers.
4. Modular Arithmetic: The congruence results are derived by substituting an odd prime $p$ into the derived formulas and applying known theorems regarding Stirling numbers modulo $p$ (e.g., $\left\{ \begin{matrix} p \\ k \end{matrix} \right\} \equiv 0 \pmod p$ for $1 < k < p$), Babbage's theorem, and the von Staudt-Clausen theorem.

Key Contributions and Results

• Main Theorem (Theorem 1): The paper establishes a new explicit formula for the aforementioned convolution:
  $$\sum_{k=j}^{n} (-1)^{k-j} \left\{ \begin{matrix} n \\ k \end{matrix} \right\} \left[ \begin{matrix} k \\ j \end{matrix} \right] H_k = \left( \binom{n}{j} - 1 \right) \frac{B_{n-j}}{n-j}$$
  for integers $0 \le j \le n$. The authors note that for $j=1$ and $j=2$, this recovers previously known identities involving $B_{n-1}$ and $B_{n-2}$.

• Recurrence Relations:

  • The authors reprove a linear recurrence relation for Bernoulli numbers previously given by Agoh.
  • A new recurrence relation is derived:
    $$\sum_{j=1}^{n} \left( \binom{n}{j} + 1 \right) \frac{B_j}{j} (1 - 2^j) = 1$$

• Cumulative Sum Formula: A closed formula for the cumulative sum of Bernoulli numbers is presented, linking them to Bernoulli numbers, di-Bernoulli numbers ($B_n^{(2)}$), and di-Bernoulli polynomials evaluated at $x=1$:
  $$\sum_{j=0}^{n} B_j = B_n^{(2)}(1) + B_n - B_n^{(2)} - 1$$

• Congruences: The paper establishes several congruences for sums of Bernoulli and Euler numbers modulo an odd prime $p$ and its square:

  • $\sum_{j=0}^{p} p B_j \equiv -1 \pmod p$
  • $\sum_{j=0}^{p-3} B_j \equiv -1 \pmod p$ (for $p \ge 5$)
  • $\sum_{j=0}^{p} E_j \equiv \frac{3}{2} \pmod p$
  • $\sum_{j=0}^{p} \frac{p B_j}{p-j+1} \equiv -1 \pmod p$

Significance and Claims
The paper claims that the primary significance of this work lies in the unification of various known identities into a single, general expression (Theorem 1) involving Stirling numbers and harmonic numbers. The authors state that this expression serves as a powerful tool to:

1. Reprove existing results (such as Agoh's recurrence) through a different methodological lens.
2. Derive new recurrence relations and closed formulas for cumulative sums that were not previously available in this specific form.
3. Provide a systematic approach to establishing congruences for sums of Bernoulli and Euler numbers, extending the understanding of their modular properties.

The work does not propose experimental applications or future theoretical directions beyond the immediate mathematical derivations presented. The contributions are strictly confined to the realm of number theory and combinatorics, specifically focusing on the properties of Bernoulli, Euler, and related special numbers.