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Faulhaber's formula, Bernoulli numbers and the equation $f(x)+x^k=f(x+1)$

Chai Wah Wu

Abstract

In modern usage the Bernoulli numbers and polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation $f(x)+x^k=f(x+1)$ and show that a solution can be derived from Faulhaber's formula for the sum of powers. We show how these solutions provide a characterization of Bernoulli numbers and related results.

Faulhaber's formula, Bernoulli numbers and the equation $f(x)+x^k=f(x+1)$

Abstract

In modern usage the Bernoulli numbers and polynomials follow Euler's approach and are defined using generating functions. We consider the functional equation and show that a solution can be derived from Faulhaber's formula for the sum of powers. We show how these solutions provide a characterization of Bernoulli numbers and related results.
Paper Structure (4 sections, 4 theorems, 7 equations)

This paper contains 4 sections, 4 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Faulhaber's formula
  3. A variant of Faulhaber's formula
  4. A solution to the equation $f(x)+x^k=f(x+1)$

Key Result

Lemma 1

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • proof
  • Proposition 1
  • Proposition 2