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The denominators of power sums of arithmetic progressions

Bernd C. Kellner, Jonathan Sondow

Abstract

In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit criterion for integrality of the coefficients of these polynomials. As applications, we obtain new results on the sequence of denominators of the Bernoulli polynomials. A consequence is that certain quotients of successive denominators are infinitely often integers, which we characterize.

The denominators of power sums of arithmetic progressions

Abstract

In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit criterion for integrality of the coefficients of these polynomials. As applications, we obtain new results on the sequence of denominators of the Bernoulli polynomials. A consequence is that certain quotients of successive denominators are infinitely often integers, which we characterize.

Paper Structure

This paper contains 6 sections, 16 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem 2
  4. Proofs of Theorem 3 and Corollary 2
  5. Proofs of Theorems 4 and 5
  6. Conclusion

Key Result

Theorem \oldthetheorem

For $n \geq 1$, denote Then we have the relation and the remarkable formula where $s_p(n)$ denotes the sum of the base-$p$ digits of $n$, as defined in Section sec:proofs2. Moreover,

Theorems & Definitions (32)

  • Remark
  • Theorem \oldthetheorem: Kellner and Sondow KellnerSondow:2017
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Example 1
  • Remark
  • Corollary 1
  • Remark
  • Corollary 2
  • Theorem \oldthetheorem
  • ...and 22 more