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In this paper we solve the famous problem \( \sum _{i=1}^n i^k\) in the basis \( n^l\) . We use the following approach. Change the basis in \( i^k\) into the basis \( (i)_k\) using Stirling numbers. Then solve the problem in the basis \( (i)_k \) and return into the original basis \( i^k \) using again Stirling numbers. As a consequence we solve the problem represent the Bernoulli numbers in terms of Stirling numbers. Here we use the notation \((i)_k=i(i+1)(i+2)\dots (i+k-1) \) .

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Sum of Powers. Bernoulli Numbers in Terms of Stirling Numbers

  • 会议论文

  • 发表日期：

    2025年

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  • Inna Nikolova,
  • Pencho Marinov

摘要

In this paper we solve the famous problem \( \sum _{i=1}^n i^k\) in the basis \( n^l\) . We use the following approach. Change the basis in \( i^k\) into the basis \( (i)_k\) using Stirling numbers. Then solve the problem in the basis \( (i)_k \) and return into the original basis \( i^k \) using again Stirling numbers. As a consequence we solve the problem represent the Bernoulli numbers in terms of Stirling numbers. Here we use the notation \((i)_k=i(i+1)(i+2)\dots (i+k-1) \) .