On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers

2017/07/14 by Wolfdieter Lang, Lang, Wolfdieter · 1 citation
Mathematics · Physics and Astronomy · #11B68 11B73 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1707.04451

openalex publication_date 2017/07/14 · openalex created_date 2017/07/21 · openalex updated_date 2026/07/28

Abstract

For finite sums of non-negative powers of arithmetic progressions the generating functions (ordinary and exponential ones) for given powers are computed. This leads to a two parameter generalization of Stirling and Eulerian numbers. A direct generalization of Bernoulli numbers and their polynomials follows. On the way to find the Faulhaber formula for these sums of powers in terms of generalized Bernoulli polynomials one is led to a one parameter generalization of Bernoulli numbers and their polynomials. Generalized Lah numbers are also considered.

Cited by

• An introduction to the Bernoulli function

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