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

TL;DR

This paper develops generalized Stirling, Eulerian, and Bernoulli numbers to compute finite sums of powers of arithmetic progressions, providing new formulas and connections between these classical number sequences.

Contribution

It introduces a two-parameter generalization of classical numbers and polynomials, extending their applicability to sums of powers of arithmetic progressions.

Findings

Derived generating functions for sums of powers of arithmetic progressions.

Established a two-parameter generalization of Stirling and Eulerian numbers.

Formulated a one-parameter generalization of Bernoulli numbers and polynomials.

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.