Multiplication formulas for q-Appell polynomials and the multiple q-power sums

Thomas Ernst

doi:10.17951/a.2016.70.1.1

Abstract

In the first article on q-analogues of two Appell polynomials, the generalized Apostol-Bernoulli and Apostol-Euler polynomials, focus was on generalizations, symmetries, and complementary argument theorems. In this second article, we focus on a recent paper by Luo, and one paper on power sums by Wang and Wang. Most of the proofs are made by using generating functions, and the (multiple) q-addition plays a fundamental role. The introduction of the q-rational numbers in formulas with q-additions enables natural q-extension of vector forms of Raabes multiplication formulas. As special cases, new formulas for q-Bernoulli and q-Euler polynomials are obtained.

Highlights

In 2006, Luo and Srivastava [8, p. 635-636] found new relationships between Apostol–Bernoulli and Apostol–Euler polynomials. This was followed by the pioneering article by Luo [10], where multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, together with λ-multiple power sums were introduced
In [5] it was proved that the q-Appell polynomials form a commutative ring; in this paper we show what this means in practice
The beautiful symmetry of the formulas comes from the ring structure of the q-Appell polynomials

Summary

Introduction

In 2006, Luo and Srivastava [8, p. 635-636] found new relationships between Apostol–Bernoulli and Apostol–Euler polynomials. 635-636] found new relationships between Apostol–Bernoulli and Apostol–Euler polynomials. This was followed by the pioneering article by Luo [10], where multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, together with λ-multiple power sums were introduced. Luo expressed these λ-multiple power sums as sums of the above polynomials. Wang and Wang [12] introduced generating functions for λ-power sums, some of the proofs use a symmetry reasoning, which lead. Raabes multiplication formulas, q-Appell polynomials, multiple q-power sum, symmetry, q-rational number

Objectives

Conclusion