The q-umbral calculus and semigroups. The Nørlund calculus of finite differences

  • Thomas Ernst
Chapter

Abstract

In this chapter we focus on formal power series. In the important Section 4.1, which contains the algebraic rules for the two q-additions and the infinite alphabet, we introduce the q-umbral calculus in the spirit of Rota. We present tables of the important Ward numbers, which will later occur in matrix computations. We continue with a q-analogue of Nørlund’s and Jordan’s finite difference calculus. In Section 4.3, we systematically analyse q-Appell polynomials in the spirit of Milne-Thomson, and it’s special cases q-Bernoulli and q-Euler polynomials. We show the unification of finite differences and differential calculus in the shape of q-Appell polynomials. Because of the complementary argument theorem, we define two dual types of q-Bernoulli and q-Euler polynomials, NWA and JHC. This is a characteristic phenomenon, which we will often encounter in further computations. We present tables of q-Bernoulli and q-Euler numbers and show simple symmetry relations for these, corresponding to the classical case q=1. As suggested by Ward, we introduce q-Lucas and G polynomials and show their corresponding expansions. These q-Appell polynomials will occur in many further publications. Chapter 4, except for the first section, is not necessary for the rest of the book.

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© Springer Basel 2012

Authors and Affiliations

  • Thomas Ernst
    • 1
  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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