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

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.