q-Bernoulli and q-Euler Polynomials, an Umbral Approach
Abstract
We present a variety of q-formulas linked together by the q-umbral calculus introduced here, equivalent to the approach to q-hypergeometric functions and q-Appell functions given earlier, influenced by Rota and Cigler. This q-umbral calculus is connected to formal power series; three q-Taylor formulas occur; one of these forms the basis of an umbral formula influenced by Nörlund, which enables q-Euler-Maclaurin- and q-Euler-Boole formulas. q-Appell polynomials in the spirit of Milne-Thomson are also treated before specialization to q-Bernoulli and q-Euler polynomials. A brief survey of the theory and history of finite differences and umbral calculus is given.
Keywords
Quantum calculus, q-Bernoulli polynomials, q-Euler polynomials, q-umbral method, Euler–Boole theorem, Euler–Maclaurin theorem, equivalence relation, Ward number, letter, q-multinomial coefficient, formal power series, finite differences, q-Bernoulli operator, field of fractions
