A Kind Of Series Expansion-Transformation Formulas About Operators

The symbolic operators△、E and D play important role in the finite operators calculus, whose academic foundation is Heaviside Calculus. what’s more, this theory has been approximate perfection now. Here mainly studies some kinds of expansion-transformation formulas of power series with symbolic operators, and the way of power series’s expansion.The matter of this thesis are listed as follows. The first is to communicate the symbolic operator theory with the theory of Sheffer polynomial, and replace the variable t of Sheffer polynomial’s definition by the difference operator△, in order to get series expansion-transformation formulas by expanding the Sheffer polynomial’s definition of variable△, the relation of the operator are also useful. Under the above result, it discusses the applications of the formulas in some special cases. Secondly, here presents the operator g(x△)in different ways, so as to construct some series-transformation formulas,and simplify these formulas by importing a new definition of Euler polynomial with two variables. It studies the applications of these formulas too. What’s more, here gives a new way of power series’s expansion by the discussing the properties of operator(xD)n, it’s application are also discussed.