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The iterated logarithmic algebra

We generalize the Umbral Calculus of G-C. Rota by studying not only sequences of polynomials and inverse power series, or even the logarithms studied in, but instead we study sequences of formal expressions involving the iterated logarithms and x to an arbitrary real power. Using a theory of formal power series with real exponents, and a more general definition of factorial, binomial coefficient, and Stirling numbers to all the real numbers, we define the Iterated Logarithmic Algebra I. Its elements are the formal representations of the asymptotic expansions of a large class of real functions, and we define the harmonic logarithm basis of I which will be interpreted as a generalization of the powers x^n since it behaves nicely with respect to the derivative We classify all operators over I which commute with the derivative (classically these are known as shift-invariant operators), and formulate several equivalent definitions of a sequence of binomial type. We then derive many formulas useful towards the calculation of these sequences including the Recurrence Formula, the Transfer Formula, and the Lagrange Inversion Formula. Finally, we study Sheffer sequences, and give many examples.