16-4. Formulas for Other Difficult Functions

Let us have a closer look at what Willans and Wormell have done. We postulate the rules below as defining what we mean by the class of functions that can be represented by “formulas,” which we will call “formula functions.” Here, x¯ is shorthand for x1, x2, …, xn for any n ≥ 1. The domain of values is the integers … −2, −1, 0, 1, 2, ….

  1. The constants … −1, 0, 1, … are formula functions.

  2. The projection functions f(x¯) = xi, for 1 ≤ i ≤ n, are formula functions.

  3. The expressions x + y, x − y, and xy are formula functions, if x and y are.

  4. The class of formula functions is closed under composition (substitution). That is, f(g1(x¯), g2(x¯), …, gm(x¯)) is a formula function if f and gi are, for i = 1, …, m

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