(Ordinary primitive recursive functions are always total functions, that give definite values for every possible input. But general recursive functions can be partial functions, that do not terminate for some inputs.) … (Note that multiple arguments to a recursive function can be encoded as a single argument using functions like the β of page 1120 —though the irregularity of such functions tends to make it difficult then to tell what is going on in the underlying recursive function.)

Primitive [Boolean] functions There are several possible choices of primitive functions that can be combined to represent any Boolean function. … The functions And , Xor and Not are equivalent to Times , Plus and 1 - # & for variables modulo 2, and in this case algebraic functions like PolynomialReduce can be used for minimization.

Minimal representations in terms of Nand functions of the first two steps in the evolution of the same cellular automata as on the facing page . … The picture at the top left shows the action of a single Nand function.

So is it in fact possible to get formulas for the colors of squares that involve only such functions? … The succession of polynomials above can be obtained by expanding the generating functions 1/(1 - (1 + x) y) and 1/(1 - (1 + x + x 2 ) y) . … GegenbauerC is a so-called orthogonal polynomial—a higher mathematical function.

[Boolean] formula sizes There are a total of 2 2 n possible Boolean functions of n variables. The maximum number of terms needed to represent any of these functions in DNF is 2 n - 1 . … (Other functions with maximal length are never additive, at least for n ≤ 4 .)

But in the so-called lambda calculus of Alonzo Church from around 1930 what were instead used were pure functions such as s = Function[x, x + 1] and plus = Function[{x, y}, If[x  0, y, s[plus[x - 1, y]]]] —of just the kind now familiar from Mathematica. … The definitions in the note above involve both symbolic functions and literal integers. … The typical way this is done is to say that a function f n corresponds to an integer n if f n [a][b] yields Nest[a, b, n] (see note below ).

Growth rates [of functions] One can characterize most functions by their ultimate rates of growth. … Yet it is still possible to define functions with even more rapid rates of growth. … In general this function must grow faster than any computable function, and is not itself computable.

Nested continuous functions Most standard continuous mathematical functions never show any kind of nested behavior. Elliptic theta and elliptic modular functions are exceptions. … The function can be obtained as the solution to a second-order nonlinear ordinary differential equation.

Generating functions [for regular languages] The sequences in a regular language can be thought of as corresponding to products of non-commuting variables that appear as coefficients in a formal power series expansion of a generating function. A basic result is that for regular languages this generating function is always rational.

[Generating functions for] 1D sequences Generating functions that are rational always lead to sequences which after reduction modulo 2 are purely repetitive. Algebraic generating functions can also lead to nested sequences. … For any sequence with an algebraic generating function and thus for any nested sequence the n th element can always be expressed in terms of hypergeometric functions.