## Monday, October 4, 2021

### Simple Generating Functions

Generating Functions

Generating a Sequence Via a Function

A generating function, g(x), can be expressed as a power series polynomial.  The coefficients of such polynomials make up the generating sequence.

g(x) = Σ( c_k * x^k, k=0 to ∞) = c_0 + c_1 * x + c_2 * x^2 + c_3 * x^3 + ....

The generated sequence is {c_0, c_1, c_2, c_3, ... }.

A simple approach is find the Maclaurin Series (Taylor Series about the point 0) of g(x).

Example:

g(x) = 2 ÷ (1 - 3*x)

The Maclaurin Series of g(x) is:

2 + 6*x + 18*x^2 + 54*x^3 + 162*x^4 + 486*x^5 + ...

with the sequence of { 2, 6, 18, 54, 162, 486 ... }

Some Simple Generating Functions

1 ÷ (1 - x) = 1 + x + x^2 + x^3 + x^4 + ....

÷ (1 - x) = b + b + b^2 + b^3 + b^4 + ....

1 ÷ (1 - a*x) = 1 + a*x + a^2*x^2 + a^3*x^3 + a^4*x^4 + ....

1 ÷ (1 - x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + ....

1 ÷ (1 + x) = 1 - x + x^2 - x^3 + x^4 - x^5 + ....

(as pictures the size of 3" x 5" index cards:  let me know if you want more posts like this - Eddie)

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