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)





All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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