Generating Functions Homework: Q1 & Q2

[saubbie]

Homework Statement

I have two questions. The first is the generating function for a(n)=5a(n-1) - 6(an-2) where a(0)=0 and a(1)=1. The explicit equation for the sequence is 3^n-2^n.

The second one is finding the generating function for the sequence of perfect squares {n^2} for n greater than or equal to 0.

Homework Equations

The Attempt at a Solution

I am really confused about finding generating functions, and don't know where to start, so any help would be greatly appreciated. Thanks a lot.

 

[mighty2000]

Generating functions are a useful tool in combinatorics and number theory, allowing us to represent a sequence as a polynomial in a variable x. To find the generating function for a sequence, we first need to determine a recursive formula or explicit equation for the sequence.

For the first question, the recursive formula is a(n) = 5a(n-1) - 6a(n-2) and the initial values are a(0) = 0 and a(1) = 1. To find the generating function, we can use the formula F(x) = a(0) + a(1)x + a(2)x^2 + a(3)x^3 + ..., where a(n) is the nth term in the sequence.

Substituting in the recursive formula, we get F(x) = 0 + 1x + (5a(1) - 6a(0))x^2 + (5a(2) - 6a(1))x^3 + ...

Simplifying, we get F(x) = x + 5x^2 - 6x^3 + 5a(3)x^4 + ...

To find the explicit equation for the sequence, we can use the method of characteristic equations. The characteristic equation for this recursive formula is x^2 - 5x + 6 = 0. Solving for the roots, we get x = 2 and x = 3.

Thus, the explicit equation for the sequence is a(n) = A(2)^n + B(3)^n, where A and B are constants that can be determined using the initial values a(0) = 0 and a(1) = 1.

For the second question, the sequence of perfect squares is 0, 1, 4, 9, 16, ... and the explicit equation is a(n) = n^2. To find the generating function, we can use the same formula F(x) = a(0) + a(1)x + a(2)x^2 + a(3)x^3 + ..., where a(n) is the nth term in the sequence.

Substituting in the explicit equation, we get F(x) = 0 + 1x + 2x^2 + 3x^3 + 4x^4 + ...

Simplifying, we