Generating Functions from Recurrence Relations
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when they need a concrete, step-by-step example of finding a generating function from a recurrence relation.
Recommended by 1 people across 1 different months, 2013 to 2023.
What it is
A short worked tutorial from the Mathematics Exemplified channel showing how to turn a recursively defined sequence into a generating function. It walks through examples, including a second-order linear recurrence with a closed-form rational generating function.
What people said
Recommends Joel · 2019-11 · Mathematics Stack Exchange · on “Recurence function inside a double sum”
1.- let $B_k= (1-\beta)^{k}$ 2.- Modify into a polynomic form: $$P_N = (A_1 +P_0B_1) + (A_2 + P_0B_2 + P_1B_1) + (A_3 + P_0B_3 + P_1B_2 + P_2B_1) ... + (A_N + P_0B_N + P_1B_{N-1} + P_2B_{N-2} ... + P_{N-1}B_1)$$ $$-> P_N= \sum_{n=1}^N ((\sum_{m=1}^{n}B_m)P_{N-n}+A_n)= \sum_{n=1}^N(C_nP_{N-n} + A_n)$$ 3.- Create a "Generating function": [LINK] [this video] $$P_1 = C_1P_0 + A_1, P_2 = C_1(C_1P_0 + A_1)+C_2P_0 + A_2, P_3 = C_1(C_1(C_1P_0 + A_1)+C_2P_0 + A_2) + C_2(C_1P_0 + A_1) + C_3P_0 + A_3 ... $$ Let: $$F(x) = P_0 +P_1x + P_2x^2 + P_3x^3...+P_Nx^N $$ Multiply F(x) by $1-\sum_{n=1}^NC_nx^n$ gives: $$(1-\sum_{n=1}^NC_nx^n)F(x) = P_0 + \sum_{n=1}^NA_nx^n$$ $$->F(x) = \frac{P_0 + \sum_{n=1}^NA_nx^n}{
Cites Remi.b · 2013-08 · Mathematics Stack Exchange · on “The meaning of Generating functions”
I had a look to this [this video] video on the field of series and sequences which I know not much about! This guy looked for the generating function of $a_n = 2a_{n-1} + 4a_{n-2}$ with $a_0=1$ and $a_1=3$. The generating function is $$A(x)=\frac{1+x}{1-2x-4x^2}$$ (solution showed at 04:33 in the video) I've been trying to understand what is the real meaning of this generating function he found with wikipedia but I got lost into thousands of pages and definitions. Can we find with this formula the 18th number of the series? What does the x represents? How does this generating function "represents" the series? Hope my question is not too broad and that you will easily find where is my gap!
Cites Gaurang Tandon · 2017-08 · Mathematics Stack Exchange · on “Find degree of polynomial satisfying given condition”
Well, I was viewing this after that video - [this video] - in this he uses a method which is completely different. I used the method I learnt in the previous video - of using characteristic equations - I got a weird expression involving $\sqrt{5}$. What am I doing wrong?
Cites sibillalazzerini · 2023-02 · Mathematics Stack Exchange · on “Generalised algorithm to find generating function for any linear recurrance relation.”
Problem: Formally speaking lets say we have, $ F_n = a_{k}F_{n-1} + a_{k-1}F_{n-2} + a_{k-2}F_{n-3} + \cdots + a_1F_{n-k} $ Base case being, $F_0 = b_0\\F_1 = b_1\\ \cdots\\F_{k-1} = b_{k-1}$ We are to find the generating function of this recurrence relation in terms of the coefficients $a_i$ and $b_j$ . My attempts: From symmetry the solution I get is $f(x)(1 - a_k - a_{k-1}x - a_{k-2}x^2 \cdots - a_{k-(k-1)}x^{k-1}) = b_0 + (b_0+b_1)x + (b_0+b_1+b_2)x^2 + \cdots + (b_0+b_1+b_2+ \cdots b_{k-1})x^{k-1}$ $\implies f(x)= \frac {b_0 + (b_0+b_1)x + (b_0+b_1+b_2)x^2 + \cdots + (b_0+b_1+b_2+ \cdots b_{k-1})x^{k-1}}{1-a_k-a_{k-1}x-a_{k-2}x^2- \cdots -a_{k-(k-1)}x^{k-1}} $ where $f(x)$ is the genera