# What is a recurrence relation

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

A recurrence relation is an equation which defines each term of a sequence as a function of preceding terms.

The most well-known are those defining the Fibonacci numbers and the binomial coefficients.

An ordinary differential equation can be considered as a recurrence relation on the sequence of nth derivatives of a function (in which the 0th derivative is the function itself), and if linear can be solved using the characteristic polynomial method.

Equations

Fibonacci numbers:

Each Fibonacci number is the sum of the two preceding numbers, starting with 1 and 1:

$$F_n\ =\ F_{n-1}\ +\ F_{n-2}$$

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ...

Binomial coefficients (Pascal's triangle):

Each binomial coefficient is the sum of the two coefficients "diagonally below", starting with 1:

$$\left(\begin{array}{c}n\\k\end{array}\right)\ =\ \left(\begin{array}{c}n - 1\\k\end{array}\right)\ +\ \left(\begin{array}{c}n-1\\k-1\end{array}\right)$$

1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1; 1 5 10 10 5 1; ...

Extended explanation

Characteristic polynomial:

This is not the same as the characteristic polynomial of a matrix or matroid

In a linear differential equation, the derivative may be replaced by an operator, D, giving a polynomial equation in D:

$$\sum_{n\,=\,0}^m\,a_n\,\frac{d^ny}{dx^n}\ =\ 0\ \mapsto\ \left(\sum_{n\,=\,0}^m\,a_n\,D^n\right)y\ =\ 0$$

Similarly, in a linear recurrence relation, the nth term may be replaced by an nth power:

$$\sum_{n\,=\,0}^m\,a_{k+n}\,P_{k+n}\ =\ 0\ \mapsto\ \sum_{n\,=\,0}^m\,a_n\,x^n\ =\ 0$$

If this polynomial has distinct (different) roots $r_1,\dots,r_m$:

$$\prod_{n\,=\,1}^m(D\,-\,r_n)\ =\ 0$$ or $$\prod_{n\,=\,1}^m(x\,-\,r_n)\ =\ 0$$

then the general solution is a linear combination of the solutions of each of the equations:

$$\left(D\,-\,r_n\right)y\ =\ 0$$ or $$P_{k+1}\ -\ r_nP_k\ =\ 0$$

which are the same as $$\frac{dy}{dx}\ =\ r_n\,y$$ or $$\frac{P_{k+1}}{P_k}\ =\ r_n$$

and so is of the form:

$$y\ =\ \sum_{n\,=\,1}^m\,C_n\,e^{r_nx}$$ or $$P_k\ =\ \sum_{n\,=\,1}^m\,C_n\,(r_n)^k$$

For a pair of complex roots (they always come in conjugate pairs) $p\ \pm\ iq$ or $r\,e^{\pm is}$, a pair of $$C_ne^{r_nx}$$ or $C_n\,(r_n)^k$may be replaced by $$e^{px}(A\,cos(qx)\,+\,iB\,sin(qx))$$ or $$r^k(A\,cos(sk)\,+\,iB\,sin(sk))$$

However, if the polynomial has some repeated roots:

$$\prod_{p\,=\,1}^q\prod_{n\,=\,1}^p(D\,-\,r_n)^p\,y\ =\ 0$$ or $$\prod_{p\,=\,1}^q\prod_{n\,=\,1}^p(x\,-\,r_n)^p\ =\ 0$$

then the general solution is of the form:

$$y\ =\ \sum_{p\,=\,1}^q\sum_{n\,=\,1}^pC_{n,p}\,x^{p-1}\,e^{r_nx}$$ or $$P_k\ =\ \sum_{p\,=\,1}^q\sum_{n\,=\,1}^pC_{n,p}\,n^{p-1}\,(r_n)^k$$

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