📂Lemmas

Recurrence Relation

Definition

Let’s consider a sequence $\left\{ a_{n} \right\}$. At this time, expressing $a_{n}$ as a function of $a_{n-1}$, $a_{n-2}$, $\cdots$, and $a_{1}$ is called a recurrence relation.

Explanation

For instance, the sequence of natural numbers $\left\{ 1, 2, 3, 4, \dots \right\}$ can be expressed by the following recurrence relation.

$$a_{n} = a_{n-1} + 1, \qquad a_{1} = 1$$

The coefficients of the Legendre polynomial are expressed by the following recurrence relation. Therefore, knowing only $a_{0}$ and $a_{1}$, all coefficients can be obtained.

$$a_{n+2} = -\dfrac{(\ell + n + 1)(\ell - n)}{(n+1)(n+2)} a_{n}$$