17 - Three-Term Recurrence Relation for Orthogonal Polynomials

Three-Term Recurrence Relation for Orthogonal Polynomials

Let $\{{\varphi }_0,{\varphi }_1,\cdots {\varphi }_n,\cdots \}$ be a set of orthogonal polynomials

There exists sequences of real numbers

$$({\alpha }_k{)}_{k=1}^{\infty },({\beta }_k{)}_{k=1}^{\infty },({\gamma }_k{)}_{k=1}^{\infty }$$

such that a three-term recurrence relation holds of form

$${\varphi }_{k+1}(x)={\alpha }_k(x-{\beta }_k){\varphi }_k(x)-{\gamma }_k{\varphi }_{k-1}(x),k=1,2,\cdots$$

Proof

Polynomial $x{\varphi }_k\in {\Pi }_{k+1}$ so exists real numbers

$${\sigma }_{k,0},{\sigma }_{k,1},\cdots ,{\sigma }_{k,k+1}$$

such that

$$x{\varphi }_k(x)=\sum _{i=0}^{k+1}{\sigma }_{k,i}{\varphi }_i(x)$$

as $\{{\varphi }_0,{\varphi }_1,{\varphi }_{k+1}\}$ is a basis for ${\Pi }_{k+1}$

For $j\le k-2$ then

$$⟨x{\varphi }_k,{\varphi }_j⟩={\int }_a^{b}w(x)x{\varphi }_k(x){\varphi }_j(x)dx={\int }_a^{b}w(x){\varphi }_k(x)x{\varphi }_j(x)dx=⟨{\varphi }_k,x{\varphi }_j⟩=0$$

as $x{\varphi }_j\in {\Pi }_{k-1}$ and

$$⟨\sum _{i=0}^{k+1}{\sigma }_{k,i}{\varphi }_i,{\varphi }_j⟩=\sum _{i=0}^{k+1}{\sigma }_{k,i}⟨{\varphi }_i,{\varphi }_j⟩={\sigma }_{k,j}⟨{\varphi }_j,{\varphi }_j⟩$$

by linearity of $⟨\cdot ,\cdot ⟩$ and orthogonality of ${\varphi }_i$ and ${\varphi }_j$ for $i\ne j$

Hence ${\sigma }_{k,j}=0$ for $j\le k-2$ hence

$$x{\varphi }_k(x)={\sigma }_{k,k+1}{\varphi }_{k+1}(x)+{\sigma }_{k,k}{\varphi }_k(x)+{\sigma }_{k,k-1}{\varphi }_{k-1}(x)$$

Thus

$$⟨x{\varphi }_k,{\varphi }_{k+1}⟩={\sigma }_{k,k+1}⟨{\varphi }_{k+1},{\varphi }_{k+1}⟩$$

So ${\sigma }_{k,k+1}\ne 0$ as $x{\varphi }_k$ is of exact degree $k+1$

Thus

$${\varphi }_{k+1}(x)=\frac{1}{{\sigma }_{k,k+1}}(x-{\sigma }_{k,k}){\varphi }_k(x)-\frac{{\sigma }_{k,k-1}}{{\sigma }_{k,k+1}}{\varphi }_{k-1}(x)$$

so

$${\alpha }_k=\frac{1}{{\sigma }_{k,k+1}},{\beta }_k={\sigma }_{k,k},{\gamma }_k=\frac{{\sigma }_{k,k-1}}{{\sigma }_{k,k+1}}\text{ for }k=1,2,\cdots$$