Three-term recurrence relation for orthogonal polynomials.

roposition

(Three-term recurrence relation)
Let
be the polynomials related to measure
as in the definition (
Orthogonal
polynomials
) and the inner product
is positive definite. We
have

Next, we apply the operation
to both sides and use consequences of orthogonality to calculate
and
.

For
we
get
For
we
get
Note
that
where
is
-th
degree polynomial with leading coefficient equal to one. Thus, by the
proposition (
Basic
property of orthogonal
polynomials
),
for some numbers
.
We continue transformation of
using orthogonality:
We combine the last result with the relationship
and
obtain
For
we use the
property
and orthogonality to
find

Definition

(Jacobi matrix) We introduce the following notation

1.

2. The numbers
are zeros of
:
for each
.

Proposition

(Zeros of orthogonal polynomials)
Let
be the polynomials related to the measure
as in the definition (
Orthogonal
polynomials
)-2 and the inner product
is positive definite. The zeros
are eigenvalues of the matrix
for each
and
are the corresponding eigenvectors.

Proof

The proposition's (
Three-term
recurrence relation
) main statement
is
It may be rewritten as
We
substitute
then
We divide the last relationship by
then
or
We restate the last result in matrix form as
where
and
.
The statement is apparent after the substitution
.