Let
denote the sequence of polynomials orthogonal with respect to (1.3) normalized by the condition that they have the same leading coefficient as
, that is,
.
The following relation between
and
holds.
Proposition 3.1.
For
,
where, for
,
Moreover,
and
for
.
Proof.
Expanding
with respect to the basis
of the linear space of polynomials with degree at most
, we get
where, for
,
For
,
Therefore,
As a conclusion,
Using the extremal property for monic orthogonal polynomials with respect to the corresponding norm (see [12, Theorem 3.1.2]),
we get
Thus,
Finally, from (3.8), we find that
and from Schwarz inequality,
Thus,
Using (3.1) in a recursive way, we get the representation of the polynomial
in terms of the elements of the sequence
. More precisely we get the following.
Proposition 3.2.
For
,
, it holds that
where
,
, and
,
,
. Moreover,
for
, and
for
,
.
Proof.
Let denote by
,
, and
,
,
. First, we prove that
where
, and, by convention,
,
.
We will prove (3.16) by induction. When
, it is a trivial result. On the other hand, applying (3.1) in a recursive way, we get
Taking into account (3.7), we have
. Thus, (3.16) follows for
. Now, we assume (3.16) holds for
. Again, from (3.1),
Now, we prove that
for
. For
, this follows from (3.7). Since
and
, for
the statement follows by induction. Thus, (3.16) holds for
. Now taking
in (3.16), we get (3.15).
Finally, we prove that
for
, and
for
,
. First, the following inequality holds:
. Indeed, for
, (3.19) follows from Proposition 3.1 and (3.7). Now, we assume that the relation (3.19) holds for
. Thus, for
,
for 
for 
and for 
Therefore, from
the relation (3.19) holds for
. As consequence,
for
and
.
Now, we will prove by induction that
for
and
.
The case
follows from (3.19). We assume that
for
and
. For
,
and for 
Therefore, from
the statement holds for
.
Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.
Proposition 3.3.
-
(a)
For the polynomials
, we get
where
, and
.
-
(b)
For the polynomials
, we get
where
, and
,
.
Proof.
-
(a)
Using Proposition 3.2, we have
Therefore, from Proposition 2.3
, the statement follows immediately.
On the other hand, taking into account Proposition 2.1
, Proposition 2.2, (2.14), and (3.15), the proof of
can be done in a similar way.
Now, we show that, like for the classical Jacobi polynomials, the polynomial
attains its maximum in
at the end-points. More precisely,
Proposition 3.4.
-
(a)
For
,
, and
where
if
and
if
.
-
(b)
For
,
and
where
if
and
if
.
Proof.
Here, we will prove only the case when
. The case when
can be done in a similar way.
-
(a)
From Proposition 2.3
,
for
and
. Therefore, according to (3.30),
for
and
. From Proposition 3.1, we get
Finally, from Proposition 2.3(a), the statement follows.
-
(b)
Taking into account Proposition 2.1
, Proposition 2.2, (2.14), (3.1), and (3.15), we can conclude the proof in the same way as we did in (a).
Corollary 3.5.
For
,
,
and for
,
,
where
Proof.
The inequality
holds for
, as well as
for
. Therefore, from Propositions 3.3 and 3.4, the statement follows immediately.
Next, we deduce a Mehler-Heine-type formula for
and
(see Theorem 4.1 in [10]).
Proposition 3.6.
Uniformly on compact subsets of
,
-
(a)
-
(b)
where
are real numbers, and
is the Bessel function of the first kind.
Proof.
To prove the proposition, we use the same technique as in [17].
-
(a)
Multiplying in (3.1) by
, we obtain
where
,
and
,
. Moreover,
and
for
.
Using the above relation in a recursive way as well as the same argument of Proposition 3.2, we have
where
,
for
, and
for
,
. Thus,
On the other hand, from Proposition 2.3(c),
is uniformly bounded on compact subsets of
. Thus, for a fixed compact set
, there exists a constant
, depending only on
, such that when
,
Thus, the sequence
is uniformly bounded on
. As a conclusion,
and from Proposition 2.3(c), we obtain the result.
-
(b)
Since we have uniform convergence in (3.41), taking derivatives and using a well known property of Bessel functions of the first kind (see [12, formula 1.71.5]), we obtain (3.42).
Now, we give the inner strong asymptotics of
on
.
Proposition 3.7.
For
and
,
where
,
,
,
, and
.
Proof.
From Proposition 3.3(a), the sequence
is uniformly bounded on compact subsets of
; thus, from Proposition 3.1,
Now, using Proposition 2.3(d), the relation (3.48) follows.
Concerning (3.49), it can be obtained in a similar way by using Propositions 2.1(f) and 2.2, (2.14), Propositions 3.1 and 3.3(b).
Now, we can give the sharp estimate for the Sobolev norms of the Jacobi-Sobolev polynomials.
Proposition 3.8.
For
and
,
Proof.
Clearly, if
, then we get Proposition 3.4(b). Thus, in the proof, we will assume
. Since by Proposition 3.2 and(2.14)
where
,
, and
are bounded because of the orthonormality condition, we obtain
where
, and
.
On the other hand, using (3.30), Minkowski's inequality, and Proposition 2.3(e), we deduce
In the same way as above, we get
Thus, from (3.53), (3.54), and (3.55), we have
Notice that the upper estimate in (3.54) and (3.55) can also be proved using the bounds for Jacobi-Sobolev polynomials given in Corollary 3.5.
In order to prove the lower bound in (3.51) we will need the following.
Proposition 3.9.
For
and
,
Proof.
We will use a technique similar to [12, Theorem 7.34]. According to (3.42),
On the other hand, from (see [18, Lemma 2.1]), if
and
, we have
Thus, for
and
large enough, (3.57) follows.
Finally, from (3.49), we obtain
The proof of Proposition 3.9 is complete.
From (3.57), for
and
,
Thus, using (3.56) and (3.61), the statement follows.