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 JacobiSobolev 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 endpoints. 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 MehlerHeinetype 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 JacobiSobolev 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 JacobiSobolev 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.