- Research Article
- Open Access
Some Properties of Orthogonal Polynomials for Laguerre-Type Weights
© H. Jung and R. Sakai 2011
Received: 3 November 2010
Accepted: 21 February 2011
Published: 14 March 2011
1. Introduction and Main Results
Let and . Let be a continuous, nonnegative, and increasing function. Consider the exponential weights , , and then we construct the orthonormal polynomials with the weight . In this paper, for the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted -norms ( ) of .
We say that is quasi-increasing if there exists such that for . The notation means that there are positive constants such that for the relevant range of , . The similar notation is used for sequences and sequences of functions.
First, we introduce some classes of weights.
Let us consider the weight in Definition 1.2. Levin and Lubinsky [2, Theorem 1.3] have given the following theorem.
Theorem A (see [3, Theorem 1.3]).
In this paper, we will consider the orthonormal polynomials with respect to the weight class . Our main themes in this paper are to estimate the higher-order derivatives of at the zeros of and to investigate the various weighted -norms ( ) of . More precisely, we will estimate the higher-order derivatives of at all zeros of for two cases of an odd order and of an even order. In addition, we will give asymptotic relation of the odd order derivatives of at the zeros of in a certain finite interval. These estimations will play an important role in investigating convergence or divergence of higher-order Hermite-Fejér interpolation polynomials (see [3, 5–17]).
Theorem B (see [18, Theorem 13.6]).
This paper is organized as follows. In Section 2, we will introduce the weight class as an analogy of the class and the known results of orthonormal polynomials with respect to in order to prove the main results. In Section 3, we will prove Theorems 1.5, 1.6, and 1.7. Finally, we will prove the results for the various weighted -norms ( ) of , that is, Theorems 1.8, 1.9, and 1.10, in Section 4.
Definition 2.1 (see ).
Then we see that and from [1, Lemma 2.2]. In addition, we easily have the following.
Definition 2.3 (cf. ).
Jung and Sakai [5, Theorems 3.3 and 3.6] estimate , , , and we will obtain analogous estimations with respect to , , in Theorems 1.6 and 1.7.
There are many properties of with respect to , , of Definition 2.3 in [4–6, 19–21]. They were obtained by transformations from the results in [1, 2]. In this paper, we consider and . In  we got the estimations of , , with the weight . By a transformation of the results with respect to , we estimate , , . In order to it we will give the transformation theorems in this section. In the following, we will give some applications of them.
Theorem 2.4 (see [21, Theorem 2.1]).
3. Proofs of Theorems 1.5, 1.6, and 1.7
In the following, we introduce some useful notations.
Then, one has the following.
Lemma 3.1 (see [1, (2.5), (2.7), (2.9)]).
To prove Theorem 1.6, we need some lemmas as follows.
Lemma 3.2 (see [21, Theorem 2.2, Lemma 3.7]).
Lemma 3.3 (see [6, Theorem 2.5]).
Lemma 3.4 (see [5, Theorem 3.6 and Lemma 3.7 (3.20)]).
Proof of Theorem 1.5.
Proof of Theorem 1.6.
Next, we will prove Theorem 1.7. To prove it, we need two lemmas as follows.
Lemma 3.7 ([5, Theorem 3.3]).
From Lemma 3.3, we easily have the following.
Proof of Theorem 1.7..
4. Proofs of Theorems 1.8, 1.9, and 1.10
But, seeing our proof of [21, Theorem 2.6] carefully, we can easily prove the first equivalence.
Lemma 4.2 (see [21, Theorem 2.4]).
Proof of Theorem 1.8.
Consequently, using Lemma 3.1, one has the result.
Proof of Theorem 1.9.
Lemma 4.4 (see [21, Theorem 2.7]).
Proof of Theorem 1.10.
The authors thank the referees for many kind suggestions and comments. H. S. Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.
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