Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 482935 | 25 pages | https://doi.org/10.1155/2012/482935

Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Academic Editor: Jin L. Kuang
Received16 Mar 2012
Accepted24 May 2012
Published09 Aug 2012

Abstract

Let π‘€πœ†(π‘₯)∢=(1βˆ’π‘₯2)πœ†βˆ’1/2 and π‘ƒπœ†,𝑛(π‘₯) be the ultraspherical polynomials with respect to π‘€πœ†(π‘₯). Then, we denote the Stieltjes polynomials with respect to π‘€πœ†(π‘₯) by πΈπœ†,𝑛+1(π‘₯) satisfying ∫1βˆ’1π‘€πœ†(π‘₯)π‘ƒπœ†,𝑛(π‘₯)πΈπœ†,𝑛+1(π‘₯)π‘₯π‘šπ‘‘π‘₯=0, 0β‰€π‘š<𝑛+1, ∫1βˆ’1π‘€πœ†(π‘₯)π‘ƒπœ†,𝑛(π‘₯)πΈπœ†,𝑛+1(π‘₯)π‘₯π‘šπ‘‘π‘₯β‰ 0, π‘š=𝑛+1. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials πΈπœ†,𝑛+1(π‘₯) and the product πΈπœ†,𝑛+1(π‘₯)π‘ƒπœ†,𝑛(π‘₯). Especially, we estimate the even-order derivative values of πΈπœ†,𝑛+1(π‘₯) and πΈπœ†,𝑛+1(π‘₯)π‘ƒπœ†,𝑛(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and the product πΈπœ†,𝑛+1(π‘₯)π‘ƒπœ†,𝑛(π‘₯), respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of πΈπœ†,𝑛+1(π‘₯) and πΈπœ†,𝑛+1(π‘₯)π‘ƒπœ†,𝑛(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and πΈπœ†,𝑛+1(π‘₯)π‘ƒπœ†,𝑛(π‘₯) on a closed subset of (βˆ’1,1), respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-FejΓ©r interpolation polynomials.

1. Introduction

Consider the generalized Stieltjes polynomials πΈπœ†,𝑛+1(π‘₯) defined (up to a multiplicative constant) by ξ€œ1βˆ’1π‘€πœ†(π‘₯)π‘ƒπœ†,𝑛(π‘₯)πΈπœ†,𝑛+1(π‘₯)π‘₯π‘˜π‘‘π‘₯=0,π‘˜=0,1,2,…,𝑛,𝑛⩾1,(1.1) where π‘€πœ†(π‘₯)=(1βˆ’π‘₯2)πœ†βˆ’1/2, πœ†>βˆ’1/2, and π‘ƒπœ†,𝑛(π‘₯) is the 𝑛th ultraspherical polynomial for the weight function π‘€πœ†(π‘₯).

The polynomials πΈπœ†,𝑛+1(π‘₯), introduced by Stieltjes and studied by SzegΓΆ, have been used in numerical integration, whereas the polynomials π‘ƒπœ†,𝑛(π‘₯)πΈπœ†,𝑛+1(π‘₯) have been used in extended Lagrange interpolation. In this paper, we will prove pointwise and asymptotic estimates for the higher-order derivatives of πΈπœ†,𝑛+1(π‘₯) and π‘ƒπœ†,𝑛(π‘₯)πΈπœ†,𝑛+1(π‘₯). It is well known that these kind of estimates are useful for studying interpolation processes with multiple nodes.

In 1934, G. SzegΓΆ [1] showed that the zeros of the generalized Stieltjes polynomials πΈπœ†,𝑛+1(π‘₯) are real and inside [βˆ’1,1] and interlace with the zeros of π‘ƒπœ†,𝑛(π‘₯) whenever 0β©½πœ†β©½2. Recently, several authors [2–8] studied further interesting properties for these Stieltjes polynomials. Ehrich and Mastroianni [3, 4] gave accurate pointwise bounds of πΈπœ†,𝑛+1(π‘₯)(0β©½πœ†β©½1) and the product πΉπœ†,2𝑛+1∢=πΈπœ†,𝑛+1(π‘₯)π‘ƒπœ†,𝑛(π‘₯)(0β©½πœ†β©½1) on [βˆ’1,1], and they estimated asymptotic representations for πΈξ…žπœ†,𝑛+1(π‘₯) and πΉξ…žπœ†,2𝑛+1(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and πΉπœ†,2𝑛+1(π‘₯), respectively. In [6], pointwise upper bounds of πΈξ…žπœ†,𝑛+1(π‘₯), πΈξ…žξ…žπœ†,𝑛+1(π‘₯), πΉξ…žπœ†,2𝑛+1(π‘₯), and πΉξ…žξ…žπœ†,2𝑛+1(π‘₯) are obtained using the asymptotic differential relations of the first and the second order for the Stieltjes polynomials πΈπœ†,𝑛+1(π‘₯)(0β©½πœ†β©½1) and πΉπœ†,2𝑛+1(π‘₯)(0β©½πœ†β©½1). Also the values of πΈξ…žξ…žπœ†,𝑛+1(π‘₯) and πΉξ…žξ…žπœ†,2𝑛+1(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and πΉπœ†,2𝑛+1(π‘₯) are estimated in [6]. Moreover, using the results of [6], the Lebesgue constants of Hermite-FejΓ©r interpolatory process are estimated in [7].

In this paper, we find pointwise upper bounds of 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯) and 𝐹(π‘Ÿ)πœ†,2𝑛+1(π‘₯) for two cases of an odd order and of even order. Using these relations, we investigate asymptotic properties of derivatives of the Stieltjes polynomials πΈπœ†,𝑛+1(π‘₯) and πΉπœ†,2𝑛+1(π‘₯) and we also estimate the values of 𝐸(2β„“)πœ†,𝑛+1(π‘₯) and 𝐹(2β„“)πœ†,2𝑛+1(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and πΉπœ†,2𝑛+1(π‘₯), respectively. Especially, for the value of 𝐹(2β„“)πœ†,2𝑛+1(π‘₯) at the zeros of πΉπœ†,2𝑛+1(π‘₯), we will estimate 𝑃(π‘Ÿ)πœ†,𝑛(π‘₯) and 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯) for an odd π‘Ÿ at the zeros of πΈπœ†,𝑛+1(π‘₯) and π‘ƒπœ†,𝑛(π‘₯), respectively. Finally, we investigate asymptotic representations for the values of 𝐸(2β„“+1)πœ†,𝑛+1 and 𝐹(2β„“+1)πœ†,2𝑛+1(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and πΉπœ†,2𝑛+1(π‘₯) on a closed subset of (βˆ’1,1), respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-FejΓ©r interpolation polynomials.

This paper is organized as follows. In Section 2, we will introduce the main results. In Section 3, we will introduce the known results in order to prove the main results. Finally, we will prove the results in Section 4.

2. Main Results

We first introduce some notations, which we use in the following. For the ultraspherical polynomials π‘ƒπœ†,𝑛, πœ†β‰ 0, we use the normalization π‘ƒπœ†,𝑛(1)=𝑛𝑛+2πœ†βˆ’1ξ€Έ and then we know that π‘ƒπœ†,𝑛(1)βˆΌπ‘›2πœ†βˆ’1. We denote the zeros of π‘ƒπœ†,𝑛 by π‘₯(πœ†)𝜈,𝑛, 𝜈=1,…,𝑛, and the zeros of Stieltjes polynomials πΈπœ†,𝑛+1 by πœ‰(πœ†)πœ‡,𝑛+1, πœ‡=1,…,𝑛+1. We denote the zeros of πΉπœ†,2𝑛+1∢=π‘ƒπœ†,π‘›πΈπœ†,𝑛+1 by 𝑦(πœ†)𝜈,2𝑛+1, 𝜈=1,…,2𝑛+1. All nodes are ordered by increasing magnitude. We set βˆšπœ‘(π‘₯)∢=1βˆ’π‘₯2, and, for any two sequences {𝑏𝑛}𝑛 and {𝑐𝑛}𝑛 of nonzero real numbers (or functions), we write 𝑏𝑛≲𝑐𝑛, if there exists a constant 𝐢>0, independent of 𝑛 (and π‘₯) such that 𝑏𝑛⩽𝐢𝑐𝑛 for 𝑛 large enough and write π‘π‘›βˆΌπ‘π‘› if 𝑏𝑛≲𝑐𝑛 and 𝑐𝑛≲𝑏𝑛. We denote by 𝒫𝑛 the space of polynomials of degree at most 𝑛.

For the Chebyshev polynomial 𝑇𝑛(π‘₯), note that for πœ†=0 and πœ†=1𝐸0,𝑛+1(π‘₯)=2π‘›πœ‹ξ€·π‘‡π‘›+1(π‘₯)βˆ’π‘‡π‘›βˆ’1𝐸(π‘₯)1,𝑛+12(π‘₯)=πœ‹π‘‡π‘›+1(π‘₯).(2.1) Therefore, we will consider πΈπœ†,𝑛+1(π‘₯) for 0<πœ†<1.

Theorem 2.1. Let 0<πœ†<1 and π‘Ÿβ©Ύ1 be a positive integer. Then, for all π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)β‰²π‘›π‘Ÿ+1βˆ’πœ†πœ‘1βˆ’π‘Ÿβˆ’πœ†(π‘₯).(2.2) Moreover, one has maxπ‘₯∈[βˆ’1,1]||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)βˆΌπ‘›2π‘Ÿ,(2.3) and especially one has, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)βˆΌπ‘›2π‘Ÿ.(2.4)

Theorem 2.2. Let 0<πœ†<1 and π‘Ÿβ©Ύ1 be a positive integer. Then, for all π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||𝐹(π‘Ÿ)πœ†,2𝑛+1||(π‘₯)β‰²π‘›π‘Ÿπœ‘1βˆ’2πœ†βˆ’π‘Ÿ(π‘₯).(2.5) Moreover, one has maxπ‘₯∈[βˆ’1,1]||𝐹(π‘Ÿ)πœ†,2𝑛+1||(π‘₯)≲𝑛2πœ†+2π‘Ÿβˆ’1,(2.6) and especially, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||𝐹(π‘Ÿ)πœ†,2𝑛+1||(π‘₯)βˆΌπ‘›2πœ†+2π‘Ÿβˆ’1.(2.7)

In the following, we also estimate the values of 𝐸(2β„“)πœ†,𝑛+1(π‘₯) and 𝐹(2β„“)πœ†,2𝑛+1(π‘₯), β„“β©Ύ1 at the zeros {πœ‰(πœ†)πœ‡,𝑛+1} of πΈπœ†,𝑛+1(π‘₯) and the zeros {𝑦(πœ†)𝜈,2𝑛+1} of πΉπœ†,2𝑛+1(π‘₯), respectively.

Theorem 2.3. Let 0<πœ†<1 and π‘Ÿβ©Ύ2 be an even integer. For 1β©½πœ‡β©½π‘›+1, one has |||𝐸(π‘Ÿ)πœ†,𝑛+1ξ‚€πœ‰(πœ†)πœ‡,𝑛+1|||β‰²π‘›π‘Ÿπœ‘βˆ’π‘Ÿξ‚€πœ‰(πœ†)πœ‡,𝑛+1.(2.8)

Theorem 2.4. Let 0<πœ†<1 and π‘Ÿβ©Ύ2 be an even integer. For 1⩽𝜈⩽2𝑛+1, one has |||𝐹(π‘Ÿ)πœ†,2𝑛+1𝑦(πœ†)𝜈,2𝑛+1|||β‰²π‘›π‘Ÿβˆ’1+πœ†πœ‘βˆ’π‘Ÿβˆ’πœ†ξ‚€π‘¦(πœ†)𝜈,2𝑛+1.(2.9)

Finally, we obtain the asymptotic representations for the values of 𝐸(2β„“+1)πœ†,𝑛+1(π‘₯) and 𝐹(2β„“+1)πœ†,2𝑛+1(π‘₯) at the zeros of πΈπœ†,𝑛+1(π‘₯) and πΉπœ†,2𝑛+1(π‘₯) on a closed subset of (βˆ’1,1), respectively.

Theorem 2.5. Let 0<πœ†<1 and 0<πœ€<1. Suppose |πœ‰(πœ†)πœ‡,𝑛+1|β©½1βˆ’πœ€. Then, (a)𝐸(2β„“+1)πœ†,𝑛+1ξ‚€πœ‰(πœ†)πœ‡,𝑛+1=(βˆ’1)β„“(𝑛+1)2β„“πœ‘βˆ’2β„“ξ‚€πœ‰(πœ†)πœ‡,𝑛+1ξ‚πΈξ…žπœ†,𝑛+1ξ‚€πœ‰(πœ†)πœ‡,𝑛+1𝑛+𝑂2β„“+1ξ€Έ,(2.10)(b)𝑃(2β„“)πœ†,π‘›ξ‚€πœ‰(πœ†)πœ‡,𝑛+1=(βˆ’1)ℓ𝑛ℓ(𝑛+2πœ†)β„“πœ‘βˆ’2β„“ξ‚€πœ‰(πœ†)πœ‡,𝑛+1ξ‚π‘ƒπœ†,π‘›ξ‚€πœ‰(πœ†)πœ‡,𝑛+1𝑛+𝑂2πœ†+2β„“βˆ’3ξ€Έ.(2.11) In addition, 𝑃(2β„“)πœ†,π‘›ξ‚€πœ‰(πœ†)πœ‡,𝑛+1=(βˆ’1)β„“(𝑛+1)2β„“πœ‘βˆ’2β„“ξ‚€πœ‰(πœ†)πœ‡,𝑛+1ξ‚π‘ƒπœ†,π‘›ξ‚€πœ‰(πœ†)πœ‡,𝑛+1𝑛+𝑂2β„“+πœ†βˆ’2ξ€Έ.(2.12)

Theorem 2.6. Let 0<πœ†<1 and 0<πœ€<1. Suppose |π‘₯(πœ†)𝜈,𝑛|β©½1βˆ’πœ€. Then, (a)𝐸(2β„“)πœ†,𝑛+1ξ‚€π‘₯(πœ†)𝜈,𝑛=(βˆ’1)β„“(𝑛+1)2β„“πœ‘βˆ’2β„“ξ‚€π‘₯(πœ†)𝜈,π‘›ξ‚πΈπœ†,𝑛+1ξ‚€π‘₯(πœ†)𝜈,𝑛𝑛+𝑂2β„“ξ€Έ,(2.13)(b)𝑃(2β„“+1)πœ†,𝑛π‘₯(πœ†)𝜈,𝑛=(βˆ’1)ℓ𝑛ℓ(𝑛+2πœ†)β„“πœ‘βˆ’2β„“ξ‚€π‘₯(πœ†)𝜈,π‘›ξ‚π‘ƒξ…žπœ†,𝑛π‘₯(πœ†)𝜈,𝑛𝑛+π‘‚πœ†+2β„“βˆ’2ξ€Έ.(2.14) In addition, 𝑃(2β„“+1)πœ†,𝑛π‘₯(πœ†)𝜈,𝑛=(βˆ’1)β„“(𝑛+1)2β„“πœ‘βˆ’2β„“ξ‚€π‘₯(πœ†)𝜈,π‘›ξ‚π‘ƒξ…žπœ†,𝑛π‘₯(πœ†)𝜈,𝑛𝑛+π‘‚πœ†+2β„“βˆ’1ξ€Έ.(2.15)

Theorem 2.7. Let 0<πœ†<1 and 0<πœ€<1. Suppose |𝑦(πœ†)𝜈,2𝑛+1|β©½1βˆ’πœ€. Then, one has, for a positive integer β„“β©Ύ1, 𝐹(2β„“+1)πœ†,2𝑛+1𝑦(πœ†)𝜈,2𝑛+1=𝑐ℓ(βˆ’1)β„“(𝑛+1)2β„“πœ‘βˆ’2ℓ𝑦(πœ†)𝜈,2𝑛+1ξ‚πΉξ…žπœ†,2𝑛+1𝑦(πœ†)𝜈,2𝑛+1𝑛+π‘‚πœ†+2β„“ξ€Έ,(2.16) where 𝑐ℓ=4β„“βˆ’2β„“βˆ’1.

3. The Known Results

In this section, we will introduce the known results in [4, 6, 9] to prove main results.

Proposition 3.1. (a) Let πœ†>βˆ’1/2. Then, π‘ƒπœ†,𝑛(π‘₯) satisfies the second-order differential equation as follows: ξ€·1βˆ’π‘₯2ξ€Έπ‘ƒξ…žξ…žπœ†,𝑛(π‘₯)βˆ’(2πœ†+1)π‘₯π‘ƒξ…žπœ†,𝑛(π‘₯)+𝑛(𝑛+2πœ†)π‘ƒπœ†,𝑛(π‘₯)=0.(3.1)
(b) Let πœ†>βˆ’1/2. Then, π‘ƒξ…žπœ†,𝑛(π‘₯)=2πœ†π‘ƒπœ†+1,π‘›βˆ’1(π‘₯).(3.2)
(c) Let πœ†>βˆ’1/2. Then, for 1β©½πœ‡β©½π‘›+1, |||π‘ƒξ…žπœ†,π‘›ξ‚€πœ‰(πœ†)πœ‡,𝑛+1|||≲𝑛2πœ†βˆ’1πœ‘βˆ’2ξ‚€πœ‰(πœ†)πœ‡,𝑛+1.(3.3)
(d) Let πœ†>βˆ’1/2. Then, for 1β©½πœˆβ©½π‘›, |||π‘ƒξ…žπœ†,𝑛π‘₯(πœ†)𝜈,𝑛|||βˆΌπ‘›πœ†πœ‘βˆ’πœ†βˆ’1ξ‚€π‘₯(πœ†)𝜈,𝑛.(3.4)
(e) Let πœ†>βˆ’1/2. Then, for π‘₯∈(βˆ’1,1) and π‘Ÿβ©Ύ0, ||𝑃(π‘Ÿ)πœ†,𝑛||(π‘₯)β‰²π‘›πœ†+π‘Ÿβˆ’1πœ‘βˆ’πœ†βˆ’π‘Ÿ(π‘₯).(3.5)
(f) Let 0β©½πœ†β©½1. Then, for 1β©½πœˆβ©½π‘›, |||πΈξ…žπœ†,𝑛+1ξ‚€π‘₯(πœ†)𝜈,𝑛|||β‰²π‘›πœ‘βˆ’1ξ‚€π‘₯(πœ†)𝜈,𝑛.(3.6)
(g) Let πœ†>βˆ’1/2 and π‘Ÿβ©Ύ0. Then, π‘ƒπœ†,𝑛(π‘₯) satisfies the higher-order differential equation as follows: ξ€·1βˆ’π‘₯2𝑃(π‘Ÿ+2)πœ†,𝑛(π‘₯)βˆ’(2πœ†+2π‘Ÿ+1)π‘₯𝑃(π‘Ÿ+1)πœ†,𝑛𝑛(π‘₯)+2𝑃+2πœ†π‘›βˆ’π‘Ÿ(2πœ†βˆ’π‘Ÿ+2)(π‘Ÿ)πœ†,𝑛(π‘₯)=0.(3.7)

Proof. (a) It is from [9, (4.2.1)]. (b) It is from [9, (4.7.14)]. (c) It is from [6, Lemma 3.4]. (d) It is from [9, (8.9.7)]. (e) For π‘Ÿ=0, it follows from [9, (7.33.5)], and, for π‘Ÿβ©Ύ1, it comes from (b) and the case of π‘Ÿ=0. (f) It is from [6, Lemma 3.3 (3.23)]. (g) Equation (3.7) comes from (a).

Proposition 3.2 (see [4]). Let 0<πœ†<1. Let πœ‰(πœ†)πœ‡,𝑛+1∢=cosπœƒ(πœ†)πœ‡,𝑛+1, πœ‡=1,…,𝑛+1 and 𝑦(πœ†)𝜈,2𝑛+1∢=cosπœ“(πœ†)𝜈,2𝑛+1, 𝜈=1,…,2𝑛+1. Then, for πœ‡=0,1,…,𝑛+2 and 𝜈=0,1,…,2𝑛+2, |||πœƒ(πœ†)πœ‡,𝑛+1βˆ’πœƒ(πœ†)πœ‡+1,𝑛+1|||∼||πœ“(πœ†)𝜈,2𝑛+1βˆ’πœ“(πœ†)𝜈+1,2𝑛+1||βˆΌπ‘›βˆ’1,(3.8) where πœ“(πœ†)0,2𝑛+1∢=πœƒ(πœ†)0,𝑛+1∢=πœ‹ and πœ“(πœ†)2𝑛+2,2𝑛+1∢=πœƒ(πœ†)𝑛+2,𝑛+1∢=0.

Proposition 3.3 ([6, Proposition  2.3]). Let 0<πœ†<1. Then, for all π‘₯∈[βˆ’1,1], ξ€·1βˆ’π‘₯2ξ€ΈπΈξ…žξ…žπœ†,𝑛+1(π‘₯)βˆ’π‘₯πΈξ…žπœ†,𝑛+1(π‘₯)+(𝑛+1)2πΈπœ†,𝑛+1(π‘₯)=πΌπœ†,𝑛(π‘₯),(3.9) where πΌπœ†,𝑛8(π‘₯)=𝛾𝑛(πœ†)[(𝑛+1)/2]ξ“πœˆ=1(𝑛+1βˆ’πœˆ)πœˆπ›Ό(πœ†)𝜈,𝑛𝑇𝑛+1βˆ’2𝜈(π‘₯).(3.10) Then πΌπœ†,𝑛(π‘₯) is a polynomial of degree π‘›βˆ’1 satisfying maxπ‘₯∈[βˆ’1,1]||πΌπœ†,𝑛||(π‘₯)≲𝑛2.(3.11)

Proposition 3.4 ([4, Theorem  2.1]). Let 0<πœ†<1. Then, for 𝑛⩾0, ||πΈπœ†,𝑛+1||(π‘₯)≲𝑛1βˆ’πœ†πœ‘1βˆ’πœ†(π‘₯)+1βˆ’1β©½π‘₯β©½1.(3.12) Furthermore, πΈπœ†,𝑛+1(1)≳1.

Proposition 3.5 ([6, Theorem  2.5]). Let 0<πœ†<1. (a)For all π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||πΈξ…žπœ†,𝑛+1||(π‘₯)≲𝑛2βˆ’πœ†πœ‘βˆ’πœ†(π‘₯).(3.13) Moreover, one has, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||πΈξ…žπœ†,𝑛+1||(π‘₯)βˆΌπ‘›2.(3.14)(b)For all π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||πΈξ…žξ…žπœ†,𝑛+1||(π‘₯)≲𝑛3βˆ’πœ†πœ‘βˆ’1βˆ’πœ†(π‘₯).(3.15) Moreover, one has, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||πΈξ…žξ…žπœ†,𝑛+1||(π‘₯)βˆΌπ‘›4.(3.16)

Proposition 3.6 ([6, Corollary  2.6]). Let 0<πœ†<1. Then, for all π‘₯∈[βˆ’1,1], ξ€·1βˆ’π‘₯2ξ€ΈπΉξ…žξ…žπœ†,2𝑛+1(π‘₯)βˆ’π‘₯πΉξ…žπœ†,2𝑛+1ξ€·(π‘₯)+2𝑛2𝐹+2(1+πœ†)𝑛+1πœ†,2𝑛+1(π‘₯)=π½πœ†,𝑛(π‘₯).(3.17) Here, π½πœ†,𝑛(π‘₯) is a polynomial of degree of 2𝑛+1 defined in (4.37) such that, for π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||π½πœ†,𝑛||(π‘₯)≲𝑛2πœ‘1βˆ’2πœ†(π‘₯)(3.18) and, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||π½πœ†,𝑛||(π‘₯)≲𝑛1+2πœ†.(3.19)

Proposition 3.7 ([6, Corollary  2.7]). Let 0<πœ†<1. (a)For all π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||πΉξ…žπœ†,2𝑛+1||(π‘₯)β‰²π‘›πœ‘βˆ’2πœ†(π‘₯).(3.20) Moreover, one has, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||πΉξ…žπœ†,2𝑛+1||(π‘₯)βˆΌπ‘›1+2πœ†.(3.21)(b)For all π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||πΉξ…žξ…žπœ†,2𝑛+1||(π‘₯)≲𝑛2πœ‘βˆ’1βˆ’2πœ†(π‘₯).(3.22) Moreover, one has, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1], ||πΉξ…žξ…žπœ†,2𝑛+1||(π‘₯)βˆΌπ‘›3+2πœ†.(3.23)

Proposition 3.8 ([4, Lemma  5.5]). Let 0<πœ†<1. Then, for πœ‡=1,2,…,𝑛+1, |||πΈξ…žπœ†,𝑛+1ξ‚€πœ‰(πœ†)πœ‡,𝑛+1|||βˆΌπ‘›2βˆ’πœ†πœ‘βˆ’πœ†ξ‚€πœ‰(πœ†)πœ‡,𝑛+1(3.24) and, for 𝜈=1,2,…,2𝑛+1, |||πΉξ…žπœ†,2𝑛+1𝑦(πœ†)𝜈,2𝑛+1|||βˆΌπ‘›πœ‘βˆ’2πœ†ξ‚€π‘¦(πœ†)𝜈,2𝑛+1.(3.25)

We now estimate the second derivatives at the zeros of πΈπœ†,𝑛+1 and πΉπœ†,2𝑛+1.

Proposition 3.9 ([6, Theorem  2.9]). Let 0<πœ†<1. Then, for πœ‡=1,2,…,𝑛+1, |||πΈξ…žξ…žπœ†,𝑛+1ξ‚€πœ‰(πœ†)πœ‡,𝑛+1|||≲𝑛2πœ‘βˆ’2ξ‚€πœ‰(πœ†)πœ‡,𝑛+1(3.26) and, for 𝜈=1,2,…,2𝑛+1, |||πΉξ…žξ…žπœ†,2𝑛+1𝑦(πœ†)𝜈,2𝑛+1|||≲𝑛1+πœ†πœ‘βˆ’2βˆ’πœ†ξ‚€π‘¦(πœ†)𝜈,2𝑛+1.(3.27)

4. The Proofs of Main Results

In this section, we let 0<πœ†<1 and π‘š=⌊(𝑛+1/2)βŒ‹. A representation of Stieltjes polynomials πΈπœ†,𝑛+1(π‘₯) is (cf. [1, 10]) 𝛾𝑛(πœ†)2πΈπœ†,𝑛+1(cosπœƒ)=𝛼(πœ†)0,𝑛cos(𝑛+1)πœƒ+𝛼(πœ†)1,𝑛𝛼cos(π‘›βˆ’1)πœƒ+β‹―+(πœ†)𝑛/2,𝑛1cosπœƒ,𝑛even2𝛼(πœ†)𝑛+1/2,𝑛,𝑛odd,(4.1) where 𝛼(πœ†)0,𝑛=𝑓(πœ†)0,𝑛=1,πœˆξ“πœ‡=0𝛼(πœ†)πœ‡,𝑛𝑓(πœ†)πœˆβˆ’πœ‡,𝑛𝑓=0,𝜈=1,2,…,(πœ†)𝜈,π‘›ξ‚€πœ†βˆΆ=1βˆ’πœˆπœ†ξ‚ξ‚€1βˆ’ξ‚π›Ύπ‘›+𝜈+πœ†,𝜈=1,2,…,𝑛(πœ†)=βˆšπœ‹Ξ“(𝑛+2πœ†)βˆΌβˆšΞ“(𝑛+πœ†+1)πœ‹π‘›πœ†βˆ’1.(4.2)

In the following, we state the asymptotic differential relation of the higher order of πΈπœ†,𝑛+1.

Lemma 4.1. Let 0<πœ†<1. Then, for all π‘₯∈[βˆ’1,1] and π‘Ÿβ©Ύ2, ξ€·1βˆ’π‘₯2𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯)=(2π‘Ÿβˆ’3)π‘₯𝐸(π‘Ÿβˆ’1)πœ†,𝑛+1ξ€·(π‘₯)+(π‘Ÿβˆ’2)2βˆ’(𝑛+1)2𝐸(π‘Ÿβˆ’2)πœ†,𝑛+1(π‘₯)+𝐼(π‘Ÿβˆ’2)πœ†,𝑛(π‘₯)(4.3) and, for π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||𝐼(π‘Ÿβˆ’2)πœ†,𝑛||(π‘₯)β‰²π‘›π‘Ÿπœ‘2βˆ’π‘Ÿ(π‘₯).(4.4) Here, πΌπœ†,𝑛(π‘₯) is a polynomial of degree π‘›βˆ’1 defined in (3.10); πΌπœ†,𝑛8(π‘₯)=π›Ύπ‘›π‘š(πœ†)ξ“πœˆ=1(𝑛+1βˆ’πœˆ)πœˆπ›Ό(πœ†)𝜈,𝑛𝑇𝑛+1βˆ’2𝜈(π‘₯),(4.5) such that maxπ‘₯∈[βˆ’1,1]||𝐼(π‘Ÿβˆ’2)πœ†,𝑛||(π‘₯)≲𝑛2π‘Ÿβˆ’2.(4.6)

Proof. For π‘Ÿβ©Ύ2, (4.3) is obtained by π‘Ÿβˆ’2 times differentiation of (3.9). Equation (4.6) follows by (3.11) and the use of Markov-Bernstein inequality. Now, we prove (4.4). We know that the Chebyshev polynomial 𝑇𝑛(π‘₯) satisfies the second-order differential equation ξ€·1βˆ’π‘₯2ξ€Έπ‘‡π‘›ξ…žξ…ž(π‘₯)βˆ’π‘₯π‘‡ξ…žπ‘›(π‘₯)+𝑛2𝑇𝑛(π‘₯)=0,(4.7) so we have, by π‘Ÿβˆ’2 times differentiation of (4.7), ξ€·1βˆ’π‘₯2𝑇𝑛(π‘Ÿ)(π‘₯)βˆ’(2π‘Ÿβˆ’3)π‘₯𝑇𝑛(π‘Ÿβˆ’1)ξ€·(π‘₯)βˆ’(π‘Ÿβˆ’2)2βˆ’π‘›2𝑇𝑛(π‘Ÿβˆ’2)(π‘₯)=0.(4.8) Let for a nonnegative integer 𝑗⩾0, πΌπœ†,𝑛,𝑗8(π‘₯)∢=βˆ’π›Ύπ‘›π‘š(πœ†)ξ“πœˆ=1(𝑛+1βˆ’πœˆ)πœˆπ›Ό(πœ†)𝜈,𝑛||𝑇(𝑗)𝑛+1βˆ’2𝜈||.(π‘₯)(4.9) Observe that in the view of Szegâ’s result (cf. [1]) 𝛼(πœ†)1,𝑛<𝛼(πœ†)2,𝑛<𝛼(πœ†)3,𝑛<β‹―<0,0β©½βˆžξ“πœˆ=0𝛼(πœ†)𝜈,𝑛<1.(4.10) Then, since |𝐼(𝑗)πœ†,𝑛(π‘₯)|β©½πΌπœ†,𝑛,𝑗(π‘₯) (note (4.10)), we will prove that, for π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1] and 𝑗⩾0, πΌπœ†,𝑛,𝑗(π‘₯)≲𝑛𝑗+2πœ‘βˆ’π‘—(π‘₯)(4.11) instead of (4.4). Since, from the proof of [6, Proposition  2.3], 80<βˆ’π›Ύπ‘›π‘š(πœ†)ξ“πœˆ=1(𝑛+1βˆ’πœˆ)πœˆπ›Ό(πœ†)𝜈,𝑛≲𝑛2(4.12) and, for π‘₯=cosπœƒ, ||π‘‡ξ…žπ‘›+1βˆ’2𝜈(||=|||(π‘₯)𝑛+1βˆ’2𝜈)sin(𝑛+1βˆ’2𝜈)πœƒ|||sinπœƒβ‰²π‘›πœ‘βˆ’1(π‘₯),(4.13) we obtain that πΌπœ†,𝑛,0(π‘₯)≲𝑛2 and πΌπœ†,𝑛,1(π‘₯)≲𝑛3πœ‘βˆ’1(π‘₯). Using (4.8), we have, for 2⩽𝑗⩽𝑛, πΌπœ†,𝑛,𝑗8(π‘₯)=βˆ’π›Ύπ‘›π‘š(πœ†)ξ“πœˆ=1(𝑛+1βˆ’πœˆ)πœˆπ›Ό(πœ†)𝜈,𝑛||𝑇(𝑗)𝑛+1βˆ’2𝜈||8(π‘₯)β‰€βˆ’π›Ύπ‘›π‘š(πœ†)ξ“πœˆ=1(𝑛+1βˆ’πœˆ)πœˆπ›Ό(πœ†)𝜈,𝑛×(2π‘—βˆ’3)|π‘₯|1βˆ’π‘₯2||𝑇(π‘—βˆ’1)𝑛+1βˆ’2𝜈||+||((π‘₯)π‘—βˆ’2)2βˆ’(𝑛+1βˆ’2𝜈)2||1βˆ’π‘₯2||𝑇(π‘—βˆ’2)𝑛+1βˆ’2𝜈||ξƒͺ≲(π‘₯)(2π‘—βˆ’3)|π‘₯|1βˆ’π‘₯2πΌπœ†,𝑛,π‘—βˆ’1(π‘₯)+(𝑛+1)21βˆ’π‘₯2πΌπœ†,𝑛,π‘—βˆ’2(π‘₯).(4.14) Therefore, (4.11) is proved by the mathematical induction on 𝑗. Consequently, we have (4.4).

We obtain pointwise upper bounds of 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯) for two cases of an odd order and an even order in the following.

Lemma 4.2. Let 0<πœ†<1 and π‘Ÿβ©Ύ2. Let π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1]. If π‘Ÿ is even, then one has ||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)β‰²π‘›π‘Ÿβˆ’2πœ‘βˆ’π‘Ÿ||𝐸(π‘₯)ξ…žπœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿπœ‘βˆ’π‘Ÿ||𝐸(π‘₯)πœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿπœ‘βˆ’π‘Ÿ(π‘₯),(4.15) and, if π‘Ÿ is odd, then one has ||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)β‰²π‘›π‘Ÿβˆ’1πœ‘1βˆ’π‘Ÿ||𝐸(π‘₯)ξ…žπœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿβˆ’1πœ‘βˆ’1βˆ’π‘Ÿ||𝐸(π‘₯)πœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿπœ‘βˆ’π‘Ÿ(π‘₯).(4.16)

Proof. Let π‘Ÿβ©Ύ2. From (4.3) and (4.4), we have, for π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)β‰²πœ‘βˆ’2||𝐸(π‘₯)(π‘Ÿβˆ’1)πœ†,𝑛+1||(π‘₯)+𝑛2πœ‘βˆ’2||𝐸(π‘₯)(π‘Ÿβˆ’2)πœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿπœ‘βˆ’π‘Ÿ(π‘₯)(4.17) and especially ||πΈξ…žξ…žπœ†,𝑛+1||(π‘₯)β‰²πœ‘βˆ’2||𝐸(π‘₯)ξ…žπœ†,𝑛+1||(π‘₯)+𝑛2πœ‘βˆ’2||𝐸(π‘₯)πœ†,𝑛+1||(π‘₯)+𝑛2πœ‘βˆ’2(π‘₯),(4.18) that is, we have (4.15) for π‘Ÿ=2. From Proposition 3.2, we see 1+πœ‰(πœ†)1,𝑛+1, 1βˆ’πœ‰(πœ†)𝑛+1,𝑛+1≳1/𝑛, so we have, for π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], πœ‘βˆ’1(π‘₯)≲𝑛.(4.19) Then, from (4.17) with π‘Ÿ=3 and (4.18), we know that ||𝐸(3)πœ†,𝑛+1||(π‘₯)≲𝑛2πœ‘βˆ’2||𝐸(π‘₯)ξ…žπœ†,𝑛+1||(π‘₯)+𝑛2πœ‘βˆ’4||𝐸(π‘₯)πœ†,𝑛+1||(π‘₯)+𝑛3πœ‘βˆ’3(π‘₯),(4.20) that is, we have (4.16) for π‘Ÿ=3. Assume that (4.15) and (4.16) hold for 3,4,…,π‘Ÿβˆ’1 times differentiation. Let π‘Ÿ be an even number. Then, we have, from (4.17), (4.19), and the assumptions for π‘Ÿβˆ’1 and π‘Ÿβˆ’2, ||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)β‰²π‘›π‘Ÿβˆ’2πœ‘βˆ’π‘Ÿ||𝐸(π‘₯)ξ…žπœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿπœ‘βˆ’π‘Ÿ||𝐸(π‘₯)πœ†,𝑛+1||(π‘₯)+π‘›π‘Ÿπœ‘βˆ’π‘Ÿ(π‘₯),(4.21) that is, we have (4.15). Similarly, we also have (4.16) for an odd π‘Ÿ.

Lemma 4.3. Let βˆ’1<π‘₯1<π‘₯2<β‹―<π‘₯𝑛<1 and 𝑃(π‘₯)∢=π‘₯βˆ’π‘₯1ξ€Έξ€·π‘₯βˆ’π‘₯2ξ€Έβ‹―ξ€·π‘₯βˆ’π‘₯𝑛.(4.22) Then, π‘ƒξ…ž(π‘₯) is a polynomial of degree of π‘›βˆ’1 and has distinct real π‘›βˆ’1 zeros in (βˆ’1,1). Moreover, if one lets βˆ’1<𝑦1<𝑦2<β‹―<π‘¦π‘›βˆ’1<1 be the zeros of π‘ƒξ…ž(π‘₯), then {𝑦𝑖}𝑛𝑖=1 is interlaced with the zeros of 𝑃(π‘₯) that is, π‘₯𝑖<𝑦𝑖<π‘₯𝑖+1, 𝑖=1,2,…,π‘›βˆ’1.

Proof. Since the sign of π‘ƒξ…ž(π‘₯𝑖) is (βˆ’1)π‘›βˆ’π‘–, it is proved.

Lemma 4.4. Let π‘Ÿ be a nonnegative integer. Then, 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯) has distinct 𝑛+1βˆ’π‘Ÿ real zeros on (βˆ’1,1). If one lets {π‘₯𝑛+1(π‘Ÿ,𝑖)}𝑛+1βˆ’π‘Ÿπ‘–=1 be the zeros of the polynomial 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯) with βˆ’1<π‘₯𝑛+1(π‘Ÿ,1)<π‘₯𝑛+1(π‘Ÿ,2)<β‹―<π‘₯𝑛+1(π‘Ÿ,𝑛+1βˆ’π‘Ÿ)<1,(4.23) then one has, for 1β©½π‘Ÿβ©½π‘› and π‘˜=1,…,𝑛+1βˆ’π‘Ÿ, π‘₯𝑛+1(0,π‘˜)<π‘₯𝑛+1(π‘Ÿ,π‘˜).(4.24)

Proof. From Lemma 4.3, we know that 𝐸(π‘Ÿ)πœ†,𝑛+1 has distinct real 𝑛+1βˆ’π‘Ÿ zeros on (βˆ’1,1). By the interlaced zeros property of Lemma 4.3, we see that, for π‘˜=1,…,𝑛+1βˆ’π‘Ÿ, π‘₯𝑛+1(π‘Ÿβˆ’1,π‘˜)<π‘₯𝑛+1(π‘Ÿ,π‘˜)<π‘₯𝑛+1(π‘Ÿβˆ’1,π‘˜+1).(4.25) Thus, (4.24) is proved.

Proof of Theorem 2.1. Let π‘Ÿβ©Ύ1. Equation (2.2) comes from (4.15), (4.16), (3.12), and (3.13). From Propositions 3.4, 3.5, and (4.19), we have maxπ‘₯∈[βˆ’1,1]||𝐸(π‘Ÿ)πœ†,𝑛+1||(π‘₯)βˆΌπ‘›2π‘Ÿ,π‘Ÿ=1,2.(4.26) Hence, using the Markov-Bernstein inequality, we have (2.3). To prove (2.4), we will use the mathematical induction. We use (3.14). The formula (2.3) holds for π‘Ÿ=1 from (3.14). We suppose that, for π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1]βˆͺ[πœ‰(πœ†)𝑛+1,𝑛+1,1] and π‘Ÿβ©Ύ2, ||𝐸(π‘Ÿβˆ’1)πœ†,𝑛+1||(π‘₯)βˆΌπ‘›2(π‘Ÿβˆ’1).(4.27) Then, by Lemma 4.4 and (3.8), we have, for π‘₯∈[πœ‰(πœ†)𝑛+1,𝑛+1,1] and π‘Ÿβ©Ύ2, 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯)𝐸(π‘Ÿβˆ’1)πœ†,𝑛+1=(π‘₯)𝑛+2βˆ’π‘Ÿξ“π‘˜=11π‘₯βˆ’π‘₯𝑛+1β©Ύ1(π‘Ÿβˆ’1,π‘˜)πœ‰(πœ†)𝑛+1,𝑛+1βˆ’π‘₯𝑛+1β©Ύ1(π‘Ÿβˆ’1,𝑛+2βˆ’π‘Ÿ)πœ‰(πœ†)𝑛+1,𝑛+1βˆ’π‘₯𝑛+1=1(0,𝑛+2βˆ’π‘Ÿ)πœ‰(πœ†)𝑛+1,𝑛+1βˆ’πœ‰(πœ†)𝑛+2βˆ’π‘Ÿ,𝑛+1≳1πœ‰(πœ†)𝑛+1,𝑛+1βˆ’πœ‰(πœ†)𝑛,𝑛+1≳𝑛2.(4.28) Here, the last inequality is obtained by Proposition 3.2, that is, πœ‰(πœ†)𝑛+1,𝑛+1βˆ’πœ‰(πœ†)𝑛,𝑛+1=cosπœƒ(πœ†)𝑛+1,𝑛+1βˆ’cosπœƒ(πœ†)𝑛,𝑛+1βˆΌπ‘›βˆ’2.(4.29) Therefore, we have, for π‘₯∈[πœ‰(πœ†)𝑛+1,𝑛+1,1] and π‘Ÿβ©Ύ2, 𝐸(π‘Ÿ)πœ†,𝑛+1(π‘₯)≳𝐸(π‘Ÿβˆ’1)πœ†,𝑛+1(π‘₯)𝑛2βˆΌπ‘›2π‘Ÿ.(4.30) Hence, from (2.3), we have (2.4). For π‘₯∈[βˆ’1,πœ‰(πœ†)1,𝑛+1], the proof is similar.

Lemma 4.5. Let β„“ be a nonnegative integer and π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1]. Then,

|||ξ€·π‘₯πΈπœ†,𝑛+1(π‘₯)π‘ƒξ…žπœ†,𝑛(π‘₯)(β„“)|||≲𝑛ℓ+1πœ‘βˆ’β„“βˆ’2πœ†|||𝐼(π‘₯),(4.31a)πœ†,𝑛(π‘₯)π‘ƒπœ†,𝑛(π‘₯)(β„“)|||≲𝑛ℓ+1+πœ†πœ‘βˆ’β„“βˆ’πœ†||||𝐸(π‘₯),(4.31b)ξ…žπœ†,𝑛+1(π‘₯)π‘ƒξ…žπœ†,𝑛(π‘₯)(β„“)||||≲𝑛ℓ+2πœ‘βˆ’2πœ†βˆ’β„“βˆ’1(π‘₯).(4.31c)

Proof. (a) When β„“=0, it is obvious from (3.5) and (3.12). Now, suppose β„“β©Ύ1. From (3.12), (2.2), and (3.5), we have |||ξ€·π‘₯πΈπœ†,𝑛+1(π‘₯)π‘ƒξ…žπœ†,𝑛(π‘₯)(β„“)|||β‰²ξ“β„“βˆ’1β©½π‘ž+π‘Ÿβ©½β„“||𝐸(π‘ž)πœ†,𝑛+1(π‘₯)𝑃(π‘Ÿ+1)πœ†,𝑛||≲(π‘₯)β„“βˆ’1β©½π‘ž+π‘Ÿβ©½β„“π‘›π‘ž+π‘Ÿ+1πœ‘βˆ’(π‘ž+π‘Ÿ)βˆ’2πœ†(π‘₯)≲𝑛ℓ+1πœ‘βˆ’β„“βˆ’2πœ†(π‘₯).(4.32)
(b) From (4.4) and (3.5), we have |||ξ€·πΌπœ†,𝑛(π‘₯)π‘ƒπœ†,𝑛(π‘₯)(β„“)|||β‰²ξ“π‘ž+π‘Ÿ=β„“||𝐼(π‘ž)πœ†,𝑛(π‘₯)𝑃(π‘Ÿ)πœ†,𝑛||(π‘₯)≲𝑛ℓ+1+πœ†πœ‘βˆ’β„“βˆ’πœ†(π‘₯).(4.33)
(c) Similarly to the proof of (a), we have, from (2.2) and (3.5), ||||ξ‚€πΈξ…žπœ†,𝑛+1(π‘₯)π‘ƒξ…žπœ†,𝑛(π‘₯)(β„“)||||β‰²ξ“π‘ž+π‘Ÿ=β„“||𝐸(π‘ž+1)πœ†,𝑛+1(π‘₯)𝑃(π‘Ÿ+1)πœ†,𝑛||(π‘₯)≲𝑛ℓ+2πœ‘βˆ’2πœ†βˆ’β„“βˆ’1(π‘₯).(4.34)

Lemma 4.6. Let 0<πœ†<1. Then, for all π‘₯∈[βˆ’1,1] and π‘Ÿβ©Ύ2, ξ€·1βˆ’π‘₯2𝐹(π‘Ÿ)πœ†,2𝑛+1(π‘₯)=(2π‘Ÿβˆ’5)π‘₯𝐹(π‘Ÿβˆ’1)πœ†,2𝑛+1+ξ€·(π‘₯)(π‘Ÿβˆ’2)2βˆ’(𝑛+1)2ξ€ΈπΉβˆ’π‘›(𝑛+2πœ†)(π‘Ÿβˆ’2)πœ†,2𝑛+1(π‘₯)+𝐽(π‘Ÿβˆ’2)πœ†,𝑛(π‘₯)(4.35) and, for π‘₯∈[πœ‰(πœ†)1,𝑛+1,πœ‰(πœ†)𝑛+1,𝑛+1], ||𝐽(π‘Ÿβˆ’2)πœ†,𝑛||(π‘₯)β‰²π‘›π‘Ÿπœ‘βˆ’2πœ†βˆ’π‘Ÿ+3(π‘₯).(4.36) Here, π½πœ†,𝑛(π‘₯) is a polynomial of degree of 2𝑛+1 defined as follows: π½πœ†,𝑛(π‘₯)=2πœ†π‘₯πΈπœ†,𝑛+1(π‘₯)π‘ƒξ…žπœ†,𝑛(π‘₯)+21βˆ’π‘₯2ξ€ΈπΈξ…žπœ†,𝑛+1(π‘₯)π‘ƒξ…žπœ†,𝑛(π‘₯)+πΌπœ†,𝑛(π‘₯)π‘ƒπœ†,𝑛(π‘₯).(4.37) Furthermore, one has ||𝐽(π‘Ÿβˆ’2)πœ†,𝑛||(1)≲𝑛2πœ†+2π‘Ÿβˆ’3.(4.38)

Proof. Similarly to the proof of Lemma 4.1, (4.35) is obtained by π‘Ÿβˆ’2 times differentiation of the second-order differential relation with respect to πΉπœ†,2𝑛+1(π‘₯), that is, (3.17). So it is sufficient to prove (4.36) and (4.38). From (4.37), we know that 𝐽(π‘Ÿβˆ’