Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Abstract

Let 𝑤_𝜆(𝑥)∶=(1−𝑥²)^𝜆−1/2 and 𝑃_𝜆,𝑛(𝑥) be the ultraspherical polynomials with respect to 𝑤_𝜆(𝑥). Then, we denote the Stieltjes polynomials with respect to 𝑤_𝜆(𝑥) by 𝐸_𝜆,𝑛+1(𝑥) satisfying ∫¹⁻¹𝑤_𝜆(𝑥)𝑃_𝜆,𝑛(𝑥)𝐸_𝜆,𝑛+1(𝑥)𝑥^𝑚𝑑𝑥=0, 0≤𝑚<𝑛+1, ∫¹⁻¹𝑤_𝜆(𝑥)𝑃_𝜆,𝑛(𝑥)𝐸_𝜆,𝑛+1(𝑥)𝑥^𝑚𝑑𝑥≠0, 𝑚=𝑛+1. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials 𝐸_𝜆,𝑛+1(𝑥) and the product 𝐸_𝜆,𝑛+1(𝑥)𝑃_𝜆,𝑛(𝑥). Especially, we estimate the even-order derivative values of 𝐸_𝜆,𝑛+1(𝑥) and 𝐸_𝜆,𝑛+1(𝑥)𝑃_𝜆,𝑛(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and the product 𝐸_𝜆,𝑛+1(𝑥)𝑃_𝜆,𝑛(𝑥), respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of 𝐸_𝜆,𝑛+1(𝑥) and 𝐸_𝜆,𝑛+1(𝑥)𝑃_𝜆,𝑛(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝐸_𝜆,𝑛+1(𝑥)𝑃_𝜆,𝑛(𝑥) on a closed subset of (−1,1), respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation polynomials.

1. Introduction

Consider the generalized Stieltjes polynomials 𝐸_𝜆,𝑛+1(𝑥) defined (up to a multiplicative constant) by ¹⁻¹𝑤_𝜆(𝑥)𝑃_𝜆,𝑛(𝑥)𝐸_𝜆,𝑛+1(𝑥)𝑥^𝑘𝑑𝑥=0,𝑘=0,1,2,…,𝑛,𝑛⩾1,(1.1) where 𝑤_𝜆(𝑥)=(1−𝑥²)^𝜆−1/2, 𝜆>−1/2, and 𝑃_𝜆,𝑛(𝑥) is the 𝑛th ultraspherical polynomial for the weight function 𝑤_𝜆(𝑥).

The polynomials 𝐸_𝜆,𝑛+1(𝑥), introduced by Stieltjes and studied by Szegö, have been used in numerical integration, whereas the polynomials 𝑃_𝜆,𝑛(𝑥)𝐸_𝜆,𝑛+1(𝑥) have been used in extended Lagrange interpolation. In this paper, we will prove pointwise and asymptotic estimates for the higher-order derivatives of 𝐸_𝜆,𝑛+1(𝑥) and 𝑃_𝜆,𝑛(𝑥)𝐸_𝜆,𝑛+1(𝑥). It is well known that these kind of estimates are useful for studying interpolation processes with multiple nodes.

In 1934, G. Szegö [1] showed that the zeros of the generalized Stieltjes polynomials 𝐸_𝜆,𝑛+1(𝑥) are real and inside [−1,1] and interlace with the zeros of 𝑃_𝜆,𝑛(𝑥) whenever 0⩽𝜆⩽2. Recently, several authors [2–8] studied further interesting properties for these Stieltjes polynomials. Ehrich and Mastroianni [3, 4] gave accurate pointwise bounds of 𝐸_𝜆,𝑛+1(𝑥)(0⩽𝜆⩽1) and the product 𝐹_𝜆,2𝑛+1∶=𝐸_𝜆,𝑛+1(𝑥)𝑃_𝜆,𝑛(𝑥)(0⩽𝜆⩽1) on [−1,1], and they estimated asymptotic representations for 𝐸^{𝜆,𝑛+1}(𝑥) and 𝐹^{𝜆,2𝑛+1}(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝐹_𝜆,2𝑛+1(𝑥), respectively. In [6], pointwise upper bounds of 𝐸^{𝜆,𝑛+1}(𝑥), 𝐸^{𝜆,𝑛+1}(𝑥), 𝐹^{𝜆,2𝑛+1}(𝑥), and 𝐹^{𝜆,2𝑛+1}(𝑥) are obtained using the asymptotic differential relations of the first and the second order for the Stieltjes polynomials 𝐸_𝜆,𝑛+1(𝑥)(0⩽𝜆⩽1) and 𝐹_𝜆,2𝑛+1(𝑥)(0⩽𝜆⩽1). Also the values of 𝐸^{𝜆,𝑛+1}(𝑥) and 𝐹^{𝜆,2𝑛+1}(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝐹_𝜆,2𝑛+1(𝑥) are estimated in [6]. Moreover, using the results of [6], the Lebesgue constants of Hermite-Fejér interpolatory process are estimated in [7].

In this paper, we find pointwise upper bounds of 𝐸^{(𝑟)𝜆,𝑛+1}(𝑥) and 𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥) for two cases of an odd order and of even order. Using these relations, we investigate asymptotic properties of derivatives of the Stieltjes polynomials 𝐸_𝜆,𝑛+1(𝑥) and 𝐹_𝜆,2𝑛+1(𝑥) and we also estimate the values of 𝐸^{(2ℓ)𝜆,𝑛+1}(𝑥) and 𝐹^{(2ℓ)𝜆,2𝑛+1}(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝐹_𝜆,2𝑛+1(𝑥), respectively. Especially, for the value of 𝐹^{(2ℓ)𝜆,2𝑛+1}(𝑥) at the zeros of 𝐹_𝜆,2𝑛+1(𝑥), we will estimate 𝑃^{(𝑟)𝜆,𝑛}(𝑥) and 𝐸^{(𝑟)𝜆,𝑛+1}(𝑥) for an odd 𝑟 at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝑃_𝜆,𝑛(𝑥), respectively. Finally, we investigate asymptotic representations for the values of 𝐸^{(2ℓ+1)𝜆,𝑛+1} and 𝐹^{(2ℓ+1)𝜆,2𝑛+1}(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝐹_𝜆,2𝑛+1(𝑥) on a closed subset of (−1,1), respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér 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 𝑃_𝜆,𝑛, 𝜆≠0, we use the normalization 𝑃_𝜆,𝑛(1)=^{𝑛𝑛+2𝜆−1} and then we know that 𝑃_𝜆,𝑛(1)∼𝑛^2𝜆−1. We denote the zeros of 𝑃_𝜆,𝑛 by 𝑥^{(𝜆)𝜈,𝑛}, 𝜈=1,…,𝑛, and the zeros of Stieltjes polynomials 𝐸_𝜆,𝑛+1 by 𝜉^{(𝜆)𝜇,𝑛+1}, 𝜇=1,…,𝑛+1. We denote the zeros of 𝐹_𝜆,2𝑛+1∶=𝑃_𝜆,𝑛𝐸_𝜆,𝑛+1 by 𝑦^{(𝜆)𝜈,2𝑛+1}, 𝜈=1,…,2𝑛+1. All nodes are ordered by increasing magnitude. We set 𝜑(𝑥)∶=√1−𝑥², and, for any two sequences {𝑏_𝑛}_𝑛 and {𝑐_𝑛}_𝑛 of nonzero real numbers (or functions), we write 𝑏_𝑛≲𝑐_𝑛, if there exists a constant 𝐶>0, independent of 𝑛 (and 𝑥) such that 𝑏_𝑛⩽𝐶𝑐_𝑛 for 𝑛 large enough and write 𝑏_𝑛∼𝑐_𝑛 if 𝑏_𝑛≲𝑐_𝑛 and 𝑐_𝑛≲𝑏_𝑛. We denote by 𝒫_𝑛 the space of polynomials of degree at most 𝑛.

For the Chebyshev polynomial 𝑇_𝑛(𝑥), note that for 𝜆=0 and 𝜆=1𝐸_0,𝑛+1(𝑥)=2𝑛𝜋𝑇^𝑛+1𝐸(𝑥)−𝑇_𝑛−1(𝑥)_1,𝑛+1(𝑥)=2𝜋𝑇^𝑛+1(𝑥).(2.1) Therefore, we will consider 𝐸_𝜆,𝑛+1(𝑥) for 0<𝜆<1.

Theorem 2.1. Let 0<𝜆<1 and 𝑟⩾1 be a positive integer. Then, for all 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||≲𝑛^{𝑟+1−𝜆}𝜑^{1−𝑟−𝜆}(𝑥).(2.2) Moreover, one has max_{𝑥∈[−1,1]}||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||∼𝑛^2𝑟,(2.3) and especially one has, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||∼𝑛^2𝑟.(2.4)

Theorem 2.2. Let 0<𝜆<1 and 𝑟⩾1 be a positive integer. Then, for all 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥)||≲𝑛^𝑟𝜑^{1−2𝜆−𝑟}(𝑥).(2.5) Moreover, one has max_{𝑥∈[−1,1]}||𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥)||≲𝑛^{2𝜆+2𝑟−1},(2.6) and especially, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥)||∼𝑛^{2𝜆+2𝑟−1}.(2.7)

In the following, we also estimate the values of 𝐸^{(2ℓ)𝜆,𝑛+1}(𝑥) and 𝐹^{(2ℓ)𝜆,2𝑛+1}(𝑥), ℓ⩾1 at the zeros {𝜉^{(𝜆)𝜇,𝑛+1}} of 𝐸_𝜆,𝑛+1(𝑥) and the zeros {𝑦^{(𝜆)𝜈,2𝑛+1}} of 𝐹_𝜆,2𝑛+1(𝑥), respectively.

Theorem 2.3. Let 0<𝜆<1 and 𝑟⩾2 be an even integer. For 1⩽𝜇⩽𝑛+1, one has |||𝐸_{(𝑟)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^𝑟𝜑^−𝑟𝜉^{(𝜆)𝜇,𝑛+1}.(2.8)

Theorem 2.4. Let 0<𝜆<1 and 𝑟⩾2 be an even integer. For 1⩽𝜈⩽2𝑛+1, one has |||𝐹_{(𝑟)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}|||≲𝑛^{𝑟−1+𝜆}𝜑^{−𝑟−𝜆}𝑦^{(𝜆)𝜈,2𝑛+1}.(2.9)

Finally, we obtain the asymptotic representations for the values of 𝐸^{(2ℓ+1)𝜆,𝑛+1}(𝑥) and 𝐹^{(2ℓ+1)𝜆,2𝑛+1}(𝑥) at the zeros of 𝐸_𝜆,𝑛+1(𝑥) and 𝐹_𝜆,2𝑛+1(𝑥) on a closed subset of (−1,1), respectively.

Theorem 2.5. Let 0<𝜆<1 and 0<𝜀<1. Suppose |𝜉^{(𝜆)𝜇,𝑛+1}|⩽1−𝜀. Then, (a)𝐸^{(2ℓ+1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}=(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^2ℓ+1,(2.10)(b)𝑃^{(2ℓ)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}=(−1)^ℓ𝑛^ℓ(𝑛+2𝜆)^ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^{2𝜆+2ℓ−3}.(2.11) In addition, 𝑃^{(2ℓ)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}=(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^{2ℓ+𝜆−2}.(2.12)

Theorem 2.6. Let 0<𝜆<1 and 0<𝜀<1. Suppose |𝑥^{(𝜆)𝜈,𝑛}|⩽1−𝜀. Then, (a)𝐸^{(2ℓ)𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}=(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ𝑥^{(𝜆)𝜈,𝑛}𝐸^𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^2ℓ,(2.13)(b)𝑃^{(2ℓ+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}=(−1)^ℓ𝑛^ℓ(𝑛+2𝜆)^ℓ𝜑^−2ℓ𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^{𝜆+2ℓ−2}.(2.14) In addition, 𝑃^{(2ℓ+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}=(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^{𝜆+2ℓ−1}.(2.15)

Theorem 2.7. Let 0<𝜆<1 and 0<𝜀<1. Suppose |𝑦^{(𝜆)𝜈,2𝑛+1}|⩽1−𝜀. Then, one has, for a positive integer ℓ⩾1, 𝐹^{(2ℓ+1)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}=𝑐^ℓ(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^𝜆+2ℓ,(2.16) where 𝑐_ℓ=4^ℓ−2^ℓ−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 𝜆>−1/2. Then, 𝑃_𝜆,𝑛(𝑥) satisfies the second-order differential equation as follows: 1−𝑥²𝑃^{𝜆,𝑛}(𝑥)−(2𝜆+1)𝑥𝑃^𝜆,𝑛(𝑥)+𝑛(𝑛+2𝜆)𝑃_𝜆,𝑛(𝑥)=0.(3.1)
(b) Let 𝜆>−1/2. Then, 𝑃^𝜆,𝑛(𝑥)=2𝜆𝑃_{𝜆+1,𝑛−1}(𝑥).(3.2)
(c) Let 𝜆>−1/2. Then, for 1⩽𝜇⩽𝑛+1, |||𝑃_𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^2𝜆−1𝜑⁻²𝜉^{(𝜆)𝜇,𝑛+1}.(3.3)
(d) Let 𝜆>−1/2. Then, for 1⩽𝜈⩽𝑛, |||𝑃_𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}|||∼𝑛^𝜆𝜑^−𝜆−1𝑥^{(𝜆)𝜈,𝑛}.(3.4)
(e) Let 𝜆>−1/2. Then, for 𝑥∈(−1,1) and 𝑟⩾0, ||𝑃^{(𝑟)𝜆,𝑛}(𝑥)||≲𝑛^{𝜆+𝑟−1}𝜑^{−𝜆−𝑟}(𝑥).(3.5)
(f) Let 0⩽𝜆⩽1. Then, for 1⩽𝜈⩽𝑛, |||𝐸_{𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛𝜑⁻¹𝑥^{(𝜆)𝜈,𝑛}.(3.6)
(g) Let 𝜆>−1/2 and 𝑟⩾0. Then, 𝑃_𝜆,𝑛(𝑥) satisfies the higher-order differential equation as follows: 1−𝑥²𝑃^{(𝑟+2)𝜆,𝑛}(𝑥)−(2𝜆+2𝑟+1)𝑥𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)+𝑛²+2𝜆𝑛−𝑟(2𝜆−𝑟+2)𝑃^{(𝑟)𝜆,𝑛}(𝑥)=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 𝑟=0, it follows from [9, (7.33.5)], and, for 𝑟⩾1, it comes from (b) and the case of 𝑟=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<𝜆<1. Let 𝜉^{(𝜆)𝜇,𝑛+1}∶=cos𝜃^{(𝜆)𝜇,𝑛+1}, 𝜇=1,…,𝑛+1 and 𝑦^{(𝜆)𝜈,2𝑛+1}∶=cos𝜓^{(𝜆)𝜈,2𝑛+1}, 𝜈=1,…,2𝑛+1. Then, for 𝜇=0,1,…,𝑛+2 and 𝜈=0,1,…,2𝑛+2, |||𝜃_{(𝜆)𝜇,𝑛+1}−𝜃^{(𝜆)𝜇+1,𝑛+1}|||∼||𝜓_{(𝜆)𝜈,2𝑛+1}−𝜓^{(𝜆)𝜈+1,2𝑛+1}||∼𝑛⁻¹,(3.8) where 𝜓^{(𝜆)0,2𝑛+1}∶=𝜃^{(𝜆)0,𝑛+1}∶=𝜋 and 𝜓^{(𝜆)2𝑛+2,2𝑛+1}∶=𝜃^{(𝜆)𝑛+2,𝑛+1}∶=0.

Proposition 3.3 ([6, Proposition  2.3]). Let 0<𝜆<1. Then, for all 𝑥∈[−1,1], 1−𝑥²𝐸^{𝜆,𝑛+1}(𝑥)−𝑥𝐸^{𝜆,𝑛+1}(𝑥)+(𝑛+1)²𝐸_𝜆,𝑛+1(𝑥)=𝐼_𝜆,𝑛(𝑥),(3.9) where 𝐼_𝜆,𝑛(𝑥)=8𝛾^{𝑛(𝜆)[(𝑛+1)/2]}_𝜈=1(𝑛+1−𝜈)𝜈𝛼^{(𝜆)𝜈,𝑛}𝑇_{𝑛+1−2𝜈}(𝑥).(3.10) Then 𝐼_𝜆,𝑛(𝑥) is a polynomial of degree 𝑛−1 satisfying max_{𝑥∈[−1,1]}||𝐼_𝜆,𝑛(𝑥)||≲𝑛².(3.11)

Proposition 3.4 ([4, Theorem  2.1]). Let 0<𝜆<1. Then, for 𝑛⩾0, ||𝐸_𝜆,𝑛+1(𝑥)||≲𝑛^1−𝜆𝜑^1−𝜆(𝑥)+1−1⩽𝑥⩽1.(3.12) Furthermore, 𝐸_𝜆,𝑛+1(1)≳1.

Proposition 3.5 ([6, Theorem  2.5]). Let 0<𝜆<1. (a)For all 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐸^{𝜆,𝑛+1}(𝑥)||≲𝑛^2−𝜆𝜑^−𝜆(𝑥).(3.13) Moreover, one has, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐸^{𝜆,𝑛+1}(𝑥)||∼𝑛².(3.14)(b)For all 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐸^{𝜆,𝑛+1}(𝑥)||≲𝑛^3−𝜆𝜑^−1−𝜆(𝑥).(3.15) Moreover, one has, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐸^{𝜆,𝑛+1}(𝑥)||∼𝑛⁴.(3.16)

Proposition 3.6 ([6, Corollary  2.6]). Let 0<𝜆<1. Then, for all 𝑥∈[−1,1], 1−𝑥²𝐹^{𝜆,2𝑛+1}(𝑥)−𝑥𝐹^{𝜆,2𝑛+1}(𝑥)+2𝑛²+2(1+𝜆)𝑛+1𝐹_𝜆,2𝑛+1(𝑥)=𝐽_𝜆,𝑛(𝑥).(3.17) Here, 𝐽_𝜆,𝑛(𝑥) is a polynomial of degree of 2𝑛+1 defined in (4.37) such that, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐽_𝜆,𝑛(𝑥)||≲𝑛²𝜑^1−2𝜆(𝑥)(3.18) and, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐽_𝜆,𝑛(𝑥)||≲𝑛^1+2𝜆.(3.19)

Proposition 3.7 ([6, Corollary  2.7]). Let 0<𝜆<1. (a)For all 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐹^{𝜆,2𝑛+1}(𝑥)||≲𝑛𝜑^−2𝜆(𝑥).(3.20) Moreover, one has, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐹^{𝜆,2𝑛+1}(𝑥)||∼𝑛^1+2𝜆.(3.21)(b)For all 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐹^{𝜆,2𝑛+1}(𝑥)||≲𝑛²𝜑^−1−2𝜆(𝑥).(3.22) Moreover, one has, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐹^{𝜆,2𝑛+1}(𝑥)||∼𝑛^3+2𝜆.(3.23)

Proposition 3.8 ([4, Lemma  5.5]). Let 0<𝜆<1. Then, for 𝜇=1,2,…,𝑛+1, |||𝐸_{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}|||∼𝑛^2−𝜆𝜑^−𝜆𝜉^{(𝜆)𝜇,𝑛+1}(3.24) and, for 𝜈=1,2,…,2𝑛+1, |||𝐹_{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}|||∼𝑛𝜑^−2𝜆𝑦^{(𝜆)𝜈,2𝑛+1}.(3.25)

We now estimate the second derivatives at the zeros of 𝐸_𝜆,𝑛+1 and 𝐹_𝜆,2𝑛+1.

Proposition 3.9 ([6, Theorem  2.9]). Let 0<𝜆<1. Then, for 𝜇=1,2,…,𝑛+1, |||𝐸_{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛²𝜑⁻²𝜉^{(𝜆)𝜇,𝑛+1}(3.26) and, for 𝜈=1,2,…,2𝑛+1, |||𝐹_{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}|||≲𝑛^1+𝜆𝜑^−2−𝜆𝑦^{(𝜆)𝜈,2𝑛+1}.(3.27)

4. The Proofs of Main Results

In this section, we let 0<𝜆<1 and 𝑚=⌊(𝑛+1/2)⌋. A representation of Stieltjes polynomials 𝐸_𝜆,𝑛+1(𝑥) is (cf. [1, 10]) 𝛾^𝑛(𝜆)2𝐸^𝜆,𝑛+1(cos𝜃)=𝛼^(𝜆)0,𝑛cos(𝑛+1)𝜃+𝛼^(𝜆)1,𝑛cos(𝑛−1)𝜃+⋯+𝛼^{(𝜆)𝑛/2,𝑛}1cos𝜃,𝑛even2𝛼^{(𝜆)𝑛+1/2,𝑛},𝑛odd,(4.1) where 𝛼^(𝜆)0,𝑛=𝑓^(𝜆)0,𝑛=1,^𝜈_𝜇=0𝑓𝛼^{(𝜆)𝜇,𝑛}𝑓^{(𝜆)𝜈−𝜇,𝑛}=0,𝜈=1,2,…,_{(𝜆)𝜈,𝑛}∶=1−𝜆𝛾𝜈1−𝑛+𝜈𝜆+𝜆,𝜈=1,2,…,_𝑛(𝜆)=√𝜋Γ(𝑛+2𝜆)Γ(𝑛+𝜆+1)∼√𝜋𝑛^𝜆−1.(4.2)

In the following, we state the asymptotic differential relation of the higher order of 𝐸_𝜆,𝑛+1.

Lemma 4.1. Let 0<𝜆<1. Then, for all 𝑥∈[−1,1] and 𝑟⩾2, 1−𝑥²𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)=(2𝑟−3)𝑥𝐸^{(𝑟−1)𝜆,𝑛+1}(𝑥)+(𝑟−2)²−(𝑛+1)²𝐸^{(𝑟−2)𝜆,𝑛+1}(𝑥)+𝐼^{(𝑟−2)𝜆,𝑛}(𝑥)(4.3) and, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐼^{(𝑟−2)𝜆,𝑛}(𝑥)||≲𝑛^𝑟𝜑^2−𝑟(𝑥).(4.4) Here, 𝐼_𝜆,𝑛(𝑥) is a polynomial of degree 𝑛−1 defined in (3.10); 𝐼_𝜆,𝑛(𝑥)=8𝛾^{𝑛𝑚(𝜆)}_𝜈=1(𝑛+1−𝜈)𝜈𝛼^{(𝜆)𝜈,𝑛}𝑇_{𝑛+1−2𝜈}(𝑥),(4.5) such that max_{𝑥∈[−1,1]}||𝐼^{(𝑟−2)𝜆,𝑛}(𝑥)||≲𝑛^2𝑟−2.(4.6)

Proof. For 𝑟⩾2, (4.3) is obtained by 𝑟−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 𝑇_𝑛(𝑥) satisfies the second-order differential equation 1−𝑥²𝑇^𝑛(𝑥)−𝑥𝑇^𝑛(𝑥)+𝑛²𝑇_𝑛(𝑥)=0,(4.7) so we have, by 𝑟−2 times differentiation of (4.7), 1−𝑥²𝑇_𝑛(𝑟)(𝑥)−(2𝑟−3)𝑥𝑇_{𝑛(𝑟−1)}(𝑥)−(𝑟−2)²−𝑛²𝑇^{𝑛(𝑟−2)}(𝑥)=0.(4.8) Let for a nonnegative integer 𝑗⩾0, 𝐼_{𝜆,𝑛,𝑗}(𝑥)∶=−8𝛾^{𝑛𝑚(𝜆)}_𝜈=1(𝑛+1−𝜈)𝜈𝛼^{(𝜆)𝜈,𝑛}||𝑇^{(𝑗)𝑛+1−2𝜈}(𝑥)||.(4.9) Observe that in the view of Szegö’s result (cf. [1]) 𝛼^(𝜆)1,𝑛<𝛼^(𝜆)2,𝑛<𝛼^(𝜆)3,𝑛<⋯<0,0⩽^∞_𝜈=0𝛼^{(𝜆)𝜈,𝑛}<1.(4.10) Then, since |𝐼^{(𝑗)𝜆,𝑛}(𝑥)|⩽𝐼_{𝜆,𝑛,𝑗}(𝑥) (note (4.10)), we will prove that, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}] and 𝑗⩾0, 𝐼_{𝜆,𝑛,𝑗}(𝑥)≲𝑛^𝑗+2𝜑^−𝑗(𝑥)(4.11) instead of (4.4). Since, from the proof of [6, Proposition  2.3], 0<−8𝛾^{𝑛𝑚(𝜆)}_𝜈=1(𝑛+1−𝜈)𝜈𝛼^{(𝜆)𝜈,𝑛}≲𝑛²(4.12) and, for 𝑥=cos𝜃, ||𝑇^{𝑛+1−2𝜈}(𝑥)||=|||(𝑛+1−2𝜈)sin(𝑛+1−2𝜈)𝜃sin𝜃|||≲𝑛𝜑⁻¹(𝑥),(4.13) we obtain that 𝐼_𝜆,𝑛,0(𝑥)≲𝑛² and 𝐼_𝜆,𝑛,1(𝑥)≲𝑛³𝜑⁻¹(𝑥). Using (4.8), we have, for 2⩽𝑗⩽𝑛, 𝐼_{𝜆,𝑛,𝑗}(𝑥)=−8𝛾^{𝑛𝑚(𝜆)}_𝜈=1≤−8(𝑛+1−𝜈)𝜈𝛼^{(𝜆)𝜈,𝑛}||𝑇^{(𝑗)𝑛+1−2𝜈}(𝑥)||𝛾^{𝑛𝑚(𝜆)}_𝜈=1(𝑛+1−𝜈)𝜈𝛼^{(𝜆)𝜈,𝑛}×(2𝑗−3)|𝑥|1−𝑥²||𝑇^{(𝑗−1)𝑛+1−2𝜈}(𝑥)||+||(𝑗−2)²−(𝑛+1−2𝜈)²||1−𝑥²≲(2𝑗−3)|𝑥|||𝑇^{(𝑗−2)𝑛+1−2𝜈}(𝑥)||1−𝑥²𝐼_{𝜆,𝑛,𝑗−1}(𝑥)+(𝑛+1)²1−𝑥²𝐼_{𝜆,𝑛,𝑗−2}(𝑥).(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^{(𝑟)𝜆,𝑛+1}(𝑥) for two cases of an odd order and an even order in the following.

Lemma 4.2. Let 0<𝜆<1 and 𝑟⩾2. Let 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}]. If 𝑟 is even, then one has ||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||≲𝑛^𝑟−2𝜑^−𝑟(𝑥)||𝐸^{𝜆,𝑛+1}(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥)||𝐸_𝜆,𝑛+1(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥),(4.15) and, if 𝑟 is odd, then one has ||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||≲𝑛^𝑟−1𝜑^1−𝑟(𝑥)||𝐸^{𝜆,𝑛+1}(𝑥)||+𝑛^𝑟−1𝜑^−1−𝑟(𝑥)||𝐸_𝜆,𝑛+1(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥).(4.16)

Proof. Let 𝑟⩾2. From (4.3) and (4.4), we have, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||≲𝜑⁻²(𝑥)||𝐸^{(𝑟−1)𝜆,𝑛+1}(𝑥)||+𝑛²𝜑⁻²(𝑥)||𝐸^{(𝑟−2)𝜆,𝑛+1}(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥)(4.17) and especially ||𝐸^{𝜆,𝑛+1}(𝑥)||≲𝜑⁻²(𝑥)||𝐸^{𝜆,𝑛+1}(𝑥)||+𝑛²𝜑⁻²(𝑥)||𝐸_𝜆,𝑛+1(𝑥)||+𝑛²𝜑⁻²(𝑥),(4.18) that is, we have (4.15) for 𝑟=2. From Proposition 3.2, we see 1+𝜉^{(𝜆)1,𝑛+1}, 1−𝜉^{(𝜆)𝑛+1,𝑛+1}≳1/𝑛, so we have, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], 𝜑⁻¹(𝑥)≲𝑛.(4.19) Then, from (4.17) with 𝑟=3 and (4.18), we know that ||𝐸^{(3)𝜆,𝑛+1}(𝑥)||≲𝑛²𝜑⁻²(𝑥)||𝐸^{𝜆,𝑛+1}(𝑥)||+𝑛²𝜑⁻⁴(𝑥)||𝐸_𝜆,𝑛+1(𝑥)||+𝑛³𝜑⁻³(𝑥),(4.20) that is, we have (4.16) for 𝑟=3. Assume that (4.15) and (4.16) hold for 3,4,…,𝑟−1 times differentiation. Let 𝑟 be an even number. Then, we have, from (4.17), (4.19), and the assumptions for 𝑟−1 and 𝑟−2, ||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||≲𝑛^𝑟−2𝜑^−𝑟(𝑥)||𝐸^{𝜆,𝑛+1}(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥)||𝐸_𝜆,𝑛+1(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥),(4.21) that is, we have (4.15). Similarly, we also have (4.16) for an odd 𝑟.

Lemma 4.3. Let −1<𝑥₁<𝑥₂<⋯<𝑥_𝑛<1 and 𝑃(𝑥)∶=𝑥−𝑥₁𝑥−𝑥₂⋯𝑥−𝑥_𝑛.(4.22) Then, 𝑃^(𝑥) is a polynomial of degree of 𝑛−1 and has distinct real 𝑛−1 zeros in (−1,1). Moreover, if one lets −1<𝑦₁<𝑦₂<⋯<𝑦_𝑛−1<1 be the zeros of 𝑃^(𝑥), then {𝑦_𝑖}^𝑛𝑖=1 is interlaced with the zeros of 𝑃(𝑥) that is, 𝑥_𝑖<𝑦_𝑖<𝑥_𝑖+1, 𝑖=1,2,…,𝑛−1.

Proof. Since the sign of 𝑃^(𝑥_𝑖) is (−1)^𝑛−𝑖, it is proved.

Lemma 4.4. Let 𝑟 be a nonnegative integer. Then, 𝐸^{(𝑟)𝜆,𝑛+1}(𝑥) has distinct 𝑛+1−𝑟 real zeros on (−1,1). If one lets {𝑥_𝑛+1(𝑟,𝑖)}^{𝑛+1−𝑟𝑖=1} be the zeros of the polynomial 𝐸^{(𝑟)𝜆,𝑛+1}(𝑥) with −1<𝑥_𝑛+1(𝑟,1)<𝑥_𝑛+1(𝑟,2)<⋯<𝑥_𝑛+1(𝑟,𝑛+1−𝑟)<1,(4.23) then one has, for 1⩽𝑟⩽𝑛 and 𝑘=1,…,𝑛+1−𝑟, 𝑥_𝑛+1(0,𝑘)<𝑥_𝑛+1(𝑟,𝑘).(4.24)

Proof. From Lemma 4.3, we know that 𝐸^{(𝑟)𝜆,𝑛+1} has distinct real 𝑛+1−𝑟 zeros on (−1,1). By the interlaced zeros property of Lemma 4.3, we see that, for 𝑘=1,…,𝑛+1−𝑟, 𝑥_𝑛+1(𝑟−1,𝑘)<𝑥_𝑛+1(𝑟,𝑘)<𝑥_𝑛+1(𝑟−1,𝑘+1).(4.25) Thus, (4.24) is proved.

Proof of Theorem 2.1. Let 𝑟⩾1. 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 max_{𝑥∈[−1,1]}||𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)||∼𝑛^2𝑟,𝑟=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 𝑟=1 from (3.14). We suppose that, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1] and 𝑟⩾2, ||𝐸^{(𝑟−1)𝜆,𝑛+1}(𝑥)||∼𝑛^2(𝑟−1).(4.27) Then, by Lemma 4.4 and (3.8), we have, for 𝑥∈[𝜉^{(𝜆)𝑛+1,𝑛+1},1] and 𝑟⩾2, 𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)𝐸=^{(𝑟−1)𝜆,𝑛+1}(𝑥)_{𝑛+2−𝑟}_𝑘=11⩾𝑥−𝑥_𝑛+11(𝑟−1,𝑘)⩾𝜉^{(𝜆)𝑛+1,𝑛+1}−𝑥_𝑛+1(1𝑟−1,𝑛+2−𝑟)𝜉^{(𝜆)𝑛+1,𝑛+1}−𝑥_𝑛+1(0,𝑛+2−𝑟)=𝜉^{(𝜆)𝑛+1,𝑛+1}−1𝜉^{(𝜆)𝑛+2−𝑟,𝑛+1}≳1𝜉^{(𝜆)𝑛+1,𝑛+1}−𝜉^{(𝜆)𝑛,𝑛+1}≳𝑛².(4.28) Here, the last inequality is obtained by Proposition 3.2, that is, 𝜉^{(𝜆)𝑛+1,𝑛+1}−𝜉^{(𝜆)𝑛,𝑛+1}=cos𝜃^{(𝜆)𝑛+1,𝑛+1}−cos𝜃^{(𝜆)𝑛,𝑛+1}∼𝑛⁻².(4.29) Therefore, we have, for 𝑥∈[𝜉^{(𝜆)𝑛+1,𝑛+1},1] and 𝑟⩾2, 𝐸^{(𝑟)𝜆,𝑛+1}(𝑥)≳𝐸^{(𝑟−1)𝜆,𝑛+1}(𝑥)𝑛²∼𝑛^2𝑟.(4.30) Hence, from (2.3), we have (2.4). For 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}], the proof is similar.

Lemma 4.5. Let ℓ be a nonnegative integer and 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}]. Then,

|||𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(ℓ)|||≲𝑛_ℓ+1|||𝐼𝜑^{−ℓ−2𝜆}(𝑥),(4.31a)_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(ℓ)|||≲𝑛_ℓ+1+𝜆||||𝐸𝜑^{−ℓ−𝜆}(𝑥),(4.31b)_{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(ℓ)||||≲𝑛_ℓ+2𝜑^{−2𝜆−ℓ−1}(𝑥).(4.31c)

Proof. (a) When ℓ=0, it is obvious from (3.5) and (3.12). Now, suppose ℓ⩾1. From (3.12), (2.2), and (3.5), we have |||𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(ℓ)|||≲_{ℓ−1⩽𝑞+𝑟⩽ℓ}||𝐸^{(𝑞)𝜆,𝑛+1}≲(𝑥)𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)_{ℓ−1⩽𝑞+𝑟⩽ℓ}||𝑛^𝑞+𝑟+1𝜑^{−(𝑞+𝑟)−2𝜆}(𝑥)≲𝑛^ℓ+1𝜑^{−ℓ−2𝜆}(𝑥).(4.32)
(b) From (4.4) and (3.5), we have |||𝐼_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(ℓ)|||≲_{𝑞+𝑟=ℓ}||𝐼^{(𝑞)𝜆,𝑛}≲𝑛_ℓ+1+𝜆(𝑥)𝑃^{(𝑟)𝜆,𝑛}(𝑥)||𝜑^{−ℓ−𝜆}(𝑥).(4.33)
(c) Similarly to the proof of (a), we have, from (2.2) and (3.5), ||||𝐸_{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(ℓ)||||≲_{𝑞+𝑟=ℓ}≲𝑛_ℓ+2||𝐸^{(𝑞+1)𝜆,𝑛+1}(𝑥)𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)||𝜑^{−2𝜆−ℓ−1}(𝑥).(4.34)

Lemma 4.6. Let 0<𝜆<1. Then, for all 𝑥∈[−1,1] and 𝑟⩾2, +1−(𝑟𝑥−²2𝐹)^{(𝑟)𝜆,2𝑛+1}₂(𝑥)=(2𝑟−5)𝑥𝐹^{(𝑟−1)𝜆,2𝑛+1}(𝑥)−(𝑛+1)²−𝑛(𝑛+2𝜆)𝐹^{(𝑟−2)𝜆,2𝑛+1}(𝑥)+𝐽^{(𝑟−2)𝜆,𝑛}(𝑥)(4.35) and, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐽^{(𝑟−2)𝜆,𝑛}(𝑥)||≲𝑛^𝑟𝜑^{−2𝜆−𝑟+3}(𝑥).(4.36) Here, 𝐽_𝜆,𝑛(𝑥) is a polynomial of degree of 2𝑛+1 defined as follows: 𝐽+_𝜆,𝑛2(𝑥1)−=𝑥2₂𝜆𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)+𝐼_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥).(4.37) Furthermore, one has ||𝐽^{(𝑟−2)𝜆,𝑛}(1)||≲𝑛^{2𝜆+2𝑟−3}.(4.38)

Proof. Similarly to the proof of Lemma 4.1, (4.35) is obtained by 𝑟−2 times differentiation of the second-order differential relation with respect to 𝐹_𝜆,2𝑛+1(𝑥), that is, (3.17). So it is sufficient to prove (4.36) and (4.38). From (4.37), we know that 𝐽^{(𝑟−2)𝜆,𝑛}(𝑥)=2𝜆𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−2)+21−𝑥²𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−2)+𝐼_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(𝑟−2).(4.39) By Lemma 4.5 (a) and (b), we have, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], |||2𝜆𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−2)|||≲𝑛_𝑟−1|||𝐼𝜑^{−𝑟+2−2𝜆}(𝑥),_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(𝑟−2)|||≲𝑛_{𝑟−1+𝜆}𝜑^{−𝑟+2−𝜆}(𝑥).(4.40) From (4.19) and Lemma 4.5 (c), we have, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}] and 𝑟⩾4, ||||21−𝑥₂𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−2)≲||||1−𝑥₂||||𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−2)||||+||||𝐸_{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−3)+||||𝐸_{𝜆,𝑛+1}||||(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−4)≲𝑛_𝑟||||≲𝑛_𝑟𝜑^{−2𝜆−𝑟+3}(𝑥)+𝑛^𝑟−1𝜑^{−2𝜆−𝑟+2}(𝑥)+𝑛^𝑟−2𝜑^{−2𝜆−𝑟+3}(𝑥)𝜑^{−2𝜆−𝑟+3}(𝑥).(4.41) When 𝑟=2,3, we can similarly obtain that ||||21−𝑥₂𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(𝑟−2)||||≲𝑛_𝑟𝜑^{−2𝜆−𝑟+3}(𝑥).(4.42) Therefore, we have (4.36). On the other hand, from (2.4), (4.6), and (3.2), we know that, for a nonnegative integer ℓ, 𝐸𝑃^{(ℓ)𝜆,𝑛+1}_{(ℓ)𝜆,𝑛}(1)∼𝑛^2ℓ,||𝐼^{(ℓ)𝜆,𝑛}(1)||≲𝑛^2ℓ+2,(1)∼𝑃_{𝜆+ℓ,𝑛−ℓ}(1)∼𝑛^{2(𝜆+ℓ)−1}.(4.43) Then, similarly to the proof of Lemma 4.5, we obtain that |||𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(ℓ)|||_𝑥=1≲𝑛^2ℓ+2𝜆+1,|||𝐼_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(ℓ)|||_𝑥=1||||𝐸_{𝜆,𝑛+1}≲𝑛^2ℓ+2𝜆+1,(𝑥)𝑃^𝜆,𝑛(𝑥)^(ℓ)||||_𝑥=1≲𝑛^2ℓ+2𝜆+3.(4.44) Therefore, we have (4.38).

Lemma 4.7. Let 0<𝜆<1. Then, for 𝑟⩾2, if 𝑟 is even, one has, for 𝑥∈[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}], ||𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥)||≲𝑛^𝑟−2𝜑^−𝑟(𝑥)||𝐹^{𝜆,2𝑛+1}(𝑥)||+𝑛^𝑟𝜑^−𝑟(𝑥)||𝐹_𝜆,2𝑛+1(𝑥)||+𝑛^𝑟𝜑^{1−2𝜆−𝑟}(𝑥)(4.45) and, if 𝑟 is odd, one has ||𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥)||≲𝑛^𝑟−1𝜑^1−𝑟(𝑥)||𝐹^{𝜆,2𝑛+1}(𝑥)||+𝑛^𝑟−1𝜑^−1−𝑟(𝑥)||𝐹_𝜆,2𝑛+1(𝑥)||+𝑛^𝑟𝜑^{1−2𝜆−𝑟}(𝑥).(4.46)

Proof. Using (4.35) and (4.36), we obtain the result similarly to the proof of Lemma 4.2.

Proof of Theorem 2.2. Equation (2.5) comes from Lemma 4.7 and Proposition 3.7. We will show (2.6) and (2.7). From Proposition 3.7, we see max_{𝑥∈[−1,1]}||𝐹^{𝜆,2𝑛+1}(𝑥)||∼𝑛^1+2𝜆.(4.47) Hence, using Markov-Bernstein inequality for 𝐹^{𝜆,2𝑛+1}∈𝒫_2𝑛, we have (2.6). Now, we show (2.7). By Proposition 3.7 (a), it is true for 𝑟=1. We suppose that, for 𝑟⩾2, ||𝐹^{(𝑟−1)𝜆,2𝑛+1}(𝑥)||∼𝑛^{2𝑟+2𝜆−3},𝑥∈−1,𝜉^{(𝜆)1,𝑛+1}∪𝜉^{(𝜆)𝑛+1,𝑛+1},1.(4.48) As the proof of Theorem 2.1, we have, for 𝑟⩾2 and for 𝑥∈[𝜉^{(𝜆)𝑛+1,𝑛+1},1], 𝐹^{(𝑟)𝜆,2𝑛+1}(𝑥)𝐹^{(𝑟−1)𝜆,2𝑛+1}(𝑥)≳𝑛².(4.49) Therefore, we see that by induction with Proposition 3.7, (2.7) holds for every 𝑟=1,2,3,….

Corollary 4.8. Let 0<𝜆<1 and 𝑟⩾2. Then, for 𝑥∈[−1,𝜉^{(𝜆)1,𝑛+1}]∪[𝜉^{(𝜆)𝑛+1,𝑛+1},1], ||𝐽^{(𝑟−2)𝜆,𝑛}(𝑥)||≲𝑛^{2𝑟+2𝜆−3}.(4.50)

Proof. Corollary 4.8 comes from (4.35), (2.7), and (3.8).

Proof of Theorem 2.3. Equation (2.8) comes from (4.15) and (3.24).

Lemma 4.9. For 1⩽𝜇⩽𝑛+1 and 1⩽𝜈⩽2𝑛+1, |||𝑃_𝜆,𝑛|||𝐸𝜉^{(𝜆)𝜇,𝑛+1}|||∼𝑛^𝜆−1𝜑^−𝜆𝜉^{(𝜆)𝜇,𝑛+1},(4.51)_𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}|||∼𝑛^1−𝜆𝜑^1−𝜆𝑥^{(𝜆)𝜈,𝑛}.(4.52)

Proof. Since we know from (3.24) and (3.25) that 𝑛𝜑^−2𝜆𝜉^{(𝜆)𝜇,𝑛+1}∼∼𝑛|||𝐹^{𝜆,2𝑛+1}_2−𝜆𝜉^{(𝜆)𝜇,𝑛+1}|||=|||𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}|||𝜑^−𝜆𝜉^{(𝜆)𝜇,𝑛+1}|||𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}|||,(4.53) (4.51) is obviously proved. Similarly, since we have, from (3.4) and (3.25), 𝑛𝜑^−2𝜆𝑥^{(𝜆)𝜈,𝑛}∼∼𝑛|||𝐹^{𝜆,2𝑛+1}_𝜆𝑥^{(𝜆)𝜈,𝑛}|||=|||𝐸^𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}|||𝜑^−𝜆−1𝑥^{(𝜆)𝜈,𝑛}|||𝐸^𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}|||,(4.54) (4.52) is obtained.

Lemma 4.10. Let 0<𝜆<1 and 𝑟⩾1. Let 𝑟 be an odd integer. (a)For 1⩽𝜇⩽𝑛+1, |||𝑃_{(𝑟)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^{2𝜆+𝑟−2}𝜑^−𝑟−1𝜉^{(𝜆)𝜇,𝑛+1}.(4.55)(b)For 1⩽𝜈⩽𝑛, |||𝐸_{(𝑟)𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛^𝑟𝜑^−𝑟𝑥^{(𝜆)𝜈,𝑛}.(4.56)

Proof. (a) We know, from (3.3), |||𝑃_𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^2𝜆−1𝜑⁻²𝜉^{(𝜆)𝜇,𝑛+1}.(4.57) So (4.55) holds for 𝑘=1. Assume that, for 𝑘=1,2,…,ℓ, |||𝑃_{(2𝑘−1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^{2𝜆+2𝑘−3}𝜑^−2𝑘𝜉^{(𝜆)𝜇,𝑛+1}.(4.58) Then, we have from (3.5), (3.7), and (4.58) that |||𝑃_{(2ℓ+1)𝜆,𝑛}≲𝜑₋₂𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛𝜉^{(𝜆)𝜇,𝑛+1}_{2𝜆+2ℓ−1}|||𝑃^{(2ℓ)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}|||+𝑛²𝜑⁻²𝜉^{(𝜆)𝜇,𝑛+1}|||𝑃^{(2ℓ−1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}|||𝜑^−2ℓ−2𝜉^{(𝜆)𝜇,𝑛+1}.(4.59) Therefore, we have the result using the mathematical induction.
(b) For an odd integer 𝑟⩾1, we have from (3.6), (4.16), and (4.52) |||𝐸_{(𝑟)𝜆,𝑛+1}≲𝑛_𝑟−1𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛_𝑟𝜑^1−𝑟(𝑥)|||𝐸^{𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}|||+|||𝐸^𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}|||+𝑛^𝑟𝜑^−𝑟𝑥^{(𝜆)𝜈,𝑛}𝜑^−𝑟𝑥^{(𝜆)𝜈,𝑛}.(4.60)

Lemma 4.11. Let 𝑟⩾2. If 𝑟 is even, then |||𝑃_{(𝑟)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛^{𝜆+𝑟−2}𝜑^{−𝜆−𝑟−1}𝑥^{(𝜆)𝜈,𝑛}.(4.61)

Proof. It is easily proved from (3.4) and (3.7).

Lemma 4.12. Let 𝑘 be a positive integer. Then, one has, for 1⩽𝜈⩽2𝑛+1, |||𝐽_{(2𝑘)𝜆,𝑛}𝑦^{(𝜆)𝜈,2𝑛+1}|||≲𝑛^2𝑘+1+𝜆𝜑^{−2𝑘−𝜆}𝑦^{(𝜆)𝜈,2𝑛+1}.(4.62)

Proof. From (4.37), we know that 𝐽+_𝜆,𝑛2(𝑥1)−=𝑥2₂𝜆𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)+𝐼_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥).(4.63) From Lemma 4.5 (a) and (b), we know that for 𝑥∈[𝑦^{(𝜆)1,2𝑛+1},𝑦^{(𝜆)2𝑛+1,2𝑛+1}] (=[𝜉^{(𝜆)1,𝑛+1},𝜉^{(𝜆)𝑛+1,𝑛+1}]) |||𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(2𝑘)|||≲𝑛_2𝑘+1|||𝐼𝜑^{−2𝑘−2𝜆}(𝑥)_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(2𝑘)|||≲𝑛_2𝑘+1+𝜆𝜑^{−2𝑘−𝜆}(𝑥).(4.64) On the other hand, we estimate |((1−𝑥²)𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥))^(2𝑘)| splitting into three terms as follows: ||||1−𝑥₂𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2𝑘)≲_{𝑞+𝑟=2𝑘}||||+𝑂(1)||||||1𝐸−𝑥²_{𝜆,𝑛+1}𝐸^{(𝑞+1)𝜆,𝑛+1}(𝑥)𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)||(𝑥)𝑃^𝜆,𝑛(𝑥)^(2𝑘−1)||||+||||𝐸_{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2𝑘−2)||||.(4.65) Here, from Lemma 4.5 (c), we have for 𝑥∈[𝑦^{(𝜆)1,2𝑛+1},𝑦^{(𝜆)2𝑛+1,2𝑛+1}]||||𝐸_{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2𝑘−1)||||≲𝑛_2𝑘+1||||𝐸𝜑^{−2𝜆−2𝑘}(𝑥),_{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2𝑘−2)||||≲𝑛_2𝑘𝜑^{−2𝜆−2𝑘+1}(𝑥)≲𝑛^2𝑘+1𝜑^{−2𝜆−2𝑘}(𝑥).(4.66) For the first term, we also split into two terms as follows: _𝑞+_𝑟=2𝑘=||1−𝑥²𝐸^{(𝑞+1)𝜆,𝑛+1}(𝑥)𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)_{𝑞+𝑟=2𝑘,𝑞∶even,𝑟∶even}||+||1−𝑥²𝐸^{(𝑞+1)𝜆,𝑛+1}(𝑥)𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)_{𝑞+𝑟=2𝑘,𝑞∶odd,𝑟∶odd}||∶=𝐴₁||1−𝑥²𝐸^{(𝑞+1)𝜆,𝑛+1}(𝑥)𝑃^{(𝑟+1)𝜆,𝑛}(𝑥)||(𝑥)+𝐴₂(𝑥).(4.67) From (2.2) and (4.55), we know that for even 𝑞 and 𝑟|||𝐸_{(𝑞+1)𝜆,𝑛+1}|||𝑃_{(𝑟+1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^{𝑞+2−𝜆}𝜑^{−𝑞−𝜆}𝜉^{(𝜆)𝜇,𝑛+1},𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^{2𝜆+𝑟−1}𝜑^−𝑟−2𝜉^{(𝜆)𝜇,𝑛+1}.(4.68) Also, we know from (4.56) and (3.5) that for even 𝑞 and 𝑟|||𝐸_{(𝑞+1)𝜆,𝑛+1}|||𝑃_{(𝑟+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛^𝑞+1𝜑^−𝑞−1𝑥^{(𝜆)𝜈,𝑛},𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛^𝜆+𝑟𝜑^{−𝜆−𝑟−1}𝑥^{(𝜆)𝜈,𝑛}.(4.69) Then, we have |||𝐴₁≲𝜑₂𝜉^{(𝜆)𝜇,𝑛+1}|||𝜉^{(𝜆)𝜇,𝑛+1}_{𝑞+𝑟=2𝑘,𝑞∶even,𝑟∶even}|||𝐸_{(𝑞+1)𝜆,𝑛+1}≲𝑛_2𝑘+1+𝜆𝜉^{(𝜆)𝜇,𝑛+1}𝑃^{(𝑟+1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}||||||𝐴₁𝜑^{−2𝑘−𝜆}𝜉^{(𝜆)𝜇,𝑛+1},𝑥^{(𝜆)𝜈,𝑛}|||≲𝜑²𝑥^{(𝜆)𝜈,𝑛}_{𝑞+𝑟=2𝑘,𝑞∶even,𝑟∶even}|||𝐸_{(𝑞+1)𝜆,𝑛+1}≲𝑛_2𝑘+1+𝜆𝑥^{(𝜆)𝜈,𝑛}𝑃^{(𝑟+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}|||𝜑^{−𝜆−2𝑘}𝑥^{(𝜆)𝜈,𝑛}.(4.70) Similarly, for odd 𝑞 and 𝑟, we have, by (2.8) and (3.5), |||𝐸_{(𝑞+1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}𝑃^{(𝑟+1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}|||≲𝑛^2𝑘+1+𝜆𝜑^{−2𝑘−𝜆−2}𝜉^{(𝜆)𝜇,𝑛+1}(4.71) and, by (2.2) and (4.61), |||𝐸_{(𝑞+1)𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}𝑃^{(𝑟+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}|||≲𝑛^2𝑘+1+𝜆𝜑^{−𝜆−2𝑘−2}𝑥^{(𝜆)𝜈,𝑛}.(4.72) Thus, we have |||𝐴₂≲𝜑₂𝜉^{(𝜆)𝜇,𝑛+1}|||𝜉^{(𝜆)𝜇,𝑛+1}_{𝑞+𝑟=2𝑘,𝑞∶odd,𝑟∶odd}|||𝐸_{(𝑞+1)𝜆,𝑛+1}≲𝑛_2𝑘+1+𝜆𝜉^{(𝜆)𝜇,𝑛+1}𝑃^{(𝑟+1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}||||||𝐴₂𝜑^{−2𝑘−𝜆}𝜉^{(𝜆)𝜇,𝑛+1},𝑥^{(𝜆)𝜈,𝑛}|||≲𝜑²𝑥^{(𝜆)𝜈,𝑛}_{𝑞+𝑟=2𝑘,𝑞∶odd,𝑟∶odd}|||𝐸_{(𝑞+1)𝜆,𝑛+1}≲𝑛_2𝑘+1+𝜆𝑥^{(𝜆)𝜈,𝑛}𝑃^{(𝑟+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}|||𝜑^{−𝜆−2𝑘}𝑥^{(𝜆)𝜈,𝑛}.(4.73) Therefore, we have the result.

Proof of Theorem 2.4. When 𝑟=2, (2.9) holds from (3.27). Let even 𝑟>2, and suppose that (2.9) holds for 𝑟−2. Since we know by (4.35), (2.5), (4.62), and (4.19) 𝜑≲²|||𝐹𝑦^{(𝜆)𝜈,2𝑛+1}_{(𝑟−1)𝜆,2𝑛+1}|||𝐹^{(𝑟)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}|||𝑦^{(𝜆)𝜈,2𝑛+1}|||+𝑛²|||𝐹_{(𝑟−2)𝜆,2𝑛+1}≲𝑛₂𝑦^{(𝜆)𝜈,2𝑛+1}|||+|||𝐽^{(𝑟−2)𝜆,𝑛}𝑦^{(𝜆)𝜈,2𝑛+1}||||||𝐹_{(𝑟−2)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}|||+𝑛^{𝑟−1+𝜆}𝜑^{−𝑟+2−𝜆}𝑦^{(𝜆)𝜈,2𝑛+1},(4.74) we obtain, using mathematical induction, 𝜑²𝑦^{(𝜆)𝜈,2𝑛+1}|||𝐹^{(𝑟)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}|||≲𝑛^{𝑟−1+𝜆}𝜑^{−𝑟+2−𝜆}𝑦^{(𝜆)𝜈,2𝑛+1}.(4.75) Therefore, (2.9) is proved.

Proof of Theorem 2.5. (a) From (4.3), we know that 𝜑+²𝑂𝜉𝐸^{(𝜆)𝜇,𝑛+1}_{𝜆,𝑛+1}𝐸^{(3)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}=−(𝑛+1)²𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝐼^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}.(4.76) Therefore, we have, by (2.2), (3.24), and (3.26), 𝐸^{(3)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}=−(𝑛+1)²𝜑⁻²𝜉^{(𝜆)𝜇,𝑛+1}𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛³.(4.77) Suppose that, for an integer ℓ⩾2, 𝐸^{(2ℓ−1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}=(−1)^ℓ−1(𝑛+1)^2(ℓ−1)𝜑^{−2(ℓ−1)}𝜉^{(𝜆)𝜇,𝑛+1}𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^2ℓ−1.(4.78) Then, from (4.3), we obtain 𝜑²𝜉^{(𝜆)𝜇,𝑛+1}𝐸^{(2ℓ+1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}=−+(𝑂𝑛+𝐸1)²𝐸^{(2ℓ−1)𝜆,𝑛+1}_{(2ℓ−1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝐸^{(2ℓ)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝐼^{(2ℓ−1)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}.(4.79) Therefore, by (2.2), (2.8), and (4.4), we have 𝐸^{(2ℓ+1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}==(−−(1𝑛)+_ℓ1)²𝜑⁻²𝜉^{(𝜆)𝜇,𝑛+1}𝐸^{(2ℓ−1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^2ℓ+1(𝑛+1)^2ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝐸^{𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^2ℓ+1.(4.80)
(b) Similarly to the proof of (a), by (4.51), (3.3), and (3.7), we can obtain 𝑃^{(2ℓ)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}=(−1)^ℓ𝑛^ℓ(𝑛+2𝜆)^ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^{2𝜆+2ℓ−3}.(4.81) In addition, we see that, from (4.51), 𝑃^{(2ℓ)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}=+(−𝑂1)^ℓ𝑛(𝑛+_2ℓ−11)^2ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}=(−1)_ℓ𝑃_𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^{2𝜆+2ℓ−3}(𝑛+1)^2ℓ𝜑^−2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝑃^𝜆,𝑛𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^{2ℓ+𝜆−2}.(4.82)

Proof of Theorem 2.6. (a) From (3.9), we know that 𝜑²𝑥^{(𝜆)𝜈,𝑛}𝐸^{𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}&=−(𝑛+1)²𝐸_𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}+𝑂𝐸^{𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}+𝑂𝐼^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}.(4.83) Therefore, by (4.4) and (4.56), we have 𝐸^{𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}=−(𝑛+1)²𝜑⁻²𝑥^{(𝜆)𝜈,𝑛}𝐸^𝜆,𝑛+1𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛².(4.84) Then, we obtain from (4.3), (2.2), and (4.56) that 𝜑²𝑥^{(𝜆)𝜈,𝑛}𝐸^{(2ℓ)𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}=−(𝑛+1)²𝐸^{(2ℓ−2)𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^2ℓ.(4.85) Therefore, we have the result inductively.
(b) From (3.7), we know that 𝜑²𝑥^{(𝜆)𝜈,𝑛}𝑃^(3)𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}=−𝑛(𝑛+2𝜆)𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑃^{𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}.(4.86) Therefore, by (3.4) and (4.61), we have 𝑃^(3)𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}=−𝑛(𝑛+2𝜆)𝜑⁻²𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^𝜆.(4.87) Suppose that, for an integer ℓ⩾2, 𝑃^{(2ℓ−1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}=(−1)^ℓ−1𝑛^ℓ−1(𝑛+2𝜆)^ℓ−1𝜑^{−2(ℓ−1)}𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^{𝜆+2ℓ−4}.(4.88) Then from (3.5), (3.7), and (4.61) 𝑃^{(2ℓ+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}=(−1)^ℓ𝑛^ℓ(𝑛+2𝜆)^ℓ𝜑^−2ℓ𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^{𝜆+2ℓ−2}.(4.89) Here, we see that for ℓ⩾2{𝑛(𝑛+2𝜆)}^ℓ=(𝑛+1)^2ℓ+𝑂𝑛^2ℓ−1.(4.90) Hence, we obtain, from (3.4), 𝑃^{(2ℓ+1)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}=+(−𝑂1)^ℓ𝑛(𝑛+_2ℓ−11)^2ℓ𝜑^−2ℓ𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}=(−1)_ℓ𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^{𝜆+2ℓ−2}(𝑛+1)^2ℓ𝜑^−2ℓ𝑥^{(𝜆)𝜈,𝑛}𝑃^𝜆,𝑛𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^{𝜆+2ℓ−1}.(4.91)

Lemma 4.13. Let 0<𝜀<1 and |𝑦^{(𝜆)𝜈,2𝑛+1}|⩽1−𝜀. Then, for a nonnegative integer ℓ⩾0, 𝐽^{(2ℓ+1)𝜆,𝑛}𝑦^{(𝜆)𝜈,2𝑛+1}=2^2ℓ+1(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^𝜆+2ℓ+2.(4.92)

Proof. Let |𝑦^{(𝜆)𝜈,2𝑛+1}|⩽1−𝜀. From (4.37) and Lemma 4.5, we see that 𝐽=^{(2ℓ+1)𝜆,𝑛}21−𝑦𝑥^{(𝜆)𝜈,2𝑛+1}₂𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)−4𝑥(2ℓ+1)𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ)−2ℓ(2ℓ+1)𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ−1)+2𝜆𝑥𝐸_𝜆,𝑛+1(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)+𝐼_𝜆,𝑛(𝑥)𝑃_𝜆,𝑛(𝑥)^(2ℓ+1)|||_{𝑥=𝑦(𝜆)𝜈,2𝑛+1}=21−𝑥²𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)||||_{𝑥=𝑦(𝜆)𝜈,2𝑛+1}+𝑂𝑛^2ℓ+2+𝜆.(4.93) Here, we let 2ℓ(2ℓ+1)(𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥))^(2ℓ−1)=0 when ℓ=0. To estimate the first term, we split it into two terms as follows: 𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)=_{0⩽𝑖⩽2ℓ+1,𝑖∶even}+_{0⩽𝑖⩽2ℓ+1,𝑖∶odd}⎛⎜⎜⎝2ℓ𝑖+1⎞⎟⎟⎠𝐸_{(𝑖+1)𝜆,𝑛+1}(𝑥)𝑃^{(2ℓ+2−i)𝜆,𝑛}(𝑥).(4.94) Then, using |𝑥^{(𝜆)𝜈,𝑛}|⩽1−𝜀, from (2.13) and (2.15), we obtain _0⩽𝑖⩽2_{ℓ+1,𝑖∶odd}⎛⎜⎜⎝2ℓ𝑖+1⎞⎟⎟⎠𝐸_{(𝑖+1)𝜆,𝑛+1}=(−1)_ℓ+1𝑥^{(𝜆)𝜈,𝑛}𝑃^{(2ℓ+2−𝑖)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}(𝑛+1)^2ℓ+2𝜑^−2ℓ−2𝑥^{(𝜆)𝜈,𝑛}𝐹^{𝜆,2𝑛+1}𝑥^{(𝜆)𝜈,𝑛}_{0⩽𝑖⩽2ℓ+1,𝑖∶odd}⎛⎜⎜⎝2ℓ𝑖+1⎞⎟⎟⎠+𝑂𝑛_𝜆+2ℓ+2=2_2ℓ(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ−2𝑥^{(𝜆)𝜈,𝑛}𝐹^{𝜆,2𝑛+1}𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^𝜆+2ℓ+2(4.95) and, from (4.56) and from (4.61), _{0⩽𝑖⩽2ℓ}_{+1,𝑖∶even}⎛⎜⎜⎝2ℓ𝑖+1⎞⎟⎟⎠𝐸_{(𝑖+1)𝜆,𝑛+1}𝑥^{(𝜆)𝜈,𝑛}𝑃^{(2ℓ+2−𝑖)𝜆,𝑛}𝑥^{(𝜆)𝜈,𝑛}=𝑂𝑛^𝜆+2ℓ+1.(4.96) Thus, we have, for |𝑥^{(𝜆)𝜈,𝑛}|⩽1−𝜀, 𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)||||_{𝑥=𝑥(𝜆)𝜈,𝑛}=2^2ℓ(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ−2𝑥^{(𝜆)𝜈,𝑛}𝐹^{𝜆,2𝑛+1}𝑥^{(𝜆)𝜈,𝑛}+𝑂𝑛^𝜆+2ℓ+2.(4.97) Similarly, noting |𝜉^{(𝜆)𝜇,𝑛}|⩽1−𝜀, from (2.10) and (2.12) _{0⩽𝑖⩽2ℓ}_{+1,𝑖∶even}⎛⎜⎜⎝2ℓ𝑖+1⎞⎟⎟⎠𝐸_{(𝑖+1)𝜆,𝑛+1}=2_2ℓ𝜉^{(𝜆)𝜇,𝑛+1}𝑃^{(2ℓ+2−𝑖)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ−2𝜉^{(𝜆)𝜇,𝑛+1}𝐹^{𝜆,2𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^𝜆+2ℓ+2(4.98) and, from (2.8) and (4.55), _{0⩽𝑖⩽2ℓ}_{+1,𝑖∶odd}⎛⎜⎜⎝2ℓ𝑖+1⎞⎟⎟⎠𝐸_{(𝑖+1)𝜆,𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}𝑃^{(2ℓ+2−𝑖)𝜆,𝑛}𝜉^{(𝜆)𝜇,𝑛+1}=𝑂𝑛^2𝜆+2ℓ+1.(4.99) Then, we have, for |𝜉^{(𝜆)𝜇,𝑛+1}|⩽1−𝜀, 𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)||||_{𝑥=𝜉(𝜆)𝜇,𝑛+1}=2^2ℓ(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ−2𝜉^{(𝜆)𝜇,𝑛+1}𝐹^{𝜆,2𝑛+1}𝜉^{(𝜆)𝜇,𝑛+1}+𝑂𝑛^𝜆+2ℓ+2.(4.100) Therefore, we have, for |𝑦^{(𝜆)𝜈,2𝑛+1}|⩽1−𝜀, 𝐸^{𝜆,𝑛+1}(𝑥)𝑃^𝜆,𝑛(𝑥)^(2ℓ+1)||||_{𝑥=𝑦(𝜆)𝜈,2𝑛+1}=2^2ℓ(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ−2𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^𝜆+2ℓ+2.(4.101) Thus, we have, for |𝑦^{(𝜆)𝜈,2𝑛+1}|⩽1−𝜀, 𝐽^{(2ℓ+1)𝜆,𝑛}𝑦^{(𝜆)𝜈,2𝑛+1}=2^2ℓ+1(−1)^ℓ+1(𝑛+1)^2ℓ+2𝜑^−2ℓ𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^𝜆+2ℓ+2.(4.102)

Proof of Theorem 2.7. From (4.35), (3.25), (3.27), and Lemma 4.13, we have 𝜑=²−𝑦2(𝑛+^{(𝜆)𝜈,2𝑛+1}1)𝐹^{(3)𝜆,2𝑛+1}₂𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛𝐹_{𝜆,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝐽^𝜆,𝑛𝑦^{(𝜆)𝜈,2𝑛+1}=−3(𝑛+1)₂𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^𝜆+2.(4.103) Suppose that 𝐹^{(2ℓ−1)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}=𝑐^ℓ−1(−1)^ℓ−1(𝑛+1)^2ℓ−2𝜑^−2ℓ+2𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^{𝜆+2ℓ−2}.(4.104) Then, we obtain from (4.35) 𝜑=²2𝑦𝑐^{(𝜆)𝜈,2𝑛+1}_ℓ−1𝐹^{(2ℓ+1)𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}=𝑐_ℓ+2^2ℓ−1(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ+2𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2𝑛+1}+𝑂𝑛^𝜆+2ℓ(−1)^ℓ(𝑛+1)^2ℓ𝜑^−2ℓ+2𝑦^{(𝜆)𝜈,2𝑛+1}𝐹^{𝜆,2𝑛+1}𝑦^{(𝜆)𝜈,2n+1}+𝑂𝑛^𝜆+2ℓ.(4.105) Therefore, (2.16) is proved.

Acknowledgment

The authors thank the referees for many kind suggestions and comments.