# Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights

## Abstract

Let + = [0, ∞) and R : ++ be a continuous function which is the Laguerre-type exponent, and pn, ρ(x), $\rho >-\frac{1}{2}$ be the orthonormal polynomials with the weight w ρ (x) = xρ e-R(x). For the zeros ${\left\{{x}_{k,n,\rho }\right\}}_{k=1}^{n}$ of ${p}_{n,\rho }\left(x\right)={p}_{n}\left({w}_{\rho }^{2};x\right)$, we consider the higher order Hermite-Fejér interpolation polynomial L n (l, m, f; x) based at the zeros ${\left\{{x}_{k,n,\rho }\right\}}_{k=1}^{n}$, where 0 ≤ lm - 1 are positive integers.

2010 Mathematics Subject Classification: 41A10.

## 1. Introduction and main results

Let = [-∞, ∞) and + = [0, ∞). Let R : ++ be a continuous, non-negative, and increasing function. Consider the exponential weights w ρ (x) = xρ exp(-R(x)), ρ > -1/2, and then we construct the orthonormal polynomials ${\left\{{p}_{n,\rho }\left(x\right)\right\}}_{n=0}^{\mathrm{\infty }}$ with the weight w ρ (x). Then, for the zeros ${\left\{{x}_{k,n,\rho }\right\}}_{k=1}^{n}$ of ${p}_{n,\rho }\left(x\right)={p}_{n}\left({w}_{\rho }^{2};x\right)$, we obtained various estimations with respect to ${p}_{n,\rho }^{\left(j\right)}\left({x}_{k,n,\rho }\right)$, k = 1, 2, ..., n, j = 1, 2, ..., ν, in [1]. Hence, in this article, we will investigate the higher order Hermite-Fejér interpolation polynomial L n (l, m, f; x) based at the zeros ${\left\{{x}_{k,n,\rho }\right\}}_{k=1}^{n}$, using the results from [1], and we will give a divergent theorem. This article is organized as follows. In Section 1, we introduce some notations, the weight classes ${\mathcal{L}}_{2}$, ${\stackrel{̃}{\mathcal{L}}}_{\nu }$ with $\mathcal{L}\left({C}^{2}\right)$, $\mathcal{L}\left({C}^{2}+\right)$, and main results. In Section 2, we will introduce the classes $\mathcal{F}\left({C}^{2}\right)$ and $\mathcal{F}\left({C}^{2}+\right)$, and then, we will obtain some relations of the factors derived from the classes $\mathcal{F}\left({C}^{2}\right)$, $\mathcal{F}\left({C}^{2}+\right)$ and the classes $\mathcal{L}\left({C}^{2}\right)$, $\mathcal{L}\left({C}^{2}+\right)$. Finally, we will prove the main theorems using known results in [15], in Section 3.

We say that f : + is quasi-increasing if there exists C > 0 such that f(x) ≤ Cf(y) for 0 < x < y. The notation f(x) ~ g(x) means that there are positive constants C1, C2 such that for the relevant range of x, C1f(x)/g(x) ≤ C2. The similar notation is used for sequences, and sequences of functions. Throughout this article, C, C1, C2, ... denote positive constants independent of n, x, t or polynomials P n (x). The same symbol does not necessarily denote the same constant in different occurrences. We denote the class of polynomials with degree n by ${\mathcal{P}}_{n}$.

First, we introduce classes of weights. Levin and Lubinsky [5, 6] introduced the class of weights on + as follows. Let I = [0, d), where 0 < d ≤ ∞.

Definition 1.1.[5, 6] We assume that R : I → [0, ∞) has the following properties: Let Q(t) = R(x) and x = t2.

1. (a)

$\sqrt{x}R\left(x\right)$ is continuous in I, with limit 0 at 0 and R(0) = 0;

2. (b)

R″(x) exists in (0, d), while Q″ is positive in $\left(0,\sqrt{d}\right)$;

3. (c)
$\underset{x\to d-}{lim}R\left(x\right)=\mathrm{\infty };$
4. (d)

The function

$T\left(x\right):=\frac{x{R}^{\prime }\left(x\right)}{R\left(x\right)}$

is quasi-increasing in (0, d), with

$T\left(x\right)\ge \Lambda >\frac{1}{2},\phantom{\rule{1em}{0ex}}x\in \left(0,d\right);$
1. (e)

There exists C 1 > 0 such that

$\frac{\mid {R}^{″}\left(x\right)\mid }{R\prime \left(x\right)}\le {C}_{1}\frac{{R}^{\prime }\left(x\right)}{R\left(x\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}x\in \left(0,d\right).$

Then, we write $w\in \mathcal{L}\left({C}^{2}\right)$. If there also exist a compact subinterval J* 0 of ${I}^{*}=\left(-\sqrt{d},\sqrt{d}\right)$ and C2 > 0 such that

$\frac{{Q}^{″}\left(t\right)}{\mid {Q}^{\prime }\left(t\right)\mid }\ge {C}_{2}\frac{\mid {Q}^{\prime }\left(t\right)\mid }{Q\left(t\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}t\in {I}^{*}\{J}^{*},$

then we write $w\in \mathcal{L}\left({C}^{2}+\right)$.

We consider the case d = ∞, that is, the space + = [0, ∞), and we strengthen Definition 1.1 slightly.

Definition 1.2. We assume that R : ++ has the following properties:

1. (a)

R(x), R'(x) are continuous, positive in +, with R(0) = 0, R'(0) = 0;

2. (b)

R″(x) > 0 exists in +\{0};

3. (c)
$\underset{x\to \mathrm{\infty }}{lim}\phantom{\rule{2.77695pt}{0ex}}R\left(x\right)=\mathrm{\infty };$
4. (d)

The function

$T\left(x\right):=\frac{x{R}^{\prime }\left(x\right)}{R\left(x\right)}$

is quasi-increasing in +\{0}, with

$T\left(x\right)\ge \Lambda >\frac{1}{2},\phantom{\rule{1em}{0ex}}x\in {ℝ}^{+}\\left\{0\right\};$
1. (e)

There exists C 1 > 0 such that

$\frac{{R}^{″}\left(x\right)}{{R}^{\prime }\left(x\right)}\le {C}_{1}\frac{{R}^{\prime }\left(x\right)}{R\left(x\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}x\in {ℝ}^{+}\\left\{0\right\}.$

There exist a compact subinterval J 0 of + and C2 > 0 such that

$\frac{{R}^{″}\left(x\right)}{{R}^{\prime }\left(x\right)}\ge {C}_{2}\frac{{R}^{\prime }\left(x\right)}{R\left(x\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}t\in {ℝ}^{+}\J,$

then we write $w\in {\mathcal{L}}_{2}$.

To obtain estimations of the coefficients of higher order Hermite-Fejér interpolation polynomial based at the zeros ${\left\{{x}_{k,n,\rho }\right\}}_{k=1}^{n}$, we need to focus on a smaller class of weights.

Definition 1.3. Let $w=exp\left(-R\right)\in {\mathcal{L}}_{2}$ and let ν ≥ 2 be an integer. For the exponent R, we assume the following:

1. (a)

R (j)(x) > 0, for 0 ≤ jν and x > 0, and R (j)(0) = 0, 0 ≤ jν - 1.

2. (b)

There exist positive constants C i > 0, i = 1, 2, ..., ν - 1 such that for i = 1, 2, ..., ν - 1

${R}^{\left(i+1\right)}\left(x\right)\le {C}_{i}{R}^{\left(i\right)}\left(x\right)\frac{{R}^{\prime }\left(x\right)}{R\left(x\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}x\in {ℝ}^{+}\\left\{0\right\}.$
3. (c)

There exist positive constants C, c 1 > 0 and 0 ≤ δ < 1 such that on x (0, c 1)

${R}^{\left(\nu \right)}\left(x\right)\le C{\left(\frac{1}{x}\right)}^{\delta }.$
(1.1)
4. (d)

There exists c 2 > 0 such that we have one among the following

(d1) $T\left(x\right)\text{/}\sqrt{x}$ is quasi-increasing on (c2, ∞),

(d2) R(ν)(x) is nondecreasing on (c2, ∞).

Then we write $w\left(x\right)={e}^{-R\left(x\right)}\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$.

Example 1.4.[6, 7] Let ν ≥ 2 be a fixed integer. There are some typical examples satisfying all conditions of Definition 1.3 as follows: Let α > 1, l ≥ 1, where l is an integer. Then we define

${R}_{l,\alpha }\left(x\right)={exp}_{l}\left({x}^{\alpha }\right)-{exp}_{l}\left(0\right),$

where exp l (x) = exp(exp(exp ... exp(x)) ...) is the l-th iterated exponential.

1. (1)

If α > ν, $w\left(x\right)={e}^{-{R}_{l,\alpha }\left(x\right)}\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$.

2. (2)

If αν and α is an integer, we define

${R}_{l,\alpha }^{*}\left(x\right)={exp}_{l}\left({x}^{\alpha }\right)-{exp}_{l}\left(0\right)-\sum _{j=1}^{r}\frac{{R}_{l,\alpha }^{\left(j\right)}\left(0\right)}{j!}{x}^{j}.$

Then $w\left(x\right)={e}^{-{R}_{l,\alpha }^{*}\left(x\right)}\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$.

In the remainder of this article, we consider the classes ${\mathcal{L}}_{2}$ and ${\stackrel{̃}{\mathcal{L}}}_{\nu }$; Let $w\in {\mathcal{L}}_{2}$ or $w\in {\stackrel{̃}{\mathcal{L}}}_{\nu }\phantom{\rule{2.77695pt}{0ex}}\nu \ge 2$. For $\rho >-\frac{1}{2}$, we set w ρ (x): = xρw(x). Then we can construct the orthonormal polynomials ${p}_{n,\rho }\left(x\right)={p}_{n}\left({w}_{\rho }^{2};x\right)$ of degree n with respect to ${w}_{\rho }^{2}\left(x\right)$. That is,

Let us denote the zeros of p n,ρ (x) by

$0<{x}_{n,n,\rho }<\cdots <{x}_{2,n,\rho }<{x}_{1,n,\rho }<\mathrm{\infty }.$

The Mhaskar-Rahmanov-Saff numbers a v is defined as follows:

$v=\frac{1}{\pi }{\int }_{0}^{1}\frac{{a}_{v}t{R}^{\prime }\left({a}_{v}t\right)}{\sqrt{t\left(1-t\right)}}\text{d}t,\phantom{\rule{1em}{0ex}}v>0.$

Let l, m be non-negative integers with 0 ≤ l < mν. For f C(l)(), we define the (l, m)-order Hermite-Fejér interpolation polynomials ${L}_{n}\left(l,m,f;x\right)\in {\mathcal{P}}_{mn-1}$ as follows: For each k = 1, 2, ..., n,

$\begin{array}{c}{L}_{n}^{\left(j\right)}\left(l,m,f;{x}_{k,n,\rho }\right)={f}^{\left(j\right)}\left({x}_{k,n,\rho }\right),\phantom{\rule{1em}{0ex}}j=0,1,2,\dots ,l,\\ {L}_{n}^{\left(j\right)}\left(l,m,f;{x}_{k,n,\rho }\right)=0,\phantom{\rule{1em}{0ex}}j=l+1,l+2,\dots ,m-1.\end{array}$

For each $P\in {\mathcal{P}}_{mn-1}$, we see L n (m - 1, m, P; x) = P(x). The fundamental polynomials ${h}_{s,k,n,\rho }\left(m;x\right)\in {\mathcal{P}}_{mn-1}$, k = 1, 2, ..., n, of L n (l, m, f; x) are defined by

${h}_{s,k,n,\rho }\left(l,m;x\right)={l}_{k,n,\rho }^{m}\left(x\right)\sum _{i=s}^{m-1}{e}_{s,i}\left(l,m,k,n\right){\left(x-{x}_{k,n,\rho }\right)}^{i}.$
(1.2)

Here, lk, n, ρ(x) is a fundamental Lagrange interpolation polynomial of degree n - 1 [[8], p. 23] given by

${l}_{k,n,\rho }\left(x\right)=\frac{{p}_{n}\left({w}_{\rho }^{2};x\right)}{\left(x-{x}_{k,n,\rho }\right){p}_{n}^{\prime }\left({w}_{\rho }^{2};{x}_{k,n,\rho }\right)}$

and hs,k, n, ρ(l, m; x) satisfies

${h}_{s,k,n,\rho }^{\left(j\right)}\left(l,m;{x}_{p,n,\rho }\right)={\delta }_{s,j}{\delta }_{k,p}\phantom{\rule{1em}{0ex}}j,s=0,1,\dots ,m-1,\phantom{\rule{2.77695pt}{0ex}}p=1,2,\dots ,n.$
(1.3)

Then

${L}_{n}\left(l,m,f;x\right)=\sum _{k=1}^{n}\sum _{s=0}^{l}{f}^{\left(s\right)}\left({x}_{k,n,\rho }\right){h}_{s,k,n,\rho }\left(l,m;x\right).$

In particular, for f C(), we define the m-order Hermite-Fejér interpolation polynomials ${L}_{n}\left(m,f;x\right)\in {\mathcal{P}}_{mn-1}$ as the (0, m)-order Hermite-Fejér interpolation polynomials L n (0, m, f; x). Then we know that

${L}_{n}\left(m,f;x\right)=\sum _{k=1}^{n}\phantom{\rule{2.77695pt}{0ex}}f\left({x}_{k,n,\rho }\right){h}_{k,n,\rho }\left(m;x\right),$

where e i (m, k, n): = e0,i(0, m, k, n) and

${h}_{k,n,\rho }\left(m;x\right)={l}_{k,n,\rho }^{m}\left(x\right)\sum _{i=0}^{m-1}{e}_{i}\left(m,k,n\right){\left(x-{x}_{k,n,\rho }\right)}^{i}.$
(1.4)

We often denote lk, n(x): = lk, n, ρ(x), hs, k, n(x): = hs, k, n, ρ(x), and xk, n: = xk, n, ρif they do not confuse us.

Theorem 1.5. Let$w\left(x\right)=exp\left(-R\left(x\right)\right)\in \mathcal{L}\left({C}^{2}+\right)$and ρ > -1/2.

1. (a)

For each m ≥ 1 and j = 0, 1, ..., we have

$\mid {\left({l}_{k,n}^{m}\right)}^{\left(j\right)}\left({x}_{k,n}\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le C{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{j}{x}_{k,n}^{-\frac{j}{2}}.$
(1.5)
2. (b)

For each m ≥ 1 and j = s, ..., m - 1, we have e s, s(l, m, k, n) = 1/s! and

$\mid {e}_{s,j}\left(l,m,k,n\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le C{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{j-s}{x}_{k,n}^{-\frac{j-s}{2}}.$
(1.6)

We remark ${\mathcal{L}}_{2}\subset \mathcal{L}\left({C}^{2}+\right)$.

Theorem 1.6. Let$w\left(x\right)=exp\left(-R\left(x\right)\right)\in {\stackrel{̃}{\mathcal{L}}}_{\nu },\nu \ge 2$and ρ > -1/2. Assume that 1 + 2ρ -δ/2 ≥ 0 for ρ < -1/4 and if T(x) is bounded, then assume that

${a}_{n}\le C{n}^{2∕\left(1+\nu -\delta \right)},$
(1.7)

where 0 ≤ δ < 1 is defined in (1.1). Then we have the following:

1. (a)

If j is odd, then we have for m ≥ 1 and j = 0, 1, ..., ν - 1,

$\begin{array}{cc}\hfill \mid {\left({l}_{k,n}^{m}\right)}^{\left(j\right)}\left({x}_{k,n}\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le & C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ ×{\left(\frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{j-1}{x}_{k,n}^{-\frac{j-1}{2}}.\hfill \end{array}$
(1.8)
2. (b)

If j - s is odd, then we have for m ≥ 1 and 0 ≤ sjm - 1,

$\begin{array}{cc}\hfill \mid {e}_{s,j}\left(l,m,k,n\right)\mid \phantom{\rule{2.77695pt}{0ex}}\le & C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ ×{\left(\frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{j-s-1}{x}_{k,n}^{-\frac{j-s-1}{2}}.\hfill \end{array}$
(1.9)

Theorem 1.7. Let 0 < ε < 1/4. Let$\frac{1}{\epsilon }\frac{{a}_{n}}{{n}^{2}}\le {x}_{k,n}\le \epsilon {a}_{n}$. Let s be a positive integer with 2 ≤ 2sν. Then under the same conditions as the assumptions of Theorem 1.6, there exists μ1(ε, n) > 0 such that

$\left|{p}_{n,\rho }^{\left(2s\right)}\left({x}_{k,n}\right)\right|\le C\delta \left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s-1}\left|{p}_{n}^{\prime }\left({x}_{k,n}\right)\right|{x}_{k,n}^{-\frac{\left(2s-1\right)}{2}}$

and δ (ε, n) → 0 as n → ∞ and ε → 0.

Theorem 1.8. [4, Lemma 10] Let 0 < ε < 1/4. Let$\frac{1}{\epsilon }\frac{{a}_{n}}{{n}^{2}}\le {x}_{k,n}\le \epsilon {a}_{n}$. Let s be a positive integer with 2 ≤ 2sν - 1. Suppose the same conditions as the assumptions of Theorem 1.6. Then

1. (a)

for 1 ≤ 2s - 1 ≤ ν - 1,

$\left|{\left({l}_{k,n}^{m}\right)}^{\left(2s-1\right)}\left({x}_{k,n}\right)\right|\le C\delta \left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s-1}{x}_{k,n}^{-\frac{2s-1}{2}},$
(1.10)

where δ(ε, n) is defined in Theorem 1.7.

1. (b)

there exists β(n, k) with 0 < D 1β(n, k) ≤ D 2 for absolute constants D 1, D 2 such that the following holds:

${\left({l}_{k,n}^{m}\right)}^{\left(2s\right)}\left({x}_{k,n}\right)={\left(-1\right)}^{s}{\varphi }_{s}\left(m\right){\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}\left(1+{\xi }_{s}\left(m,\epsilon ,{x}_{k,n},n\right)\right)$
(1.11)

and |ξ s (m, ε, xk, n, n)| → 0 as n → ∞ and ε → 0.

Theorem 1.9. [4, (4.16)], [9]Let 0 < ε < 1/4. Let$\frac{1}{\epsilon }\frac{{a}_{n}}{{n}^{2}}\le {x}_{k,n}\le \epsilon {a}_{n}$. Let s be a positive integer with 2 ≤ 2sm - 1. Suppose the same conditions as the assumptions of Theorem 1.6. Then for j = 0, 1, 2, ..., there is a polynomial Ψ j (x) of degree j such that (-1) j ψ j (-m) > 0 for m = 1, 3, 5, ... and the following relation holds:

${e}_{2s}\left(m,k,n\right)=\frac{{\left(-1\right)}^{s}}{\left(2s\right)!}{\mathrm{\Psi }}_{s}\left(-m\right){\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}\left(1+{\eta }_{s}\left(m,\epsilon ,{x}_{k,n},n\right)\right)$
(1.12)

and |η s (m, ε, xk, n, n)| → 0 as n → ∞ and ε → 0.

Theorem 1.10. Let m be an odd positive integer. Suppose the same conditions as the assumptions of Theorem 1.6. Then there is a function f in C(+) such that for any fixed interval [a, b], a > 0,

$\underset{n\to \mathrm{\infty }}{\text{lim}\phantom{\rule{2.77695pt}{0ex}}\text{sup}}\phantom{\rule{2.77695pt}{0ex}}\underset{a\le x\le b}{\text{max}}|{L}_{n}\left(m,f;x\right)|\phantom{\rule{2.77695pt}{0ex}}=\mathrm{\infty }.$

## 2. Preliminaries

Levin and Lubinsky introduced the classes $\mathcal{L}\left({C}^{2}\right)$ and $\mathcal{L}\left({C}^{2}+\right)$ as analogies of the classes $\mathcal{F}\left({C}^{2}\right)$ and $\mathcal{F}\left({C}^{2}+\right)$ defined on ${I}^{*}=\left(-\sqrt{d},\sqrt{d}\right)$. They defined the following:

Definition 2.1.[10] We assume that Q : I* → [0, ∞) has the following properties:

1. (a)

Q(t) is continuous in I*, with Q(0) = 0;

2. (b)

Q″(t) exists and is positive in I*\{0};

3. (c)
$\underset{t\to \sqrt{d}-}{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}Q\left(t\right)=\mathrm{\infty };$
4. (d)

The function

${T}^{*}\left(t\right):=\frac{t{Q}^{\prime }\left(t\right)}{Q\left(t\right)}$

is quasi-increasing in $\left(0,\sqrt{d}\right)$, with

${T}^{*}\left(t\right)\ge {\Lambda }^{*}>1,\phantom{\rule{1em}{0ex}}t\in {I}^{*}\\left\{0\right\};$
1. (e)

There exists C 1 > 0 such that

$\frac{{Q}^{″}\left(t\right)}{\mid {Q}^{\prime }\left(t\right)\mid }\le {C}_{1}\frac{\mid {Q}^{\prime }\left(t\right)\mid }{Q\left(t\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}t\in {I}^{*}\\left\{0\right\}.$

Then we write $W\in \mathcal{F}\left({C}^{2}\right)$. If there also exist a compact subinterval J* 0 of I* and C2 > 0 such that

$\frac{{Q}^{″}\left(t\right)}{\mid {Q}^{\prime }\left(t\right)\mid }\ge {C}_{2}\frac{{Q}^{\prime }\left(t\right)}{\mid Q\left(t\right)\mid },\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}t\in {I}^{*}\{J}^{*},$

then we write $W\in \mathcal{F}\left({C}^{2}+\right)$.

Then we see that $w\in \mathcal{L}\left({C}^{2}\right)⇔W\in \mathcal{F}\left({C}^{2}\right)$ and $w\in \mathcal{L}\left({C}^{2}+\right)⇔W\in \mathcal{F}\left({C}^{2}+\right)$ where W(t) = w(x), x = t2, from [6, Lemma 2.2]. In addition, we easily have the following:

Lemma 2.2.[1]Let Q(t) = R(x), x = t2. Then we have

$w\in {\mathcal{L}}_{2}⇒W\in \mathcal{F}\left({C}^{2}+\right),$

where W(t) = w(x); x = t2.

On , we can consider the corresponding class to ${\stackrel{̃}{\mathcal{L}}}_{\nu }$ as follows:

Definition 2.3.[11] Let $W=exp\left(-Q\right)\in \mathcal{F}\left({C}^{2}+\right)$ and ν ≥ 2 be an integer. Let Q be a continuous and even function on . For the exponent Q, we assume the following:

1. (a)

Q (j)(x) > 0, for 0 ≤ jν and t +/{0}.

2. (b)

There exist positive constants C i > 0 such that for i = 1, 2, ..., ν - 1,

${Q}^{\left(i+1\right)}\left(t\right)\le {C}_{i}{Q}^{\left(i\right)}\left(t\right)\frac{{Q}^{\prime }\left(t\right)}{Q\left(t\right)},\phantom{\rule{1em}{0ex}}\text{a}.\text{e}.\phantom{\rule{1em}{0ex}}x\in {ℝ}^{+}\\left\{0\right\}.$
3. (c)

There exist positive constants C, c 1 > 0, and 0 ≤ δ* < 1 such that on t (0, c 1),

${Q}^{\left(\nu \right)}\left(t\right)\le C{\left(\frac{1}{t}\right)}^{{\delta }^{*}}.$
(2.1)
4. (d)

There exists c 2 > 0 such that we have one among the following:

(d1) T* (t)/t is quasi-increasing on (c2, ∞),

(d2) Q(ν)(t) is nondecreasing on (c2, ∞).

Then we write $W\left(t\right)={e}^{-Q\left(t\right)}\in {\stackrel{̃}{\mathcal{F}}}_{\nu }$.

Let $W\in {\stackrel{̃}{\mathcal{F}}}_{\nu }$, and ν ≥ 2. For ${\rho }^{*}>-\frac{1}{2}$, we set

${W}_{{\rho *}^{}}\left(t\right):=\phantom{\rule{2.77695pt}{0ex}}\mid t{\mid }^{\rho *}W\left(t\right).$

Then we can construct the orthonormal polynomials ${P}_{{n,\rho *}^{}}\left(t\right)={P}_{n}\left({W}_{{\rho *}^{}}^{2};t\right)$ of degree n with respect to Wρ*(t). That is,

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{P}_{{n,\rho *}^{}}\left(v\right){P}_{{m,\rho *}^{}}\left(v\right){W}_{{\rho *}^{}}^{2}\left(v\right)\text{d}t={\delta }_{nm},\phantom{\rule{1em}{0ex}}n,m=0,1,2,\dots .$

Let us denote the zeros of Pn, ρ*(t) by

$-\mathrm{\infty }<{t}_{nn}<\cdots <{t}_{2n}<{t}_{1n}<\mathrm{\infty }.$

There are many properties of Pn, ρ*(t) = P n (Wρ*; t) with respect to Wρ*(t), $W\in {\stackrel{̃}{\mathcal{F}}}_{\nu },\nu =2,3,\dots$ of Definition 2.3 in [2, 3, 7, 1113]. They were obtained by transformations from the results in [5, 6]. Jung and Sakai [2, Theorem 3.3 and 3.6] estimate ${P}_{{n,\rho *}^{}}^{\left(j\right)}\left({t}_{k,n}\right)$, k = 1, 2, ..., n, j = 1, 2, ..., ν and Jung and Sakai [1, Theorem 3.2 and 3.3] obtained analogous estimations with respect to ${p}_{n,\rho }^{\left(j\right)}\left({x}_{k,n}\right)$, k = 1, 2, ..., n, j = 1, 2, ..., ν. In this article, we consider $w=exp\left(-R\right)\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$ and pn, ρ(x) = p n (w ρ ; x). In the following, we give the transformation theorems.

Theorem 2.4. [13, Theorem 2.1] Let W(t) = W(x) with x = t2. Then the orthonormal polynomials Pn, ρ*(t) on can be entirely reduced to the orthonormal polynomials p n , ρ (x) in +as follows: For n = 0, 1, 2, ...,

${P}_{2n,2\rho +\frac{1}{2}}\left(t\right)={p}_{n,\rho }\left(x\right)\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{P}_{2n+1,2\rho -\frac{1}{2}}\left(t\right)=t{p}_{n,\rho }\left(x\right).$

In this article, we will use the fact that w ρ (x) = xρ exp(-R(x)) is transformed into W2ρ+1/2(t) = |t|2ρ+1/2exp (-Q(t)) as meaning that

$\begin{array}{cc}\hfill {\int }_{0}^{\mathrm{\infty }}{p}_{n,\rho }\left(x\right){p}_{m,\rho }\left(x\right){w}_{\rho }^{2}\left(x\right)\text{d}x& =2{\int }_{0}^{\mathrm{\infty }}{p}_{n,\rho }\left({t}^{2}\right){p}_{m,\rho }\left({t}^{2}\right){t}^{4\rho +1}{W}^{2}\left(t\right)\text{d}t\hfill \\ ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{P}_{2n,2\rho +1∕2}\left(t\right){P}_{2m,2\rho +1∕2}\left(t\right){W}_{2\rho +1∕2}^{2}\left(t\right)\text{d}t.\hfill \end{array}$

Theorem 2.5. [1, Theorem 2.5] Let Q(t) = R(x), x = t2. Then we have

$w\left(x\right)=exp\left(-R\left(x\right)\right)\in {\stackrel{̃}{\mathcal{L}}}_{\nu }⇒W\left(t\right)=exp\left(-Q\left(t\right)\right)\in {\stackrel{̃}{\mathcal{F}}}_{\nu }.$
(2.2)

In particular, we have

${Q}^{\left(\nu \right)}\left(t\right)\le C{\left(\frac{1}{t}\right)}^{\delta },$

where 0 ≤ δ < 1 is defined in (1.1).

For convenience, in the remainder of this article, we set as follows:

${\rho }^{*}:=2\rho +\frac{1}{2}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\rho >-\frac{1}{2},\phantom{\rule{1em}{0ex}}{p}_{n}\left(x\right):={p}_{n,\rho }\left(x\right),\phantom{\rule{1em}{0ex}}{P}_{n}\left(t\right):={P}_{n,{\rho }^{*}}\left(t\right),$
(2.3)

and ${x}_{k,n}={x}_{k,n,\rho },\phantom{\rule{2.77695pt}{0ex}}{t}_{kn}={t}_{k,n,{\rho }^{*}}$. Then we know that ${\rho }^{*}>-\frac{1}{2}$ and

${p}_{n}\left(x\right)={P}_{2n,{\rho }^{*}}\left(t\right),\phantom{\rule{2.77695pt}{0ex}}x={t}^{2},\phantom{\rule{1em}{0ex}}{x}_{k,n}={t}_{k,2n}^{2},\phantom{\rule{1em}{0ex}}{t}_{k,2n}>0,\phantom{\rule{2.77695pt}{0ex}}k=1,2,\dots ,n.$
(2.4)

In the following, we introduce useful notations:

1. (a)

The Mhaskar-Rahmanov-Saff numbers a v and ${a}_{u}^{*}$ are defined as the positive roots of the following equations, that is,

$v=\frac{1}{\pi }{\int }_{0}^{1}{a}_{v}t{R}^{\prime }\left({a}_{v}t\right){\left\{t\left(1-t\right)\right\}}^{-\frac{1}{2}}dt,\phantom{\rule{2.77695pt}{0ex}}v>0$

and

$u=\frac{2}{\pi }{\int }_{0}^{1}{a}_{u}^{*}t{Q}^{\prime }\left({a}_{u}^{*}t\right){\left(1-{t}^{2}\right)}^{-\frac{1}{2}}dt,\phantom{\rule{2.77695pt}{0ex}}u>0.$
1. (b)

Let

${\eta }_{n}={\left\{nT\left({a}_{n}\right)\right\}}^{-\frac{2}{3}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\eta }_{n}^{*}={\left\{n{T}^{*}\left({a}_{n}^{*}\right)\right\}}^{-\frac{2}{3}}.$

Then we have the following:

Lemma 2.6. [6, (2.5),(2.7),(2.9)]

${a}_{n}={{a}_{2n}^{*}}^{2},\phantom{\rule{1em}{0ex}}{\eta }_{n}={4}^{2/3}{\eta }_{2n}^{*},\phantom{\rule{1em}{0ex}}T\left({a}_{n}\right)=\frac{1}{2}{T}^{*}\left({a}_{2n}^{*}\right).$

To prove main results, we need some lemmas as follows:

Lemma 2.7. [13, Theorem 2.2, Lemma 3.7] For the minimum positive zero, t[n/2],n([n/2] is the largest integern/2), we have

${t}_{\left[n/2\right],n}~{a}_{n}^{*}{n}^{-1},$

and for the maximum zero t1nwe have for large enough n,

$1-\frac{{t}_{1n}}{{a}_{n}^{*}}~{\eta }_{n}^{*},\phantom{\rule{1em}{0ex}}{\eta }_{n}^{*}={\left(n{T}^{*}\left({a}_{n}^{*}\right)\right)}^{-\frac{2}{3}}.$

Moreover, for some constant 0 < ε ≤ 2 we have

${T}^{*}\left({a}_{n}^{*}\right)\le C{n}^{2-\epsilon }.$

Remark 2.8. (a) Let $W\left(t\right)\in \mathcal{F}\left({C}^{2}+\right)$. Then

(a-1) T(x) is bounded T*(t) is bounded.

(a-2) T(x) is unbounded a n C(η)nη for any η > 0.

(a-3) T(a n ) ≤ Cn2-εfor some constant 0 < ε ≤ 2.

1. (b)

Let $w\left(x\right)\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$. Then

(b-1) ρ > -1/2 ρ* > -1/2.

(b-2) 1 + 2ρ - δ/2 ≥ 0 for ρ < -1/4 1 + 2ρ* - δ* ≥ 0 for ρ* < 0.

(b-3) ${a}_{n}\le C{n}^{2/\left(1+\nu -\delta \right)}⇒{a}_{n}^{*}\le C{n}^{1/\left(1+\nu -{\delta }^{*}\right)}$.

Proof of Remark 2.8. (a) (a-1) and (a-3) are easily proved from Lemma 2.6. From [11, Theorem 1.6], we know the following: When T*(t) is unbounded, for any η > 0 there exists C(η) > 0 such that

${a}_{t}^{*}\le C\left(\eta \right){t}^{\eta },\phantom{\rule{1em}{0ex}}t\ge 1.$

In addition, since T(x) = T*(t)/2 and ${a}_{n}={a}_{2n}^{*}{2}^{}$, we know that (a-2).

1. (b)

Since $w\left(x\right)\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$, we know that $W\left(t\right)\in {\stackrel{̃}{\mathcal{F}}}_{\nu }$ and δ* = δ by Theorem 2.5. Then from (2.3) and Lemma 2.6, we have (b-1), (b-2), and (b-3).    □

Lemma 2.9. [1, Lemma 3.6] For j = 1, 2, 3, ..., we have

${p}_{n}^{\left(j\right)}\left(x\right)=\sum _{i=1}^{j}{\left(-1\right)}^{j-i}{c}_{j,i}{P}_{2n}^{\left(i\right)}\left(t\right){t}^{-2j+i},$

where c j, i > 0(1 ≤ ij, j = 1, 2, ...) satisfy the following relations: for k = 1, 2, ...,

${c}_{k+1,1}=\frac{2k-1}{2}{c}_{k,1},\phantom{\rule{1em}{0ex}}{c}_{k+1,k+1}=\frac{1}{{2}^{k+1}},\phantom{\rule{1em}{0ex}}{c}_{1,1}=\frac{1}{2},$

and for 2 ≤ ik,

${c}_{k+1,i}=\frac{{c}_{k,i-1}+\left(2k-i\right){c}_{k,i}}{2}.$

## 3. Proofs of main results

Our main purpose is to obtain estimations of the coefficients e s, i (l, m, k, n), k = 1, 2, ..., 0 ≤ sl, sim - 1.

Theorem 3.1. [1, Theorem 1.5] Let$w\left(x\right)=exp\left(-R\left(x\right)\right)\in \mathcal{L}\left({C}^{2}+\right)$and let ρ > -1/2. For each k = 1, 2, ..., n and j = 0, 1, ..., we have

$\mid \phantom{\rule{0.3em}{0ex}}{p}_{n,\rho }^{\left(j\right)}\left({x}_{k,n}\right)\phantom{\rule{0.3em}{0ex}}\mid \le C{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{j-1}{x}_{k,n}^{-\frac{j-1}{2}}\mid \phantom{\rule{0.3em}{0ex}}{p}_{n,\rho }^{\prime }\left({x}_{k,n}\right)\phantom{\rule{0.3em}{0ex}}\mid .$

Proof of Theorem 1.5. (a) From Theorem 3.1 we know that

$\mid {l}_{k,n}^{\left(j\right)}\left({x}_{k,n}\right)\mid =\left|\frac{{p}_{n}^{\left(j+1\right)}\left({x}_{k,n}\right)}{\left(j+1\right){p}_{n}^{\prime }\left({x}_{k,n}\right)}\right|\le C{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{j}{x}_{k,n}^{-\frac{j}{2}}.$

Then, assuming that (a) is true for 1 ≤ m' < m, we have

$\begin{array}{cc}\hfill \mid {\left({l}_{k,n}^{m}\right)}^{\left(j\right)}\left({x}_{k,n}\right)\mid \phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\left|\sum _{s=0}^{j}\left(\begin{array}{c}\hfill j\hfill \\ \hfill s\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(s\right)}\left({x}_{k,n}\right){{l}_{k,n}}^{\left(j-s\right)}\left({x}_{k,n}\right)\right|\hfill \\ \le \phantom{\rule{1em}{0ex}}C\sum _{s=0}^{j}{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{s}{x}_{k,n}^{-\frac{s}{2}}{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{j-s}{x}_{k,n}^{-\frac{j-s}{2}}\hfill \\ \le \phantom{\rule{1em}{0ex}}{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{j}{x}_{k,n}^{-\frac{j}{2}}.\hfill \end{array}$

Therefore, the result is proved by induction with respect to m.

1. (b)

From (2) and (3), we know e s, s (l, m, k, n) = 1/s! and the following recurrence relation: for s + 1 ≤ im - 1,

${e}_{s,i}\left(l,m,k,n\right)=-\sum _{p-s}^{i-1}\frac{1}{\left(i-p\right)!}{e}_{s,p}\left(l,m,k,n\right){\left({l}_{k,n}\right)}^{\left(i-p\right)}\left({x}_{k,n}\right).$
(3.5)

Therefore, we have the result by induction on i and (3.5).

Theorem 3.2. [1, Theorem 1.6] Let$w\left(x\right)=exp\left(-R\left(x\right)\right)\in {\stackrel{̃}{\mathcal{L}}}_{\nu }$and let ρ > -1/2. Suppose the same conditions as the assumptions of Theorem 1.6. For each k = 1, 2, ..., n and j = 1, ..., ν, we have

$\mid \phantom{\rule{0.3em}{0ex}}{p}_{n,\rho }^{\left(j\right)}\left({x}_{k,n}\right)\phantom{\rule{0.3em}{0ex}}\mid \le C{\left(\frac{n}{\sqrt{{a}_{n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{j-1}{x}_{k,n}^{-\frac{j-1}{2}}\mid \phantom{\rule{0.3em}{0ex}}{p}_{n,\rho }^{\prime }\left({x}_{k,n}\right)\mid$

and in particular, if j is even, then we have

$\begin{array}{cc}\hfill \mid \phantom{\rule{0.3em}{0ex}}{p}_{n,\rho }^{\left(j\right)}\left({x}_{k,n}\right)\phantom{\rule{0.3em}{0ex}}\mid \phantom{\rule{1em}{0ex}}\le & C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ ×{\left(\frac{n}{\sqrt{{a}_{n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{j-2}{x}_{k,n}^{-\frac{j-2}{2}}\mid \phantom{\rule{0.3em}{0ex}}{p}_{n,\rho }^{\prime }\left({x}_{k,n}\right)\mid .\hfill \end{array}$

Proof of Theorem 1.6. We use the induction method on m.

1. (a)

For m = 1, we have the result because of

${l}_{k,n}^{\left(j\right)}\left({x}_{k,n}\right)=\frac{{p}_{n}^{\left(j+1\right)}\left({x}_{k,n}\right)}{\left(j+1\right){p}_{n}^{\prime }\left({x}_{k,n}\right)},\phantom{\rule{1em}{0ex}}j=1,2,3,\dots ,$

and Theorem 3.2. Now we assume the theorem for 1 ≤ m' < m. Then, we have the following: For 1 ≤ 2s - 1 ≤ ν - 1,

$\begin{array}{cc}\hfill {\left({l}_{k,n}^{m}\right)}^{\left(2s-1\right)}\left({x}_{k,n}\right)=& \sum _{r=0}^{s}\left(\begin{array}{c}\hfill 2s-1\hfill \\ \hfill 2r\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r-1\right)}\left({x}_{k,n}\right)\hfill \\ +\sum _{r=0}^{s}\left(\begin{array}{c}\hfill 2s-1\hfill \\ \hfill 2r+1\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(2r+1\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r-2\right)}\left({x}_{k,n}\right).\hfill \end{array}$

Since

$\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\le \frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}},$

we have

$\begin{array}{c}\left|{\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r-1\right)}\left({x}_{k,n}\right)\right|\hfill \\ \le C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ \phantom{\rule{1em}{0ex}}×{\left(\frac{n}{\sqrt{{a}_{2n}-{x}_{k,n}}}\right)}^{2r}{\left(\frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{2s-2r-2}{x}_{k,n}^{-s+1}\hfill \\ \le C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ \phantom{\rule{1em}{0ex}}×{\left(\frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{2s-2}{x}_{k,n}^{-s+1},\hfill \end{array}$

and similarly

$\begin{array}{cc}\hfill \left|{\left({l}_{k,n}^{m-1}\right)}^{\left(2r+1\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r-2\right)}\left({x}_{k,n}\right)\right|\phantom{\rule{1em}{0ex}}\le & C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ ×{\left(\frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{2s-2}{x}_{k,n}^{-s+1}.\hfill \end{array}$

Therefore, we have

$\begin{array}{cc}\hfill \left|{\left({l}_{k,n}^{m}\right)}^{\left(2s-1\right)}\left({x}_{k,n}\right)\right|\le & C\left(\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}{x}_{k,n}}}+{R}^{\prime }\left({x}_{k,n}\right)+\frac{1}{{x}_{k,n}}\right)\hfill \\ ×{\left(\frac{n}{\sqrt{{a}_{2n}}-\sqrt{{x}_{k,n}}}+\frac{T\left({a}_{n}\right)}{\sqrt{{a}_{n}}}\right)}^{2s-2}{x}_{k,n}^{-s+1}.\hfill \end{array}$
1. (b)

To prove the result, we proceed by induction on i. From (1.2) and (1.3) we know e s, s (l, m, k, n) = 1/s! and the following recurrence relation: for s + 1 ≤ im - 1,

${e}_{s,i}\left(l,m,k,n\right)=-\sum _{p=s}^{i-1}\frac{1}{\left(i-p\right)!}{e}_{s,p}\left(l,m,k,n\right){\left({l}_{k,n}^{m}\right)}^{\left(i-p\right)}\left({x}_{k,n}\right).$
(3.6)

When i - s is odd, we know that

$\left\{\begin{array}{c}\hfill i-p:\text{odd},\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}p-s:\text{even}\hfill \\ \hfill i-p:\text{even},\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}p-s:\text{odd}.\hfill \end{array}\right\$

Then, we have (1.9) from (1.5), (1.8), (3.6), and the assumption of induction on i.   □

Theorem 3.3. [1, Theorem 1.7] Let 0 < ε < 1/4. Let $\frac{1}{\epsilon }\frac{{a}_{n}}{{n}^{2}}\le {x}_{k,n}\le \epsilon {a}_{n}$and let s be a positive integer with 2 ≤ 2sν - 1. Suppose the same conditions as the assumptions of Theorem 1.6. Then there exists β(n, k), 0 < D1β(n, k) ≤ D2for absolute constants D1, D2such that the following equality holds:

${p}_{n,\rho }^{\left(2s+1\right)}\left({x}_{k,n}\right)={\left(-1\right)}^{s}{\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}\left(1+{\rho }_{s}\left(\epsilon ,{x}_{k,n},n\right)\right){p}_{n}^{\prime }\left({x}_{k,n}\right){x}_{k,n}^{-s}$

and |ρ s (ε, x k, n , n)| → 0 as n → ∞ and ε → 0.

Lemma 3.4. [3, Theorem 2.5] Let$W\in \mathcal{F}\left({C}^{2}+\right)$and r = 1, 2, . Then, uniformly for 1 ≤ kn,

$|\frac{{P}_{n}^{\left(r\right)}\left({t}_{k,n}\right)}{{{P}^{\prime }}_{n}\left({t}_{k,n}\right)}|\le C{\left(\frac{n}{\sqrt{{a}_{2n}^{*}{2}^{}-{t}_{k,n}^{2}}}\right)}^{r-1}.$

Lemma 3.5. [2, Theorem 3.3] Let ρ* > -1/2 and$W\left(x\right)=exp\left(-Q\left(x\right)\right)\in {\stackrel{̃}{\mathcal{F}}}_{\nu }$, ν ≥ 2. Assume that 1 + 2ρ* - δ* ≥ 0 for ρ* < 0 and if T*(t) is bounded, then assume

${a}_{n}^{*}\le C{n}^{1/\left(1+\nu -{\delta }^{*}\right)},$

where 0 ≤ δ* < 1 is defined in (2.1). Let 0 < α < 1/2. Let$\frac{1}{\epsilon }\frac{{a}_{n}^{*}}{n}\le \mid {t}_{kn}\mid \le \epsilon {a}_{n}^{*}$and let s be a positive integer with 2 ≤ 2sν. Then there exists μ(ε, n) > 0 such that

$\left|{P}_{n}^{\left(2s\right)}\left({t}_{k,n}\right)\right|\le C\mu \left(\epsilon ,n\right){\left(\frac{n}{{a}_{n}}\right)}^{2s-1}\left|{P}_{n}^{\prime }\left({t}_{k,n}\right)\right|$

and μ(ε, n) → 0 as n → ∞ and ε → 0.

Proof of Theorem 1.7. By Lemma 2.9, we have

$\begin{array}{cc}\hfill \left|{p}_{n}^{\left(2s\right)}\left({x}_{k,n}\right)\right|& =\phantom{\rule{1em}{0ex}}\left|\sum _{i=1}^{2s}{\left(-1\right)}^{2s-i}{c}_{2s,i}{P}_{2n}^{\left(i\right)}\left({t}_{k,n}\right){t}_{k,n}^{-4s+i}\right|\hfill \\ \le \phantom{\rule{1em}{0ex}}C\left|{c}_{2s,2s}{P}_{2n}^{\left(2s\right)}\left({t}_{k,n}\right){t}_{k,n}^{-2s}\right|+\left|\sum _{i=1}^{2s-1}{\left(-1\right)}^{2s-i}{c}_{2s,i}{P}_{2n}^{\left(i\right)}\left({t}_{k,n}\right){t}_{k,n}^{-4s+i}\right|.\hfill \end{array}$

Since, we have by Lemma 3.5,

$\left|{c}_{2s,2s}{P}_{2n}^{\left(2s\right)}\left({t}_{k,n}\right){t}_{k,n}^{-2s}\right|\le C\mu \left(\epsilon ,2n\right){\left(\frac{n}{{a}_{2n}^{*}}\right)}^{2s-1}\left|{P}_{2n}^{\prime }\left({t}_{k,n}\right)\right|{\left|{t}_{k,n}\right|}^{-2s}$

and by Lemma 3.4,

$\begin{array}{c}\left|\sum _{i=1}^{2s-1}{\left(-1\right)}^{2s-i}{c}_{2s,i}{P}_{2n}^{\left(i\right)}\left({t}_{k,n}\right){t}_{k,n}^{-4s+i}\right|\hfill \\ \hfill \le \phantom{\rule{1em}{0ex}}& C{\left(\frac{n}{{a}_{2n}^{*}}\right)}^{2s-1}\left|{P}_{2n}^{\prime }\left({t}_{k,n}\right)\right|{\left|{t}_{k,n}\right|}^{-2s}\sum _{i=1}^{2s-1}{\left(\frac{n}{{a}_{2n}^{*}}\mid {t}_{k,n}\mid \right)}^{-2s+i}\hfill \\ \hfill \le \phantom{\rule{1em}{0ex}}& C\epsilon {\left(\frac{n}{{a}_{2n}^{*}}\right)}^{2s-1}\left|{P}_{2n}^{\prime }\left({t}_{k,n}\right)\right|{\left|{t}_{k,n}\right|}^{-2s},\hfill \end{array}$

we have

$\begin{array}{cc}\hfill \left|{p}_{n}^{\left(2s\right)}\left({x}_{k,n}\right)\right|& \le C\delta \left(\epsilon ,n\right){\left(\frac{n}{{a}_{2n}^{*}}\right)}^{2s-1}\left|{P}_{2n}^{\prime }\left({t}_{k,n}\right)\right|{\left|{t}_{k,n}\right|}^{-2s}\hfill \\ \le C\delta \left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s-1}\left|{p}_{n}^{\prime }\left({x}_{k,n}\right)\right|{x}_{k,n}^{-\frac{\left(2s-1\right)}{2}},\hfill \end{array}$

where δ(ε, n) = μ(ε, 2n) + ε.   □

Here we can estimate the coefficients e i (ν, k, n) of the fundamental polynomials h kn (ν; x).

For j = 0, 1, ..., define ϕ j (1): = (2j + 1)-1 and for k ≥ 2,

${\phi }_{j}\left(k\right):=\sum _{r=0}^{j}\frac{1}{2j-2r+1}\left(\begin{array}{c}\hfill 2j\hfill \\ \hfill 2r\hfill \end{array}\right){\phi }_{r}\left(k-1\right).$
(3.7)

Proof of Theorem 1.8. In a manner analogous to the proof of Theorem 1.6 (a), we use mathematical induction with respect to m.

1. (a)

From Theorem 1.7, we know that for 1 ≤ 2s -1 ≤ ν - 1,

$\left|{l}_{k,n}^{\left(2s-1\right)}\left({x}_{k,n}\right)\right|=\left|\frac{{p}_{n}^{\left(2s\right)}\left({x}_{k,n}\right)}{2s{p}_{n}^{\prime }\left({x}_{k,n}\right)}\right|\le C\delta \left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s-1}{x}_{k,n}^{-\frac{2s-1}{2}}.$

From Theorem 1.5, we know that for x k, n a n /4,

$\left|{\left({l}_{k,n}^{m}\right)}^{\left(j\right)}\left({x}_{k,n}\right)\right|\le C{\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{j}{x}_{k,n}^{-\frac{j}{2}}.$
(3.8)

Then, we have by mathematical induction on m,

$\begin{array}{cc}\hfill \left|{\left({l}_{k,n}^{m}\right)}^{\left(2s-1\right)}\left({x}_{k,n}\right)\right|\le & C\sum _{r=0}^{s}\left(\begin{array}{c}\hfill 2s-1\hfill \\ \hfill 2r\hfill \end{array}\right)\left|{\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r-1\right)}\left({x}_{k,n}\right)\right|\hfill \\ +\sum _{r=0}^{s}\left(\begin{array}{c}\hfill 2s-1\hfill \\ \hfill 2r+1\hfill \end{array}\right)\left|{\left({l}_{k,n}^{m-1}\right)}^{\left(2r+1\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r-2\right)}\left({x}_{k,n}\right)\right|\hfill \\ \hfill \le & C\delta \left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s-1}{x}_{k,n}^{-\frac{2s-1}{2}}.\hfill \end{array}$
1. (b)

From Theorem 3.3, we know that for 0 ≤ 2sν - 1,

$\begin{array}{cc}\hfill {l}_{k,n}^{\left(2s\right)}\left({x}_{k,n}\right)& =\frac{{p}_{n}^{\left(2s+1\right)}\left({x}_{k,n}\right)}{\left(2s+1\right){p}_{n}^{\prime }\left({x}_{k,n}\right)}\hfill \\ ={\left(-1\right)}^{s}{\varphi }_{s}\left(1\right){\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}\left(1+{\rho }_{s}\left(\epsilon ,{x}_{k,n},n\right)\right).\hfill \end{array}$
(3.9)

If we let ξ s (1, ε, x k, n , n) = ρ s (ε, x k, n , n), then (1.11) holds for m = 1 because |ξ s (1, ε, x k, n , n)| → 0 as n → ∞ and ε → 0. Now, we split ${\left({l}_{k,n}^{m}\right)}^{\left(2s\right)}\left({x}_{k,n}\right)$ into two terms as follows:

$\begin{array}{cc}\hfill {\left({l}_{k,n}^{m}\right)}^{\left(2s\right)}\left({x}_{k,n}\right)=& \sum _{0\le 2r\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r\right)}\left({x}_{k,n}\right)\hfill \\ +\sum _{1\le 2r-1\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r-1\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(2r-1\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r+1\right)}\left({x}_{k,n}\right).\hfill \end{array}$
(3.10)

For the second term, we have from (1.10),

$\left|\sum _{1\le 2r-1\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r-1\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(2r-1\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r+1\right)}\left({x}_{k,n}\right)\right|\le C{\delta }^{2}\left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}.$
(3.11)

For the first term, we let ξ s (m) = ξ s (m, ε, x k, n , n) for convenience. Then we know that

${l}_{k,n}^{\left(2s-2r\right)}\left({x}_{k,n}\right)={\left(-1\right)}^{s-r}{\varphi }_{s-r}\left(1\right){\beta }^{s-r}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s-r}{x}_{k,n}^{-\left(s-r\right)}\left(1+{\xi }_{s-r}\left(1\right)\right)$

and |ξs-r(1)| → 0 as n → ∞ and ε → 0. By mathematical induction, we assume for 0 ≤ 2r ≤ 2s;

${\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right)={\left(-1\right)}^{r}{\varphi }_{r}\left(m-1\right){\beta }^{r}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2r}{x}_{k,n}^{-r}\left(1+{\xi }_{r}\left(m-1\right)\right)$

and |ξ r (m - 1)| → 0 as n → ∞ and ε → 0. Then, since

$\begin{array}{c}\hfill {\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r\right)}\left({x}_{k,n}\right)={\left(-1\right)}^{s}{\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}\\ \hfill \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{\varphi }_{r}\left(m-1\right){\varphi }_{s-r}\left(1\right)\left(1+{\xi }_{r}\left(m-1\right)\right)\left(1+{\xi }_{s-r}\left(1\right)\right),\end{array}$

we have for 0 ≤ 2r ≤ 2s, using the definition of (3.7),

$\begin{array}{c}\sum _{0\le 2r\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right){\left({l}_{k,n}^{m-1}\right)}^{\left(2r\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r\right)}\left({x}_{k,n}\right)\hfill \\ \hfill =& {\left(-1\right)}^{s}{\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}\hfill \\ ×\sum _{0\le 2r\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right){\varphi }_{r}\left(m-1\right){\varphi }_{s-r}\left(1\right)\left(1+{\xi }_{r}\left(m-1\right)\right)\left(1+{\xi }_{s-r}\left(1\right)\right)\hfill \\ \hfill =& {\left(-1\right)}^{s}{\varphi }_{s}\left(m\right){\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}+{\left(-1\right)}^{s}{\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}\hfill \\ ×\sum _{0\le 2r\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right){\varphi }_{r}\left(m-1\right){\varphi }_{s-r}\left(1\right)\left({\xi }_{r}\left(m-1\right)+{\xi }_{s-r}\left(1\right)+{\xi }_{r}\left(m-1\right){\xi }_{s-r}\left(1\right)\right).\hfill \end{array}$

Here, we consider (3.10). If we let

$\begin{array}{c}{\xi }_{s}\left(m,\epsilon ,{x}_{k,n},n\right)={\xi }_{s}\left(m\right)=\\ \sum _{0\le 2r\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right)\frac{{\varphi }_{r}\left(m-1\right){\varphi }_{s-r}\left(1\right)}{{\varphi }_{s}\left(m\right)}\left({\xi }_{r}\left(m-1\right)+{\xi }_{s-r}\left(1\right)+{\xi }_{r}\left(m-1\right){\xi }_{s-r}\left(1\right)\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\sum _{1\le 2r-1\le 2s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r-1\hfill \end{array}\right)\frac{{\left({l}_{k,n}^{m-1}\right)}^{\left(2r-1\right)}\left({x}_{k,n}\right){l}_{k,n}^{\left(2s-2r+1\right)}\left({x}_{k,n}\right)}{{\left(-1\right)}^{s}{\varphi }_{s}\left(m\right){\beta }^{s}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2s}{x}_{k,n}^{-s}},\end{array}$

then we have

Then, we know that (1.11) holds and |ξ i (j)| → 0 as n → ∞ and ε → 0, using mathematical induction on m. Therefore, we have the result.

We rewrite the relation (3.7) in the form for ν = 1, 2, 3 ...,

${\varphi }_{0}\left(\nu \right):=1$

and for j = 1, 2, 3 ..., ν = 2, 3, 4, ...,

${\varphi }_{j}\left(\nu \right)-{\varphi }_{j}\left(\nu -1\right)=\frac{1}{2j+1}\sum _{r=0}^{j-1}\left(\begin{array}{c}\hfill 2j+1\hfill \\ \hfill 2r\hfill \end{array}\right){\varphi }_{r}\left(\nu -1\right).$

Now, for every j we will introduce an auxiliary polynomial determined by ${\left\{{\mathrm{\Psi }}_{j}\left(y\right)\right\}}_{j=1}^{\mathrm{\infty }}$ as the following lemma:

Lemma 3.6. [4, Lemma 11] (i) For j = 0, 1, 2 ..., there exists a unique polynomial Ψ j (y) of degree j such that

${\Psi }_{j}\left(\nu \right)={\varphi }_{j}\left(\nu \right),\phantom{\rule{1em}{0ex}}\nu =1,2,3,\dots .$
1. (ii)

Ψ0(y) = 1 and Ψ j (0) = 0, j = 1, 2, ....

Since Ψ j (y) is a polynomial of degree j, we can replace ϕ j (ν) in (3.7) with Ψ j (y), that is,

${\mathrm{\Psi }}_{j}\left(y\right)=\sum _{r=0}^{j}\frac{1}{2j-2r+1}\left(\begin{array}{c}\hfill 2j\hfill \\ \hfill 2r\hfill \end{array}\right)\phantom{\rule{2.77695pt}{0ex}}{\mathrm{\Psi }}_{r}\left(y-1\right),\phantom{\rule{1em}{0ex}}j=0,1,2,...,$

for an arbitrary y and j = 0, 1, 2, .... We use the notation F kn (x, y) = (l k, n (x)) y which coincides with ${l}_{k,n}^{y}\left(x\right)$ if y is an integer. Since l k, n (x k, n ) = 1, we have F kn (x, t) > 0 for x in a neighborhood of xk, nand an arbitrary real number t.

We can show that (∂/∂x) j F kn (x k, n , y) is a polynomial of degree at most j with respect to y for j = 0, 1, 2, ..., where (∂/∂x) j F kn (x k, n , y) is the j th partial derivative of F kn (x, y) with respect to x at (x k, n , y) [14, p. 199]. We prove these facts by induction on j. For j = 0 it is trivial. Suppose that it holds for j ≥ 0. To simplify the notation, let F(x) = F kn (x, y) and l(x) = l k, n (x) for a fixed y. Then F'(x)l(x) = yl'(x)F(x). By Leibniz's rule, we easily see that

${F}^{\left(j+1\right)}\left({x}_{k,n}\right)=-\sum _{s=0}^{j-1}\left(\begin{array}{c}\hfill j\hfill \\ \hfill s\hfill \end{array}\right){F}^{\left(s+1\right)}\left({x}_{k,n}\right){l}^{\left(j-s\right)}\left({x}_{k,n}\right)+y\sum _{s=0}^{j}\left(\begin{array}{c}\hfill j\hfill \\ \hfill s\hfill \end{array}\right){l}^{\left(s+1\right)}\left({x}_{k,n}\right){F}^{\left(j-s\right)}\left({x}_{k,n}\right),$

which shows that F(j+1)(x k, n ) is a polynomial of degree at most j + 1 with respect to y. Let ${P}_{kn}^{\left[j\right]}\left(y\right)$, j = 0, 1, 2, ... be defined by

${\left(\partial ∕\partial x\right)}^{2j}{F}_{kn}\left({x}_{k,n},y\right)={\left(-1\right)}^{j}{\beta }^{j}\left(2n,k\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2j}{x}_{k,n}^{-j}{\mathrm{\Psi }}_{j}\left(y\right)+{P}_{kn}^{\left[j\right]}\left(y\right).$
(3.12)

Then ${P}_{kn}^{\left[j\right]}\left(y\right)$ is a polynomial of degree at most 2j.

By Theorem 1.8 (1.11), we have the following:

Lemma 3.7. [4, Lemma 12] Let j = 0, 1, 2, ..., and M be a positive constant. Let 0 < ε < 1/4, $\frac{1}{\epsilon }\frac{{a}_{n}}{{n}^{2}}\le {x}_{k,n}\le \epsilon {a}_{n}$, and |y| ≤ M. Then

1. (a)

there exists κ j (ε, x k, n , n) > 0 such that

$\left|{\left(\partial ∕\partial y\right)}^{s}{P}_{kn}^{\left[j\right]}\left(y\right)\right|\le C{\kappa }_{j}\left(\epsilon ,{x}_{k,n},n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2j}{x}_{k,n}^{-j},\phantom{\rule{1em}{0ex}}s=0,1$
(3.13)

and κ j (ε, x k, n , n) → 0 as n → ∞ and ε → 0.

1. (b)

there exists γ j (ε, n) > 0 such that

$\left|{\left(\partial ∕\partial x\right)}^{2j+1}{F}_{kn}\left({x}_{k,n},y\right)\right|\le C{\gamma }_{j}\left(\epsilon ,n\right){\left(\frac{n}{\sqrt{{a}_{n}}}\right)}^{2j+1}{x}_{k,n}^{-\frac{2j+1}{2}}$
(3.14)

and γ j (ε, n) → 0 as n → ∞ and ε → 0.

Lemma 3.8. [4, Lemma 13] If y < 0, then for j = 0, 1, 2 ...,

${\left(-1\right)}^{j}{\mathrm{\Psi }}_{j}\left(y\right)>0.$

Lemma 3.9. For positive integers s and m with 1 ≤ mν,

$\sum _{r=0}^{s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right){\mathrm{\Psi }}_{r}\left(-m\right){\phi }_{s-r}\left(m\right)=0.$

Proof. If we let ${C}_{s}\left(y\right)={\sum }_{r=0}^{s}\left(\begin{array}{c}\hfill 2s\hfill \\ \hfill 2r\hfill \end{array}\right){\mathrm{\Psi }}_{r}\left(-y\right){\mathrm{\Psi }}_{s-r}\left(y\right)$, then it suffices to show that C s (m) = 0. For every s,

$\begin{array}{cc}\hfill 0& ={\left({l}_{k,n}^{-m+m}\right)}^{2s}\left({x}_{k,\mathrm{n