# An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction and Background

## 2. Connection and Recurrence Formulas

#### 2.1. Connection Formula

**Proposition**

**1.**

**Proof.**

#### 2.2. Five-Term Recurrence Relation

**Lemma**

**1.**

**Theorem**

**1**(Five-term recurrence relation)

**.**

**Proof.**

#### 2.3. A Recurrence Formula with Rational Coefficients

**Theorem**

**2.**

**Proof.**

## 3. Asymptotics for the Recurrence Coefficients: The Freud Case

**Proposition**

**2.**

- $\left(i\right)$
- $\left(ii\right)$
- String equation (see (2.12) in [22]). ${\left\{{a}_{n}\right\}}_{n\ge 1}$ satisfies the following nonlinear difference equation$$4{a}_{n}^{2}\left({a}_{n+1}^{2}+{a}_{n}^{2}+{a}_{n-1}^{2}\right)=n,\phantom{\rule{1.em}{0ex}}n\ge 1.$$
- $\left(iii\right)$
- Asymptotics for ${f}_{n}\left(0\right)$ (see Th. 6 in [15]). There exists a constant $A=\sqrt[8]{12}/\sqrt{\pi}$ such that the following estimates hold$$\begin{array}{ccc}\hfill {f}_{n}\left(0\right)& =& \left\{\begin{array}{cc}0\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}odd\hfill \\ {(-1)}^{n/2}{n}^{-1/8}(A+o\left(1\right))\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}even\hfill \end{array}\right.,\hfill \\ \hfill {}_{n}{]}^{\prime}\left(0\right)& =& \left\{\begin{array}{cc}{(-1)}^{(n-1)/2}\frac{\sqrt{8}}{\sqrt[4]{27}}{n}^{5/8}(A+o\left(1\right))\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}odd\hfill \\ 0\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}even\hfill \end{array}\right.,\hfill \\ \hfill {}_{n}{]}^{\prime \prime}\left(0\right)& =& \left\{\begin{array}{cc}0\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}odd\hfill \\ {\left(-1\right)}^{\frac{n}{2}+1}\frac{8}{3\sqrt{3}}{n}^{11/8}(A+o\left(1\right))\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}even\hfill \end{array}\right.,\hfill \\ \hfill {}_{n}{]}^{\prime \prime \prime}\left(0\right)& =& \left\{\begin{array}{cc}{(-1)}^{(n-1)/2}\frac{16\sqrt{2}}{9\sqrt[4]{3}}{n}^{17/8}(A+o\left(1\right))\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}odd\hfill \\ 0\hfill & if\phantom{\rule{4.pt}{0ex}}n\phantom{\rule{4.pt}{0ex}}is\phantom{\rule{4.pt}{0ex}}even\hfill \end{array}\right..\hfill \end{array}$$

**Lemma**

**2.**

**Proof.**

#### 3.1. Asymptotics of the Recurrence Coefficients

**Lemma**

**3.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 3.2. Asymptotics for the Rational Recurrence Coefficients

**Proposition**

**4.**

**Proposition**

**5**(Modified string equation).

**Proof.**

## 4. Conclusions and Further Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Garza, L.G.; Garza, L.E.; Huertas, E.J.
An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials. *Symmetry* **2021**, *13*, 534.
https://doi.org/10.3390/sym13040534

**AMA Style**

Garza LG, Garza LE, Huertas EJ.
An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials. *Symmetry*. 2021; 13(4):534.
https://doi.org/10.3390/sym13040534

**Chicago/Turabian Style**

Garza, Lino G., Luis E. Garza, and Edmundo J. Huertas.
2021. "An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials" *Symmetry* 13, no. 4: 534.
https://doi.org/10.3390/sym13040534