In this work, we make an attempt to gather different results by many mathematicians who have made landmark achievements in studies on zeros of Laguerre polynomials. We discuss bounds for extreme zeros of some classical orthogonal polynomials, bounds for zeros of the Laguerre polynomials where we quote the works of Ilia Krasikov, among others.
The quoted works are: “Limit relation for the complex zeros of Laguerre polynomials”, “On the Asymptotic Distribution of the Zeros of Laguerre Polynomials”, “Monotonicity of zeros of Laguerre polynomials”, “Zeros of linear combinations of Laguerre polynomials from different sequences”, “Linear combinations of Laguerre polynomials of the same degree”, “Linear combinations of Laguerre polynomials of different degree” and “Convexity of the extreme zeros of Laguerre polynomials”.
Frequently asked questions about "Orthogonal polynomials"
What is Chapter 1 about in this text?
Chapter 1 introduces orthogonal polynomials, their properties, and historical context. It mentions Markov and Stieltjes' early work, Sturm's comparison theorem, Obrechkoff's theorem, and the Wall-Wetzel theorem. It also discusses the interlacing of zeros, dual polynomials, and recent developments using chain sequences and inequalities for real-root polynomials.
What topics are covered in the 'Preliminaries' section (1.2)?
Section 1.2 covers the definitions of polynomials and polynomial functions, including forms of polynomial functions, polynomial functions with multiple variables, and orthogonality of a single polynomial. It also covers how to determine what is and is not a valid polynomial function. It also discusses Kronecker delta functions.
What are some applications of orthogonal polynomials?
Applications of orthogonal polynomials include quadrature, numerical interpolation, error estimation in Padé approximations, solutions to ordinary and partial differential equations, rational approximations and equilibrium distributions, factorization of second-order difference equations, Gauss quadrature for analytic functions, and electrostatic interpretation of zeros of functions.
What does Chapter 2 focus on?
Chapter 2 is dedicated to the Laguerre Polynomials, including their recursive relations, closed form, and generating functions. It provides the first few terms of Laguerre polynomials.
What kind of information does the text provide about the zeros of Laguerre Polynomials in Chapter 3?
Chapter 3 discusses bounds for extreme zeros of some classical orthogonal polynomials and bounds for zeros of the Laguerre polynomials. It includes results from Ilia Krasikov and theorems on interlacing zeros.
What is the key result from Krasikov's work mentioned in section 3.3?
Theorem 3.3.1 provides inequalities for the least (x1) and largest (xk) zeros of the Laguerre polynomial L(α)k(x) for k ≥ 7. It also states that all zeros are confined between the two real roots of a specific equation (3.3.3).
What is the "Limit relation for the complex zeros of Laguerre polynomials" from section 3.4?
This section extracts from "Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials", and covers how to derive the complex zeros of the Laguerre polynomial, and the relationships between them.
What does section 3.5 discuss?
Section 3.5 provides data on "On the Asymptotic Distribution of the Zeros of Laguerre Polynomials"
What's the focus of section 3.6?
Section 3.6 covers "Monotonicity of zeros of Laguerre polynomials" This includes data on monotonicity with respect to parameters of certain functions involving the Laguerre equation. It also helps establish sharp upper bounds.
What are linear combinations of Laguerre polynomials of different sequences and how are the zeros established?
Section 3.7.2, "Zeros of linear combinations of Laguerre polynomials from different sequences" covers how to establish the zeros of various Laguerre Polynomial functions using known identities.
What is the major point from the section "Convexity of the extreme zeros of Laguerre polynomials"?
Section 3.8, "Convexity of the extreme zeros of Laguerre polynomials" covers the cases where the zeros of L(t) on the interval (0,1) are convex or concave.
Where can I find the references to the work cited in the text?
The bibliography, starting near the end of the HTML, lists all the cited material, sorted by reference ID. Each reference also contains the full citation text, for example: [1] Abramowitz, M. and Stegun, I. A. (Eds.) (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 19.
- Quote paper
- Ayo Odeniran (Author), 2011, On Zeros Of Laguerre Polynomials, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/339110