scholarly journals On the Galois group of Generalised Laguerre polynomials II

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Shanta Laishram ◽  
Saranya G. Nair ◽  
T. N. Shorey
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Galois Group ◽  
Laguerre Polynomials ◽  
Mathematical Physics ◽  
Negative Integer ◽  
Algebraic Properties ◽  
International Audience

International audience For real number $\alpha,$ Generalised Laguerre Polynomials (GLP) is a family of polynomials defined by$$L_n^{(\alpha)}(x)=(-1)^n\displaystyle\sum_{j=0}^{n}\binom{n+\alpha}{n-j}\frac{(-x)^j}{j!}.$$These orthogonal polynomials are extensively studied in Numerical Analysis and Mathematical Physics. In 1926, Schur initiated the study of algebraic properties of these polynomials. We consider the Galois group of Generalised Laguerre Polynomials $L_n^{(\frac{1}{2}+u)}(x)$ when $u$ is a negative integer.

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

scholarly journals On the Galois groups of generalized Laguerre Polynomials

2014 ◽  
Vol Volume 37 ◽  
Author(s):  
Shanta Laishram
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Positive Integer ◽  
Laguerre Polynomials ◽  
Galois Groups ◽  
Newton Polygons ◽  
Short Article ◽  
Generalized Laguerre Polynomials ◽  
Algebraic Properties ◽  
International Audience

International audience For a positive integer n and a real number α, the generalized Laguerre polynomials are defined by L (α) n (x) = n j=0 (n + α)(n − 1 + α) · · · (j + 1 + α)(−x) j j!(n − j)!. These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for their interesting algebraic properties. In this short article, it is shown that the Galois groups of Laguerre polynomials L(α)(x) is Sn with α ∈ {±1,±1,±2,±1,±3} except when (α,n) ∈ {(1,2),(−2,11),(2,7)}. The proof is based on ideas of p−adic Newton polygons.

scholarly journals Algebraic Properties of a Family of Generalized Laguerre Polynomials

2009 ◽  
Vol 61 (3) ◽  
pp. 583-603 ◽  
Author(s):  
Farshid Hajir
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Alternating Group ◽  
Newton Polygons ◽  
Generalized Laguerre Polynomials ◽  
Algebraic Properties

Abstract.We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that is a ℚ-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows fromrecent work of Hajir andWong that the conjecture is true when r is large with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r ≤ 8, and (iii) when n ≤ 4. The main tool is the theory of p-adic Newton Polygons.

scholarly journals Robust Stability of Hurwitz Polynomials Associated with Modified Classical Weights

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 818 ◽  
Author(s):  
Alejandro Arceo ◽  
Luis E. Garza ◽  
Gerardo Romero
Keyword(s):  
Orthogonal Polynomials ◽  
Robust Stability ◽  
Equations Of Motion ◽  
Hurwitz Polynomials ◽  
Algebraic Properties

In this contribution, we consider sequences of orthogonal polynomials associated with a perturbation of some classical weights consisting of the introduction of a parameter t, and deduce some algebraic properties related to their zeros, such as their equations of motion with respect to t. These sequences are later used to explicitly construct families of polynomials that are stable for all values of t, i.e., robust stability on these families is guaranteed. Some illustrative examples are presented.

Eigenvalues of Finite Band-Width Hilbert Space Operators and Their Application to Orthogonal Polynomials

1989 ◽  
Vol 41 (1) ◽  
pp. 106-122 ◽  
Author(s):  
Attila Máté ◽  
Paul Nevai
Keyword(s):  
Hilbert Space ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Main Diagonal ◽  
Band Width ◽  
Positive Part ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Finite Band

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that

scholarly journals Meixner polynomials of the second kind and quantum algebras representing su(1, 1)

2009 ◽  
Vol 466 (2117) ◽  
pp. 1409-1428 ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Combinatorial Theory ◽  
Quantum Algebras ◽  
Combinatorial Approach ◽  
Meixner Polynomials ◽  
Link Type ◽  
Polynomial Coefficients ◽  
The Matrix ◽  
Math Phys

We show how Viennot’s combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges (Hodges & Sukumar 2007 Proc. R. Soc. A 463 , 2401–2414 ( doi:10.1098/rspa.2007.0001 ); Sukumar & Hodges 2007 Proc. R. Soc. A 463 , 2415–2427 ( doi:10.1098/rspa.2007.0003 )) on the matrix entries in powers of certain operators in a representation of su(1, 1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and Van Der Jeugt (Van Der Jeugt 1997 J. Math. Phys. 38 , 2728–2740 ( doi:10.1063/1.531984 ); Koelink & Van Der Jeugt 1998 SIAM J. Math. Anal. 29 , 794–822 ( doi:10.1137/S003614109630673X ); Groenevelt & Koelink 2002 J. Phys. A 35 , 65–85 ( doi:10.1088/0305-4470/35/1/306 )), for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this ‘coincidence’.

On Orthogonal Polynomials of Sobolev Type: Algebraic Properties and Zeros

1992 ◽  
Vol 23 (3) ◽  
pp. 737-757 ◽  
Author(s):  
M. Alfaro ◽  
F. Marcellán ◽  
M. L. Rezola ◽  
A. Ronveaux
Keyword(s):  
Orthogonal Polynomials ◽  
Sobolev Type ◽  
Algebraic Properties
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