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 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 Classifying Galois groups of an orthogonal family of quartic polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 172-190
Author(s):  
Pradipto Banerjee ◽  
◽  
Ranjan Bera ◽  
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Galois Groups ◽  
Newton Polygons ◽  
Diophantine Problem ◽  
Generalized Laguerre Polynomials ◽  
Precise Characterization ◽  
Transitive Subgroup ◽  
Quartic Polynomials

We consider the quartic generalized Laguerre polynomials $L_{4}^{(\alpha)}(x)$ for $\alpha \in \mathbb Q$. It is shown that except $\mathbb Z/4\mathbb Z$, every transitive subgroup of $S_{4}$ appears as the Galois group of $L_{4}^{(\alpha)}(x)$ for infinitely many $\alpha \in \mathbb Q$. A precise characterization of $\alpha\in \mathbb Q$ is obtained for each of these occurrences. Our methods involve the standard use of resolvent cubics and the theory of p-adic Newton polygons. Using these, the Galois group computations are reduced to Diophantine problem of finding integer and rational points on certain curves.

scholarly journals On classifying Laguerre polynomials which have Galois group the alternating group

2013 ◽  
Vol 25 (1) ◽  
pp. 1-30 ◽  
Author(s):  
Pradipto Banerjee ◽  
Michael Filaseta ◽  
Carrie E. Finch ◽  
J. Russell Leidy
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Alternating Group

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 The Galois group of f(xr)

1984 ◽  
Vol 25 (1) ◽  
pp. 75-91 ◽  
Author(s):  
S. D. Cohen ◽  
W. W. Stothers
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Irreducible Polynomial ◽  
Alternating Group

Let f(x) be an irreducible polynomial of degree n with coefficients in a field L and r be an integer prime to the characteristic of L. The object of this paper is to describe the galois group g of f(xr) over L when the galois group G of f(x) itself over L is either the full symmetric group Snor the alternating group An. We shall call f standard if G = Sn or An with |G|>2.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

scholarly journals Rational Trinomials with the Alternating Group as Galois Group

2001 ◽  
Vol 90 (1) ◽  
pp. 113-129 ◽  
Author(s):  
Alain Hermez ◽  
Alain Salinier
Keyword(s):  
Galois Group ◽  
Alternating Group

scholarly journals On purely periodic beta-expansions of Pisot numbers

2002 ◽  
Vol 166 ◽  
pp. 183-207 ◽  
Author(s):  
Yuki Sano
Keyword(s):  
Natural Extension ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Pisot Number ◽  
Fractal Boundary ◽  
Pisot Numbers ◽  
Dimensional Domain

AbstractWe characterize numbers having purely periodic β-expansions where β is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a d-dimensional domain with a fractal boundary.

Infinite Families of A4-Sextic Polynomials

2014 ◽  
Vol 57 (3) ◽  
pp. 538-545 ◽  
Author(s):  
Joshua Ide ◽  
Lenny Jones
Keyword(s):  
Galois Group ◽  
Alternating Group

AbstractIn this article we develop a test to determine whether a sextic polynomial that is irreducible overℚhas Galois group isomorphic to the alternating groupA4. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

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