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 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 Approximate Solutions for Fractional Logistic Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
M. M. Khader ◽  
Mohammed M. Babatin
Keyword(s):  
Differential Equation ◽  
Convergence Analysis ◽  
Upper Bound ◽  
Approximate Formula ◽  
Fractional Derivatives ◽  
Laguerre Polynomials ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Generalized Laguerre Polynomials ◽  
Theoretical Results

A new approximate formula of the fractional derivatives is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. The new spectral Laguerre collocation method is presented for solving fractional Logistic differential equation (FLDE). The properties of Laguerre polynomials approximation are used to reduce FLDE to solve a system of algebraic equations which is solved using a suitable numerical method. Numerical results are provided to confirm the theoretical results and the efficiency of the proposed method.

scholarly journals On algebraic differential equations satisfied by some elliptic functions II

1984 ◽  
Vol Volume 7 ◽  
Author(s):  
P Chowla ◽  
S Chowla
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Differential Equations ◽  
Simple Proof ◽  
Elliptic Functions ◽  
Algebraic Differential Equation ◽  
Algebraic Differential Equations ◽  
International Audience

International audience In (I) we obtained the ``implicit'' algebraic differential equation for the function defined by $Y=\sum_1^{\infty}\frac{n^a x^n}{1-x^n}$ where $a$ is an odd positive integer, and conjectured that there are no algebraic differential equations for the case when $a$ is an even integer. In this note we obtain a simple proof that (this has been known for almost 200 years) $$Y=\sum_1^{\infty}x^{n^2}~~~~(\vert x\vert<1)$$ satisfies an algebraic differential equation, and conjecture that $Y=\sum_1^{\infty} x^{n^k}$ (where $k$ is a positive bigger than $2$) does not satisfy an algebraic differential equation.

Sign in / Sign up

Export Citation Format

Share Document