scholarly journals On the Galois group of generalized Laguerre polynomials

2005 ◽  
Vol 17 (2) ◽  
pp. 517-525 ◽  
Author(s):  
Farshid Hajir
Keyword(s):  
Galois Group ◽  
Laguerre Polynomials ◽  
Generalized Laguerre Polynomials

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 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.

SPATIOTEMPORAL STRONGLY NONLOCAL SPHERICAL LIGHT BULLETS

2013 ◽  
Vol 22 (01) ◽  
pp. 1350006
Author(s):  
WEI-PING ZHONG
Keyword(s):  
Numerical Simulations ◽  
Legendre Polynomials ◽  
Schrodinger Equation ◽  
Laguerre Polynomials ◽  
Three Dimensional ◽  
Integration Constant ◽  
Generalized Laguerre Polynomials ◽  
Existence And Stability ◽  
Light Bullets ◽  
Nonlocal Nonlinear

The general spherical beam solution of the three-dimensional (3D) spatiotemporal strongly nonlocal nonlinear (NN) Schrödinger equation in the form of light bullets is presented. The 3D spatiotemporal spherical beams are built by the products of generalized Laguerre polynomials and associated Legendre polynomials. By the choice of a specific integration constant, the spherical beam becomes an accessible soliton, which can exist in various forms. We confirm the existence and stability of these solutions by numerical simulations.

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