10.46298/hrj.2014.1317
https://hrj.episciences.org/1317
Laishram, Shanta
Shanta
Laishram
On the Galois groups of generalized Laguerre Polynomials
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.
episciences.org
Laguerre Polynomials
Primes
Arithmetic Progressions
Newton Polygons
Irreducibility
[MATH] Mathematics [math]
[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
2015-10-30
2014-01-01
2014-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01220303v1
2804-7370
https://hrj.episciences.org/1317/pdf
VoR
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Hardy-Ramanujan Journal
Volume 37 - 2014
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