Algebraic properties of a family of Jacobi polynomials
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 97-108

The one-parameter family of polynomials ${J}_{n}\left(x,y\right)={\sum }_{j=0}^{n}\left(\genfrac{}{}{0pt}{}{y+j}{j}\right){x}^{j}$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n\ge 6$, the polynomial ${J}_{n}\left(x,{y}_{0}\right)$ is irreducible over $ℚ$ for all but finitely many ${y}_{0}\in ℚ$. If $n$ is odd, then with the exception of a finite set of ${y}_{0}$, the Galois group of ${J}_{n}\left(x,{y}_{0}\right)$ is ${S}_{n}$; if $n$ is even, then the exceptional set is thin.

La famille des polynômes à un seul paramètre ${J}_{n}\left(x,y\right)={\sum }_{j=0}^{n}\left(\genfrac{}{}{0pt}{}{y+j}{j}\right){x}^{j}$ est une sous-famille de la famille (à deux paramètres) des polynômes de Jacobi. On montre que pour chaque $n\ge 6$, quand on spécialise en ${y}_{0}\in ℚ$, le polynôme ${J}_{n}\left(x,{y}_{0}\right)$ est irréductible sur $ℚ$, sauf pour un nombre fini des valeurs ${y}_{0}\in ℚ$. Si $n$ est impair, sauf pour un nombre fini des valeurs ${y}_{0}\in ℚ$, le groupe de Galois de ${J}_{n}\left(x,{y}_{0}\right)$ est ${S}_{n}$ ; si $n$ est pair, l’ensemble exceptionnel est mince.

DOI : https://doi.org/10.5802/jtnb.659
Keywords: Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization
@article{JTNB_2009__21_1_97_0,
author = {Cullinan, John and Hajir, Farshid and Sell, Elizabeth},
title = {Algebraic properties of a family of Jacobi polynomials},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {1},
year = {2009},
pages = {97-108},
doi = {10.5802/jtnb.659},
mrnumber = {2537705},
zbl = {pre05620670},
language = {en},
url = {http://www.numdam.org/item/JTNB_2009__21_1_97_0}
}

Cullinan, John; Hajir, Farshid; Sell, Elizabeth. Algebraic properties of a family of Jacobi polynomials. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 97-108. doi : 10.5802/jtnb.659. http://www.numdam.org/item/JTNB_2009__21_1_97_0/

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