Algebraic properties of a family of Jacobi polynomials
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 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: 10.5802/jtnb.659
Keywords: Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization
John Cullinan 1; Farshid Hajir 2; Elizabeth Sell 3

1 Department of Mathematics Bard College Annandale-On-Hudson, NY 12504
2 Department of Mathematics University of Massachusetts Amherst MA 01003
3 Department of Mathematics Millersville University P.O. Box 1002 Millersville, PA 17551
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John Cullinan; Farshid Hajir; Elizabeth Sell. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.659/

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