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For different values of parameter $r$, this family provides polynomials which are of great interest. Hajir conjectured that for integers $r\\geq 0$ and $n\\geq 1$, $L_{n}^{(-1-n-r)}(x)$ is an irreducible polynomial whose Galois group contains $A_n$, the alternating group on $n$ symbols. 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For different values of parameter $r$, this family provides polynomials which are of great interest. Hajir conjectured that for integers $r\\geq 0$ and $n\\geq 1$, $L_{n}^{(-1-n-r)}(x)$ is an irreducible polynomial whose Galois group contains $A_n$, the alternating group on $n$ symbols. 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