Extension of Laguerre polynomials with negative arguments

Abstract

We consider the irreducibility of polynomial $L_n^{(\alpha )}(x)$ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{(\alpha )}(x)$ vanishes if and only if $n\ge |\alpha |=-\alpha$. Therefore we assume that $\alpha =-n-s-1$ where $s$ is a non-negative integer. Let $$g(x)=(-1{)}^n{L}_n^{(-n-s-1)}(x)=\sum _{j=0}^n{a}_j\frac{x^j}{j!}$$ and more general polynomial, let $$G(x)=\sum _{j=0}^n{a}_j{b}_j\frac{x^j}{j!}$$ where $b_j$ with $0\le j\le n$ are integers such that $|b_0|=|b_n|=1$. Schur was the first to prove the irreducibility of $g(x)$ for $s=0$. It has been proved that $g(x)$ is irreducibile for $0\le s\le 60$. In this paper, by a different method, we prove : Apart from finitely many explicitely given posibilities, either $G(x)$ is irreducible or $G(x)$ is linear factor times irreducible polynomial. This is a consequence of the estimate $s>1.9k$ whenever $G(x)$ has a factor of degree $k\ge 2$ and $(n,k,s)\ne (10,5,4)$. This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.

Added grant number of the funding source for the author Sneh Bala Sinha