On Schur's irreducibility results and generalised $\varphi$-Hermite polynomials

Abstract

Let $c$ be a fixed integer such that $c\in \{0,2\}.$ Let $n$ be a positive integer such that either $n\ge 2$ or $2n+1\ne 3^u$ for any integer $u\ge 2$ according as $c=0$ or not. Let $\varphi (x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than $2n+c$. Let $a_i(x)$ with $0\le i\le n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\mathrm{deg}\varphi (x)$. Let $a_n\in \mathbb{Z}$ and the content of $(a_n{a}_0(x))$ is not divisible by any prime less than $2n+c$. For a positive integer $j$, if $u_j$ denotes the product of the odd numbers $\le j$, then we show that the polynomial $\frac{a_n}{u_{2n+c}}\varphi (x{)}^{2n}+\sum _{j=0}^{n-1}{a}_j(x)\frac{\varphi (x{)}^{2j}}{u_{2j+c}}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum _{j=0}^n{a}_j\frac{x^{2j}}{u_{2j+c}}$ with $a_j\in \mathbb{Z}$ and $|a_0|=|a_n|=1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.