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

Abstract: Let $c$ be a fixed integer such that $c \in {0,2}.$ Let $n$ be a positive integer such that either $n\geq 2$ or $2n+1 \neq 3^u$ for any integer $u\geq 2$ according as $c = 0$ or not. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than $2n+c$. Let $a_i(x)$ with $0\leq i\leq n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\deg\phi(x)$. Let $a_n \in \mathbb{Z}$ and the content of $(a_na_0(x))$ 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 $\leq j$, then we show that the polynomial $\frac{a_{n}}{u_{2n+c}}\phi(x)^{2n}+\sum\limits_{j=0}^{n-1}a_j(x)\frac{\phi(x)^{2j}}{u_{2j+c}}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{j=0}^{n}a_j\frac{x^{2j}}{u_{2j+c}}$ with $a_j \in \mathbb{Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.