Algebraic properties of summation of exponential Taylor polynomials

Abstract

Let $n\ge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_n(x)={\sum }_{i=0}^n\frac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $\Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $\E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $\frac{x^n}{n!}+{\sum }_{i=0}^{n-1}{c}_i\frac{x^i}{i!}$ with all $c_i(0\le i\le n-1)$ being integers is irreducible over $\Q$ if either $c_0=\pm 1$, or $n$ is not a positive power of $2$ but $|c_0|$ is a positive power of 2. This extends another theorem of Schur. We show also that $\E_n(x)$ is irreducible if $n\notin \{2,4\}$. Furthermore, we show that ${\rm Gal}_{\Q}(\E_n)$ contains $A_n$ except for $n=4$, in which case, ${\rm Gal}_{\Q}(\E_4)=S_3$. Finally, we show that the Galois group ${\rm Gal}_{\Q}(\E_n)$ is $S_n$ if $n\equiv 3(\mathrm{mod}4)$, or if $n$ is even and $v_p(n!)$ is odd for a prime divisor of $n-1$, or if $n\equiv 1(\mathrm{mod}4)$ and $n-2$ equals the product of an odd prime number $p$ which is coprime to ${\sum }_{i=1}^{p-1}{2}^{p-1-i}i!$ and a positive integer coprime to $p$.

14 pages