Some results for the irreducibility of truncated binomial expansions

For positive integers k and n with k⩽n−1, let Pn,k(x) denote the polynomial ∑j=0k(nj)xj, where (nj)=n!j!(n−j)!. In 2011, Khanduja, Khassa and Laishram proved the irreducibility of Pn,k(x) over the field Q of rational numbers for those n,k for which 2≤2k≤n<(k+1)3. In this paper, we extend the above result and prove that if 2≤2k≤n<(k+1)e+1 for some positive integer e and the smallest prime factor of k is greater than e, then there exists an explicitly constructible constant Ce depending only on e such that the polynomial Pn,k(x) is irreducible over Q for k≥Ce.