On the real-rootedness of generalized Touchard polynomials

We consider the real-rootedness of generalized Touchard polynomials recently revisited by Mansour and Schork (2013). Towards this end, we first describe the normal form of the generalized Touchard polynomials, by which recurrence relations for the polynomial part are derived. By using the recurrence relations, we prove the real-rootedness of the generalized Touchard polynomials for the parameter m∈[1,∞)∪kk+1:k∈N.