The interlacing properties of generalized Narayana polynomials

In this paper, we obtain two new interlacing properties on the zeros of a type of generalized Narayana polynomials arising in the study of the infinite log-concavity of the Boros$-$Moll polynomials. Our tools include a criterion established by Liu and Wang and two new recurrence relations for these Narayana polynomials. The new recurrences are verified with the help of ${\tt Maple}$ package ${\tt APCI}$ given by Hou. Our results also imply the real-rootedness of these Narayana polynomials.