An interlacing pattern between the types $C_n$ and $D_n$ coordinator polynomials

The study of coordinator polynomials of Weyl group lattices was initiated by Conway and Sloane. In 2013, by using a trigonometric substitution approach, Wang and Zhao proved the real-rootedness of the type $D$ coordinator polynomials. Subsequently, Xie and Zhang proved the compatibility between the types $C_n$ and $D_n$ coordinator polynomials. In this paper, we discover the interlacing pattern between the types $C_n$ and $D_n$ coordinator polynomials, which may be named as the second pattern of interlacing zeros.