Interlacing of zeros of the Poincaré series for \(\Gamma _0^+(2)\)

We consider the Poincaré series \(G_k^*(z;-m)\) $G_k^{\ast }(z;-m)$ of weight k for the Fricke group \(\Gamma _0^+(2)\) ${\mathrm{\Gamma }}_0^{+}(2)$ of level 2 for positive integers m and even integers \(k\ge 4\) $k\ge 4$ , and investigate the interlacing properties of the zeros of \(G_k^*(z;-m)\) $G_k^{\ast }(z;-m)$ and \(G_{k+8}^*(z;-m)\) $G_{k+8}^{\ast }(z;-m)$ , and those of \(G_{k}^*(z;-m-1)\) $G_k^{\ast }(z;-m-1)$ .