Gross-Zagier Type CM Value Formulas on $X_0^{\ast }(p)$

In this work, we derive Gross-Zagier type CM value formulas for Hauptmoduls $j_p^{\ast }(\tau )$ on Fricke groups ${\mathrm{\Gamma }}_0^{\ast }(p)$. We also illustrate how to employ these formulas to obtain certain Hilbert class polynomials.