Inversion formulas for the j-function around elliptic points

Recently, Hong, Mertens, Ono, and Zhang [8] proved a conjecture of Căldăraru, He, Huang [4] that expresses the Taylor series modular j-function around elliptic points i $$\rho =e^{\pi i/3}$$ as rational functions arising from signature 2 3 cases Ramanujan’s theory to alternative bases. We extend these results give inversion formulas for $$ Gauss’ hypergeometric in signatures 4 6.