Hyperbolic Belyi Maps and Shabat-Blaschke Products

We first introduce hyperbolic analogues of Belyi maps, Shabat polynomials, and Grothendieck’s dessins d’enfant. In particular, we introduce and study the Shabat-Blaschke products and the size of their hyperbolic dessin d’enfants in the unit disk. We then study a special class of Shabat-Blaschke products, namely the Chebyshev-Blaschke products. Inspired by the work of Ismail and Zhang (2007) on the coefficients of Ramanujan’s entire function, we will give similar arithmetic properties of the coefficients of the Chebyshev-Blaschke products and then use them to prove some Landen-type identities for theta functions.