  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 the 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.
