Remarks on the Rosen λ-Continued Fractions

Letting q ∈ ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq1859.tif"/> , q ≥ 3, and λ_q = 2 cos π/g, we define the Hecke group G_q = 〈S, T〉, where S = ( 1 λ q 0 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq1860.tif"/> and T = ( 0 − 1 1 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203747018/bef25d74-ef02-4914-ac4f-27ec89fc30ac/content/eq1861.tif"/> . The Hecke groups act upon the Poincaré upper half-plane via linear fractional transformations. G_q is a Fuchsian group of the first kind of signature (0;2,q,∞). When q = 3, G ₃ is the modular group.