Regular C-fractions \(f(\alpha)\equiv 1+\frac{a_ 1\alpha}{1+}\frac{a_ 2\alpha}{1+}..\). with \(a_ n=an^ 2+bn+c+V_ n,\) \(| V_ n|\) sufficiently small are examined. In the case \(V_ n=0\), exact expressions are obtained which reveal a two-sheeted Riemann structure for f(\(\alpha)\). If \(V_ n\neq 0\) analytic properties are obtained by means of perturbation theory applied to the associated difference equation. A conjecture that f(\(\alpha)\) is the ratio of entire functions of 1/\(\sqrt{\alpha}\) for an even larger class of C-fractions is proved for the case \(a_ n=\prod^{N}_{i=1}(n+r_ i)^{p_ i},\) \(r_ i>-1,\) \(\sum^{N}_{i=1}p_ i=2.\)