$$U_{678}(x, y) = \frac{\sqrt{\frac{2 x \left(1 - \sqrt{1 - 4 y}\right)}{y} + 1}}{2} + \frac{1}{2}$$

$$T_{678}(n, m, k) = \begin{cases}1&\text{if m+n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 2 n - 1}{n - 1}} {\binom{2 m + n - 1}{m}}}{m + n}&\text{if m+n>0} \end{cases} $$