$$U_{684}(x, y) = \frac{\sqrt{\frac{2 x \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{y^{2}} + 1}}{2} + \frac{1}{2}$$

$$T_{684}(n, m, k) = \begin{cases}1&\text{if n+m=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{2 m + 2 n}{m}} {\binom{- k + 2 n - 1}{n - 1}}}{m + n} \end{cases} $$