$$U_{109}(x, y) = \frac{x - y - \sqrt{x^{2} - 2 x + y^{2} + y \left(- 2 x - 2\right) + 1} + 1}{2 x}$$

$$T_{109}(n, m, k) = \begin{cases}1&\text{if m=0 and n=0} ,\ \\\frac{k {\binom{k + m + n}{m}} {\binom{m + n - 1}{m - 1}}}{k + m + n} \end{cases} $$