$$U_{810}(x, y) = \frac{\left(1 - \sqrt{1 - 4 y}\right) \left(x + y\right)}{2 y}$$

$$T_{810}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{k {\binom{k}{n}} {\binom{- k + 2 m + 2 n - 1}{- k + m + n}}}{m + n} \end{cases} $$