$$U_{811}(x, y) = \frac{\left(x + y\right) \left(- 2 y - \sqrt{1 - 4 y} + 1\right)}{2 y^{2}}$$

$$T_{811}(n, m, k) = \begin{cases}1&\text{if n=0,m=0} ,\ \\\frac{k {\binom{k}{n}} {\binom{2 m + 2 n}{- k + m + n}}}{m + n} \end{cases} $$