$$U_{566}(x, y) = \frac{\sqrt{\frac{4 x}{\left(1 - y\right)^{6}} + \frac{1}{\left(1 - y\right)^{4}}}}{2} + \frac{1}{2 \left(1 - y\right)^{2}}$$

$$T_{566}(n, m, k) = \begin{cases}\frac{k {\binom{2 k + m + 2 n - 1}{m}}}{k + n}&\text{if n==0 } ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{- k + 2 n - 1}{n - 1}} {\binom{2 k + m + 2 n - 1}{m}}}{n}&\text{if n>0} \end{cases} $$