$$U_{123}(x, y) = - \frac{2 x}{x y - x + \sqrt{x^{2} y^{2} + x^{2} - 2 x + y \left(- 2 x^{2} - 2 x\right) + 1} - 1}$$

$$T_{123}(n, m, k) = \begin{cases}1&\text{if m=0 and n= 0} ,\ \\0&\text{if m=0 or n=0} ,\ \\- \frac{k {\binom{n - 1}{- m + n}} {\binom{- k + n - 1}{m - 1}}}{m}&\text{if m>0,n>0} \end{cases} $$