$$U_{431}(x, y) = \frac{\sqrt{3} \left(- \sqrt{3 x y + \sin^{2}{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}} + \sin{\left(\frac{\operatorname{asin}{\left(\frac{3 \sqrt{3} \sqrt{y}}{2} \right)}}{3} \right)}\right)}{3 \sqrt{y}}$$

$$T_{431}(n, m, k) = \begin{cases}0&\text{if n<k} ,\ \\\frac{\left(-1\right)^{n} k {\binom{- k + 2 n - 1}{n - 1}}}{n}&\text{if m=0} ,\ \\\frac{\left(-1\right)^{m} k \left(k - 2 n\right) {\binom{k - n - 1}{n - 1}} {\binom{- k - 2 m + 2 n - 1}{m - 1}}}{m n}&\text{if m>0} \end{cases} $$