$$U_{583}(x, y) = - \frac{1}{\sqrt{1 - 4 y} \left(\frac{x}{\left(1 - 4 y\right)^{\frac{3}{2}}} - 1\right)}$$

$$T_{583}(n, m, k) = 4^{m} {\binom{k + n - 1}{n}} {\binom{\frac{k}{2} + m + \frac{3 n}{2} - 1}{m}}$$