$$U_{83}(x, y) = \frac{\left(y + 1\right) \left(\sqrt{4 x + 1} + 1\right)}{2}$$

$$T_{83}(n, m, k) = \begin{cases}{\binom{k}{m}}&\text{if n=0} ,\ \\\frac{\left(-1\right)^{n - 1} k {\binom{k}{m}} {\binom{- k + 2 n - 1}{n - 1}}}{n}&\text{if n>0} \end{cases} $$