\begin{align} U_n(x) & = \frac{(x+\sqrt{x^2-1})^{n+1} - (x-\sqrt{x^2-1})^{n+1}}{2\sqrt{x^2-1}} \\ & = \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (x^2-1)^k x^{n-2k} \\ & = x^n \sum_{k=0}^{\lfloor n/2\rfloor} \binom{n+1}{2k+1} (1 - x^{-2})^k \\ & =\sum_{k=0}^{\lfloor n/2\rfloor} \binom{2k-(n+1)}{k}~(2x)^{n-2k} \quad (n>0)\\ & =\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k \binom{n-k}{k}~(2x)^{n-2k} \quad (n>0)\\ & = \sum_{k=0}^{n}(-2)^{k} \frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k \quad (n>0)\\ & = (n+1) \, _2F_1\left(-n,n+2; \frac 3 2; \frac{1-x} 2 \right) \end{align}
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