Relationship between Chebyshev U and hypergeometric 2F1

Theorem

The following formula holds for $n \in \{0,1,2,\ldots\}$: $$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and ${}_2F_1$ denotes hypergeometric 2F1.