# Relationship between Chebyshev T and hypergeometric 2F1

## Theorem

The following formula holds for $n \in \{0,1,2,\ldots\}$: $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ where $T_n$ denotes a Chebyshev T polynomial and ${}_2F_1$ denotes the hypergeometric pFq.