Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials

Theorem

The following formula holds: $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ where $C_n$ denotes a Gegenbauer C polynomial and $P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}$ denotes a Jacobi P polynomial.