ChebyshevPolynomial

A Chebyshev polynomial of the first kind, denoted as $T_n(x)$, is a special case of Jacobi polynomials where $\alpha = \beta = -\frac{1}{2}$. Therefore, $T_n(x)$ can be written in terms of the Jacobi polynomial:

$$ T_n(x) = \frac{P_n^{(-\frac{1}{2}, -\frac{1}{2})}(x)}{P_n^{(-\frac{1}{2}, -\frac{1}{2})}(1)} $$

which can be seen to be equivalent to

$$ T_n(x) = P_n^{(-\frac{1}{2}, -\frac{1}{2})}(x) \sqrt{\pi} \frac{\Gamma(n + 1)}{\Gamma(n + \frac{1}{2})} $$

hence and since

$$ P_n^{(-\frac{1}{2}, -\frac{1}{2})}(1) = \sqrt{\pi} \frac{\Gamma(n + 1)}{\Gamma(n + \frac{1}{2})} $$