\[\frac{x^{\frac{1}{2}n}(x-1)^{\lambda+\mu-\frac{1}{2}}}{\Gamma\left(\lambda+\mu%
+\tfrac{1}{2}\right)}\frac{C^{(\lambda+\mu)}_{n}\left(x^{-\frac{1}{2}}\right)}%
{C^{(\lambda+\mu)}_{n}\left(1\right)}=\int_{1}^{x}\frac{y^{\frac{1}{2}n}(y-1)^%
{\lambda-\frac{1}{2}}}{\Gamma\left(\lambda+\tfrac{1}{2}\right)}\frac{C^{(%
\lambda)}_{n}\left(y^{-\frac{1}{2}}\right)}{C^{(\lambda)}_{n}\left(1\right)}%
\frac{(x-y)^{\mu-1}}{\Gamma\left(\mu\right)}\,\mathrm{d}y,\]