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Given the Gauss hypergeometric function ${}_{2}F_{1} (a,b,c,z)$, its $n$th derivative can be written as $\frac{\mathrm{d}^{n}}{\mathrm{d} z^{n}} {}_{2}F_{1} (a,b,c,z) = \frac{(a)_{n} (b)_{n}}{(c)_{n}} {}_{2}F_{1} (a+n,b+n,c+n,z) $ However, when...