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Problem: Generalized hypergeometric series

The generalized hypergeometric series is defined as \[\begin{align} &{_pF_q}(a_1, \ldots , a_p; c_1, \ldots, c_q; z) = \sum_{n=0}^{\infty} \frac{(a_1)_n \ldots (a_p)_n}{(c_1)_n \ldots (c_q)_n} \frac{z^n}{n!} \\[0.2cm] &\qquad\qquad = 1 + \frac{a_1 \ldots a_p}{c_1 \ldots c_q} \frac{z}{1} + \frac{a_1 (a_1 + 1) \ldots a_p (a_p + 1)}{c_1 (c_1 + 1) \ldots c_q (c_q + 1)} \frac{z^2}{1 \cdot 2} + \ldots \end{align}\] Show that \[\begin{align} {_0F_0}(z) &= e^z \\[0.3cm] {_1F_0}(-\alpha; -z) &= (1 + z)^{\alpha} \\[0.2cm] {_2F_1}(1, 1; 2; -z) &= \frac{\ln(1 + z)}{z} \\[0.2cm] {_2F_1}\left( \tfrac{1}{2}, \tfrac{1}{2}; \tfrac{3}{2}; z^2 \right) &= \frac{\arcsin z}{z} \\[0.2cm] {_1F_1}\left( \tfrac{1}{2}; \tfrac{3}{2}; -z^2 \right) &= \frac{\sqrt{\pi}}{2 z} \operatorname{erf}(z) \end{align}\]

The solution is provided in Issue #4 (premium) - Tutorial: Hypergeometric functions. Part I of Quantum Newsletter.