\begin{align*} h_{60}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(6n + 1)} z^{6n} & h_{61}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(6n + 2)} z^{6n+1} & h_{62}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(6n + 3)} z^{6n+2} \\ h_{63}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(6n + 4)} z^{6n+3} & h_{64}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(6n + 5)} z^{6n+4} & h_{65}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(6n + 6)} z^{6n+5} \end{align*}

\begin{align*} \frac{d h_{65}}{d z} &= h_{64} & \frac{d h_{64}}{d z} &= h_{63} & \frac{d h_{63}}{d z} &= h_{62} \\ \frac{d h_{62}}{d z} &= h_{61} & \frac{d h_{61}}{d z} &= h_{60} & \frac{d h_{60}}{d z} &= h_{65} \end{align*}

\begin{align*} h_{60}(z) &= \frac{1}{3} \cosh{(z)} + \frac{2}{3} \cosh{\left( \frac{z}{2} \right)} \cos{\left( \frac{\sqrt{3}}{2}z \right)} \\ h_{61}(z) &= \frac{1}{3} \sinh{(z)} + \frac{1}{3} \sinh{\left( \frac{z}{2} \right)} \cos{\left( \frac{\sqrt{3}}{2}z \right)} - \frac{\sqrt{3}}{3} \cosh{\left( \frac{z}{2} \right)} \sin{\left( \frac{\sqrt{3}}{2}z \right)} \\ h_{62}(z) &= \frac{1}{3} \cosh{(z)} - \frac{1}{3} \cosh{\left( \frac{z}{2} \right)} \cos{\left( \frac{\sqrt{3}}{2}z \right)} - \frac{\sqrt{3}}{3} \sinh{\left( \frac{z}{2} \right)} \sin{\left( \frac{\sqrt{3}}{2}z \right)} \\ h_{63}(z) &= \frac{1}{3} \sinh{(z)} - \frac{2}{3} \sinh{\left( \frac{z}{2} \right)} \cos{\left( \frac{\sqrt{3}}{2}z \right)} \\ h_{64}(z) &= \frac{1}{3} \cosh{(z)} - \frac{1}{3} \cosh{\left( \frac{z}{2} \right)} \cos{\left( \frac{\sqrt{3}}{2}z \right)} + \frac{\sqrt{3}}{3} \sinh{\left( \frac{z}{2} \right)} \sin{\left( \frac{\sqrt{3}}{2}z \right)} \\ h_{65}(z) &= \frac{1}{3} \sinh{(z)} + \frac{1}{3} \sinh{\left( \frac{z}{2} \right)} \cos{\left( \frac{\sqrt{3}}{2}z \right)} + \frac{\sqrt{3}}{3} \cosh{\left( \frac{z}{2} \right)} \sin{\left( \frac{\sqrt{3}}{2}z \right)} \end{align*}