## Appendix C Transfer Functions \(T_{1}\) and \(T_{2}\), Calculation  
The first-order hierarchy transfer function constants \(c_{1}\), \(c_{k}\), \(d_{1}\), \(d_{k}\), \(b_{1}\) and \(b_{2}\) are:  
\[c_{1,out}=-k_{0}\Big{(}iR_{1}\theta_{1}^{2}\omega+(1+iC_{1}R_{1}\omega)(k_{1}-m _{1}\omega^{2})\Big{)}\Big{(}iR_{2}\theta_{2}^{2}\omega+(1+iC_{2}R_{2}\omega)(k _{2}-m_{2}\omega^{2})\Big{)} \tag{17}\]\[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\] \[\quad-\Big{(}iR_{1}\theta_{1}^{2}\omega+(1+iC_{1}R_{1}\omega) \Big{(}k_{1}-m_{1}\omega^{2}\Big{)}\Big{)}\times\Big{(}\omega^{3}(m_{0}+m_{2}) (k_{2}+iR_{2}(k_{2}C_{2}+\theta_{2}^{2}))\] \[\quad+m_{0}m_{2}\omega^{2}(1-iC_{2}R_{2}\omega)\Big{)}-2k_{0} \Big{(}k_{2}+i\omega\Big{(}R_{2}(k_{2}C_{2}+\theta_{2}^{2})+m_{2}\omega(i-C_{2 }R_{2}\omega)\Big{)}\Big{)}\]  
\[c_{k,out} =-k_{0}\theta_{1}\theta_{2}\Big{(}\Big{(}iR_{1}\theta_{1}^{2} \omega+\big{(}1+iC_{1}R_{1}\omega\big{)}\big{(}k_{1}-m_{1}\omega^{2}\big{)} \Big{)}\Big{(}iR_{2}\theta_{2}^{2}\omega+\big{(}1+iC_{2}R_{2}\omega\big{)} \big{(}k_{2}-m_{2}\omega^{2}\big{)}\Big{)}\Big{)}. \tag{110}\]  
\[d_{k,out} =\Big{(}k_{0}(k_{2}+iR_{2}(C_{2}k_{2}+\theta_{2}^{2})\omega)- \big{(}k_{0}m_{2}+k_{2}(m_{0}+m_{2})\big{)}\omega^{2}-iR_{2}\big{(}C_{2}(k_{2 }m_{0}+m_{2}k_{0}+m_{2}k_{2})+(m_{0}+m_{2})\theta_{2}^{2}\big{)}\omega^{3} \tag{111}\] \[\quad+m_{0}m_{2}\omega^{4}(1+iC_{2}R_{2}\omega)\Big{)}\Big{(}-iR _{1}\theta_{1}^{2}\omega-\big{(}1+iC_{1}R_{1}\omega\big{)}\big{(}k_{1}-m_{1} \omega^{2}\big{)}\Big{)}+m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta _{1}^{2})\omega\Big{)}\] \[\quad\times\Big{(}k_{2}(1+iC_{2}R_{2}\omega)+i\omega\big{(}im_{2} \omega+R_{2}\big{(}\theta_{2}^{2}-m_{2}C_{2}\omega^{2}\big{)}\big{)}\Big{)}\] \[c_{1,in} =k_{0}\Big{(}\theta_{1}\theta_{2}\big{(}k_{1}+iR_{1}(C_{1}k_{1}+ \theta_{1}^{2})\omega\big{)}\big{(}iR_{2}\theta_{2}^{2}\omega-\big{(}1+C_{2}iR _{2}\omega\big{)}\big{(}k_{2}-m_{2}\omega^{2}\big{)}\big{)}-\theta_{1}\big{(}1 +C_{1}iR_{1}\omega\big{)}\] (112) \[\quad\Big{(}k_{1}\theta_{2}\big{(}k_{2}+iR_{2}(C_{2}k_{2}+\theta_ {2}^{2})\omega\big{)}-\theta_{2}\big{(}k_{2}-m_{1}\omega^{2}\big{)}\big{(}k_{2 }+C_{2}ik_{2}R_{2}\omega\] \[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]0  
\[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]1 \[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]2  
\[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]3 \[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]4 (115) \[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]5  
\[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]6 \[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]7\[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]8 \[d_{1,\omega ut} =-m_{1}\omega^{2}\Big{(}k_{1}+iR_{1}(k_{1}C_{1}+\theta_{1}^{2})\omega \Big{)}\Big{(}k_{2}+ik_{2}C_{2}R_{2}\omega+i\omega\Big{(}im_{2}\omega+R_{2}\Big{(} \theta_{2}^{2}-m_{2}C_{2}\omega^{2}\Big{)}\Big{)}\Big{)} \tag{109}\]9  
\[\quad-\Big{(}iR_{1}\theta_{1}^{2}\omega+(1+iC_{1}R_{1}\omega) \Big{(}k_{1}-m_{1}\omega^{2}\Big{)}\Big{)}\times\Big{(}\omega^{3}(m_{0}+m_{2}) (k_{2}+iR_{2}(k_{2}C_{2}+\theta_{2}^{2}))\]0 \[\quad-\Big{(}iR_{1}\theta_{1}^{2}\omega+(1+iC_{1}R_{1}\omega) \Big{(}k_{1}-m_{1}\omega^{2}\Big{)}\Big{)}\times\Big{(}\omega^{3}(m_{0}+m_{2}) (k_{2}+iR_{2}(k_{2}C_{2}+\theta_{2}^{2}))\]1  
The subscripts \(out\) and \(in\) represent the outward and inward configurations.