# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 3 Mathematical Prerequisites
### Mittag-Leffler Functions
\begin{table}
\begin{tabular}{c|c c c c c c} \hline  & \(e^{z}\) & \(E_{\alpha}(z)\) & \(E_{\alpha,\beta}(z)\) & \(E_{\alpha,\beta}^{(m)}(z)\) & \(E_{\alpha,\beta}^{\chi}(z)\) & \(E_{\alpha,\beta}^{\chi(m)}(z)\) \\ \hline \(e^{z}\) & \(e^{z}\) & \(E_{1}(z)\) & \(E_{1,1}(z)\) & \(E_{1,1}^{(m)}(z)\) & \(E_{1,1}^{1}(z)\) & \(E_{1,1}^{1}(z)\) \\ \(E_{\alpha}(z)\) & & \(\sum\limits_{k=0}^{\infty}\frac{z^{k}}{\Gamma(ak+1)}\) & \(E_{\alpha,1}(z)\) & \(E_{\alpha,1}^{(0)}(z)\) & \(E_{\alpha,1}^{1}(z)\) & \(E_{\alpha,1}^{1}(z)\) \\ \(E_{\alpha,\beta}(z)\) & & & \(\sum\limits_{k=0}^{\infty}\frac{z^{k}}{\Gamma(ak+p)}\) & \(E_{\alpha,\beta}^{(0)}(z)\) & \(E_{\alpha,\beta}^{1}(z)\) & \(E_{\alpha,\beta}^{1}(z)\) \\ \(E_{\alpha,\beta}^{(m)}(z)\) & & & & \(\sum\limits_{k=0}^{\infty}\frac{(k+m)!}{k!}\frac{z^{k}}{\Gamma(ak+m+\beta)}\) & \(m!\Gamma\!E_{\alpha,m+\beta}^{m+1}(z)\) & \(E_{\alpha,\beta}^{1}(z)\) \\ \(E_{\alpha,\beta}^{\chi}(z)\) & & & & & \(\sum\limits_{k=0}^{\infty}\frac{(\chi_{k})!}{k!}\frac{z^{k}}{\Gamma(ak+m+\beta)}\) & \(E_{\alpha,\beta}^{\chi(0)}(z)\) \\ \(E_{\alpha,\beta}^{\chi(m)}(z)\) & & & & \(\sum\limits_{k=0}^{\infty}\frac{(\chi_{k})!}{k!}\frac{z^{k}}{\Gamma(ak+m+\beta)}\) & \(\sum\limits_{k=0}^{\infty}\frac{(\chi_{k})!}{k!}\frac{z^{k}}{\Gamma(ak+m+\beta)}\) \\ \hline \end{tabular} \(E_{\alpha}\) (\(z\))—MLF with one parameter, \(\alpha\in\) C, Re(\(\alpha\)) \(>0\); \(E_{\alpha,\beta}\) (\(z\))—MLF with two parameters, \(\alpha\in\) C, \(\beta\in\) C, Re(\(\beta\)) \(>0\), Re(\(\beta\)) \(>0\); \(E_{\alpha,\beta}^{\chi}(z)\)—-th derivative of \(E_{\alpha,\beta}(z)\) with respect to \(z\), \(m\in\) N\({}_{0}\), \(\alpha\in\) C, \(\beta\in\) C, Re(\(\alpha\)) \(>0\), Re(\(\beta\)) \(>0\); Re(\(\beta\)) \(>0\); \(E_{\alpha,\beta}^{\chi}(z)\)—generalized MLF with three parameters, \(\alpha\in\) C, \(\beta\in\) C, \(\gamma\in\) C, Re(\(\alpha\)) \(>0\), Re(\(\beta\)) \(>0\), Re(\(\gamma\)) \(>0\); \(E_{\alpha,\beta}^{\chi}(z)\)—-th derivative of \(E_{\alpha,\beta}^{\chi}(z)\) with respect to \(z\), \(m\in\) N\({}_{0}\), \(\alpha\in\) C, \(\beta\in\) C, Re(\(\alpha\)) \(>0\), Re(\(\beta\)) \(>0\); \((\chi)_{k}=\Gamma(\chi+k)/\Gamma(\chi)\) is the Pochhammer symbol.  
\end{table} 
Table 1: Overview of the types of complex Mittag–Leffler functions (MLF) of the complex argument z and their relationships with each other and with exponential and gamma functions. the two-parameter MLF, the first term of the infinite series defining the function \(E_{\alpha,\beta}\) (\(z\)) is zero for \(\beta=0\) and \(k=0\). Then, the recurrence relation [26]  
\[E_{\alpha,\beta}(z)=zE_{\alpha,\alpha+\beta}(z)+\frac{1}{\Gamma(\beta)} \tag{5}\]  
can be used to derive the identity  
\[E_{\alpha,0}(z)=zE_{\alpha,\alpha}(z). \tag{6}\]  
Numerical algorithms for computing the MLF available in [27; 28; 29], as well as the Wolfram Computational Notebook, for example, evaluate the function \(E_{\alpha,0}\) (\(z\)) in accordance with (6). By successively differentiating both sides of (5), we arrive at a recurrence relation for the \(m\)-th derivative of the two-parameter MLF,  
\[E_{\alpha,\beta}^{(m)}(z)=zE_{\alpha,\alpha+\beta}^{(m)}(z)+mE_{\alpha,\alpha +\beta}^{(m-1)}(z),m>0 \tag{7}\]  
whose solution is  
\[E_{\alpha,\beta}^{(m)}(z)=m!z\sum_{k=0}^{m}\frac{E_{\alpha,\alpha+\beta}^{(k) }(z)}{k!}. \tag{8}\]  
Thus, for \(\beta=0\), arbitrary derivatives of the function \(E_{\alpha,0}\) (\(z\)) can be expressed using the derivatives of the function \(E_{\alpha,\alpha}\) (\(z\)):  
\[E_{\alpha,0}^{(m)}(z)=zE_{\alpha,\alpha}^{(m)}(z)+mE_{\alpha,\alpha}^{(m-1)}( z)=m!z\sum_{k=0}^{m}\frac{E_{\alpha,\alpha}^{(k)}(z)}{k!}. \tag{9}\]  
Note that the defining relation for the \(m\)-th derivative of the two-parameter function \(E_{\alpha,0}\) (\(z\)) exhibits no singularity for \(k=0\) when \(m>0\). Thus, the algorithms designed for this purpose [30; 31] can be used directly for numerical computations of the derivatives of the function \(E_{\alpha,0}\) (\(z\)) even for \(\beta=0\).