## 3 Differentiation of the Wright Functions of the First Kind with Respect to Parameters  
We first compare the Wright functions of the first kind with the two-parameter Mittag-Leffler functions for \(\alpha>0\) and \(\beta\geq 0\), from which they differ only by the absence of factorials. The direct differentiation of the series with respect to the \(\alpha\) parameter gives  
\[ \begin{array}{l}\frac{\partial W_{\alpha,\beta}(t)}{\partial\alpha}=-\sum_{k =\,1}^{\infty}\left(\frac{\psi(ak+\beta)}{k!\,\Gamma(ak+\beta)}\right)\,k\,t^{ k}=\\ -\sum_{k=\,1}^{\infty}\left(\frac{\psi(ak+\beta)}{(k-1)!\,\Gamma(ak+\beta)} \right)\,t^{k}\\ \frac{\partial E_{\alpha,\beta}(t)}{\partial\alpha}=-\sum_{k=\,1}^{\infty} \left(\frac{\psi(ak+\beta)}{\Gamma(ak+\beta)}\right)\,k\,t^{k}\end{array}  \tag{5}\]  
and with respect to the \(\beta\) parameter  
\[ \begin{array}{l}\frac{\partial W_{\alpha,\beta}(t)}{\partial\beta}=-\sum_{k =\,0}^{\infty}\left(\frac{\psi(ak+\beta)}{k!\,\Gamma(ak+\beta)}\right)\,t^{k} \\ \frac{\partial E_{\alpha,\beta}(t)}{\partial\beta}=-\sum_{k=\,0}^{\infty} \left(\frac{\psi(ak+\beta)}{\Gamma(ak+\beta)}\right)\,t^{k}\end{array}  \tag{6}\]  
The second derivatives are  
\[ \begin{array}{l}\frac{\partial^{2}W_{\alpha,\beta}(t)}{\partial\alpha^{2}}= \sum_{k=\,1}^{\infty}\left\{\frac{[\psi(ak+1)]^{2}-\psi^{(1)}(ak+\beta)}{k!\, \Gamma(ak+\beta)}\right\}\,k^{2}t^{k}\\ \frac{\partial^{2}E_{\alpha,\beta}(t)}{\partial\alpha^{2}}=\sum_{k=\,1}^{ \infty}\left\{\frac{[\psi(ak+1)]^{2}-\psi^{(1)}(ak+\beta)}{\Gamma(ak+\beta)} \right\}\,k^{2}t^{k}\end{array}  \tag{7}\]  
and  
\[ \begin{array}{l}\frac{\partial^{2}W_{\alpha,\beta}(t)}{\partial\beta^{2}}= \sum_{k=\,0}^{\infty}\left\{\frac{[\psi(ak+\beta)]^{2}-\psi^{(1)}(ak+\beta)}{ k!\,\Gamma(ak+\beta)}\right\}\,t^{k}\\ \frac{\partial^{2}E_{\alpha,\beta}(t)}{\partial\beta^{2}}=\sum_{k=\,0}^{ \infty}\left\{\frac{\psi(ak+\beta)]^{2}-\psi^{(1)}(ak+\beta)}{\Gamma(ak+\beta) }\right\}\,t^{k}\end{array}  \tag{8}\]  
For the Mittag-Leffler and Wright functions, we have the same functional expressions, but in the case of the Wright functions, factorials appear. Contrary to the Mittag-Leffler functions [9,10], the summation of these series by using MATHEMATICA gives only few results in a closed form in terms of the following generalized hypergeometric functions:  
\[ \begin{array}{l}\frac{\partial W_{\alpha,\beta}(t)}{\partial\alpha}\big{|} _{\alpha=\,1,\,\beta=\,0}=-\sum_{k=\,1}^{\infty}\left(\frac{\psi(k)}{(k-1)!k!} \right)\,t^{k}=t_{0}\mbox{F}_{1}(\,\,;1;t)\\ \frac{\partial W_{\alpha,\beta}(t)}{\partial\alpha}\big{|}_{\alpha=\,1,\,\beta= \,1}=-\sum_{k=\,1}^{\infty}\left(\frac{\psi(k+1)}{(k-1)!k!}\right)\,t^{k}=t_{0 }\mbox{F}_{1}(\,\,;2;t)\end{array}  \tag{9}\]  
and  
\[ \begin{array}{l}\frac{\partial W_{\alpha,\beta}(t)}{\partial\beta}\big{|}_ {\alpha=\,1,\,\beta=\,0}=-\sum_{k=\,0}^{\infty}\left(\frac{\psi(k)}{(k-1)!k!} \right)\,t^{k}=\\ \frac{1}{2}\left[t_{0}\mbox{F}_{1}(\,\,;1;t)-t_{0}\mbox{F}_{1}(\,;2;t)\,\ln t \right]+\sqrt{t}K_{1}(2\sqrt{t})\\ \frac{\partial W_{\alpha,\beta}(t)}{\partial\beta}\big{|}_{\alpha=\,1,\,\beta= \,1}=-\sum_{k=\,0}^{\infty}\left(\frac{\psi(k+1)}{(k!)^{2}}\right)\,t^{k}=\,_{0 }\mbox{F}_{1}(\,\,;1;t)\end{array}  \tag{10}\]  
In the last case, \(\alpha=\beta=1\), in the Brychkov compilation of infinite series [15], the sum is expressed in terms of the following modified Bessel functions:  
\[ \begin{array}{l}\frac{\partial W_{\alpha,\beta}(t)}{\partial\beta}\big{|}_{ \alpha=\,\beta=\,1}=-\sum_{k=\,0}^{\infty}\left(\frac{\psi(k+1)}{(k!)^{2}} \right)\,t^{k}=\\ -\frac{1}{2}\,\ln t\,\mbox{I}_{0}(2\sqrt{t})-K_{0}(2\sqrt{t})\end{array}  \tag{11}\]  
Using the MATHEMATICA programme, the values of the derivatives with respect to parameters \(\alpha\) and \(\beta\) of the Wright functions of the first kind were calculated for the argument \(0.25\leq t\leq 4.0\) and for parameters \(0\leq\alpha\leq 5.0\) and \(0\leq\beta\leq 2.0\). Figure 1 illustrates the behaviour of derivatives with respect to parameter \(\alpha\) at different values of argument \(t\).