## 7 Conclusions  
The parameters of the Wright functions that had been treated as variables and derivatives with respect to them were derived and discussed. These derivatives are expressible in terms of infinite power series with quotients of digamma and gamma functions in their coefficients. The functional form of these series resembles those that were derived for the Mittag-Leffler functions. Only in a few cases was it possible to obtain the sums of these series in a closed form. The differentiation operation when the Wright functions are treated as generalized Bessel functions leads to the Bessel functions and their derivatives with respect to the order. Simple operations with the Laplacian transforms of the Wright functions of the first kind gave explicit forms of the Mittag-Leffler functions. Applying the shifted Dirac delta function permitted to derive functional limits by using the Laplacian transforms of the Wright functions. Lastly, we would like to draw attention of the interested readers to the recent papers of [21; 22; 23] where some noteworthy applications of the Wright functions of the first and second kinds are discussed. Relevant applications can be expected in the field of special functions in fractional calculus for which we refer the interested readers to its extensive literature [24; 25; 26; 27; 28; 29; 30; 31; 32]. Conceptualization, A.A. and F.M. ; methodology, A.A. and F.M. ; validation, A.A. and F.M. ; investigation, A.A. and F.M. ; resources, A.A. and F.M. ; data curation, A.A. and F.M. ; writing--original draft preparation, A.A. and F.M. ; writing--review and editing, A.A. and F.M. ; visualization, A.A. and F.M. ; supervision, A.A. and F.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The work of [PERSON] was carried out in the framework of the activities of the National Group of Mathematical Physics (INdAM-GNFM). Both authors are grateful to Associate Professor [PERSON], Department of Mathematics, Universidad de Oviedo, Oviedo, Spain for his help with using the MATHEMATICA programme and with editing in LaTeX. The authors declare no conflict of interest. Appendix A Differentiation of the Wright Functions with Respect to Parameters versus Bessel Functions  
Initially, the Wright functions (of the first kind) were treated as generalized Bessel functions because, for parameters \(\alpha=1\) and \(\beta+1\), they become  
\[ \begin{array}{l}W_{1,\beta+1}(-\frac{t^{2}}{4})=\left(\frac{2}{7}\right)^{\beta }J_{\beta}(t)\\ W_{1,\beta+1}(\frac{t^{2}}{4})=\left(\frac{2}{7}\right)^{\beta}I_{\beta}(t) \end{array}  \tag{10}\]  
The differentiation of the Wright functions in (10) with respect to parameter \(\beta\) gives  
\[ \begin{array}{l}\frac{\partial\left(W_{1,\beta+1}(-\frac{t^{2}}{4})\right)} {\partial\beta}=\left(\frac{2}{7}\right)^{\beta}\left[\ln\left(\frac{2}{7} \right)J_{\beta}(t)+\frac{\partial j_{\beta}(t)}{\partial\beta}\right]\\ \frac{\partial\left(W_{1,\beta+1}(\frac{t^{2}}{4})\right)}{\partial\beta}= \left(\frac{2}{7}\right)^{\beta}\left[\ln\left(\frac{2}{7}\right)I_{\beta}(t) +\frac{\partial j_{\beta}(t)}{\partial\beta}\right]\end{array}  \tag{11}\]  
However, the differentiation of the [PERSON] functions with respect to the order can be expressed as follows [18]:  
\[ \begin{array}{l}\frac{\partial j_{\beta}(t)}{\partial\beta}=\pi\;\beta\; \int_{0}^{\pi/2}\tan\theta\;Y_{0}\big{(}t\left[\sin\theta\right]^{2}\big{)} \;J_{\beta}\big{(}t\left[\cos\theta\right]^{2}\big{)}\;d\theta\\ \frac{\partial j_{\beta}(t)}{\partial\beta}=-2\;\beta\;\int_{0}^{\pi/2}\tan \theta\;K_{0}\big{(}t\left[\sin\theta\right]^{2}\big{)}\;I_{\beta}\big{(}t \left[\cos\theta\right]^{2}\big{)}\;d\theta\\ Re\beta>0\end{array}  \tag{12}\]  
In particular cases, differentiation with respect to the \(\beta\) parameter can be explicitly expressed as follows [8]:  
\[