On the Laplace Transforms of Derivatives of Special Functions with Respect to Parameters

Abstract

This article is devoted to the derivation of the Laplace transforms of the derivatives with respect to parameters of certain special functions, namely, the Mittag–Leffler-type, Wright, and Le Roy-type functions. These formulas show the interconnection of these functions and lead to a better understanding of their behavior on the real line. These formulas are represented in a convoluted form and reconstructed in a more suitable form by using the Efros theorem.

1. Introduction

An interest in the differential properties of special functions with respect to their parameters has been growing recently. Respective formulas become the sources for new classes of special functions as well as differential equations of new types. They are important for a better understanding of the behavior of special functions and for their applications (see, e.g., [1]). A number of articles are devoted to the differentiation of the Bessel-type functions, see, e.g., [2,3] and references therein. In particular, the differential equation for derivatives of the Bessel function with regard to parameters was used in [4] for the derivation of an integral representation of ${\displaystyle \frac{\partial J_{\nu }\left(z\right)}{\partial \nu }}$.

Recently, Apelblat [5] and Apelblat and Mainardi proposed [6] an approach for the differentiation of the Mittag–Leffler and Wright functions, respectively (see, e.g., [7]), with respect to parameters. The obtained formulas are derived by using the formal term-by-term differentiation of the series represented by those special functions. So, the derivatives with respect to the parameters are again series, and one can ask where these series converge.

In [8], two approaches were proposed for the justification of the formulas of differentiation of the special functions (Mittag–Leffler and Le Roy-type) with respect to their parameters. These approaches are based on the uniform convergence concept and use either series representations or integral representations of the functions mentioned. The sub-products of [8] are determined and described, that is, the differential equations satisfied by the considered special functions are formulated.

This article is devoted to the derivation of the Laplace transforms (see, e.g., [9]) of these derivatives with respect to the parameters of certain special functions, namely, the Mittag–Leffler-type, Wright, and Le Roy-type functions. These formulas show the interconnection of these functions and lead to a better understanding of their behavior on the real line. In connection with this topic, we have to mention a survey paper [10] in which some examples are presented that justify the importance of the Laplace transforms technique in different branches of the applied sciences (cf. also [11]).

Special attention is paid to the reconstruction of these formulas using the Efros theorem [12] in the Wlodarski form (see [13,14]).

Throughout this paper, we assume that all parameters ($\alpha ,\beta ,\gamma ,{\alpha }_j,{\beta }_j$) are positive. It follows from [8] that under these conditions, the corresponding series converge uniformly with respect to these parameters. The detailed proof is presented in Theorem 1. Thus, we can apply the Laplace transforms to the derivatives point-wise.

2. Laplace Transforms of the Derivatives of the Mittag–Leffler Functions

We start with the two-parametric Mittag–Leffler function (which is the most useful for applications); see, e.g., [7] [Chapter 4].

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(\alpha k+\beta )}}.$$

(1)

The function $\mathsf{\Gamma }\left(z\right)$ represents the Gamma function. This function is defined via the following integral form [15] [Chapter 5]:

$$\mathsf{\Gamma }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt,$$

(2)

for all $z\in \mathbb{C}$ such that $\mathrm{Re}z>0$. It is well-known that the function $E_{\alpha ,\beta }\left(z\right)$ is an entire function of the complex variable z for all $\mathrm{Re}\alpha >0,\beta \in \mathbb{C}$. To avoid additional technical details, we limit our study to the case $\alpha >0,\beta >0$ only. The most applicable formula for the Laplace transforms of the function (1) is the following one, see, e.g., [7] [(4.9.1)]:

$$\mathcal{L}\left(t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)\left(s\right)={\displaystyle \frac{s^{\alpha -\beta }}{s^{\alpha }-\lambda }};\lambda \in \mathbb{R};\mathrm{Re}s>0;\left|\lambda s^{-\alpha }\right|<1.$$

(3)

For the formulas of the derivatives of $t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)$, we use the results obtained in [8]. In the following theorem, we prove the formula for the Laplace transform of the derivative with respect to parameter $\alpha$ of the function $t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)$.

Theorem 1.

Let $\alpha >0,\beta >0$, $\lambda \in \mathbb{R}$. Then, the following formula

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}\left[t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-st}\sum _{k=0}^{\infty }k{\lambda }^k\left[{\displaystyle \frac{t^{\alpha k+\beta -1}\left(\ln t-\psi (\alpha k+\beta )\right)}{\mathsf{\Gamma }(\alpha k+\beta )}}\right]dt=-{\displaystyle \frac{\lambda s^{\alpha -\beta }\ln s}{{(s^{\alpha }-\lambda )}^2}}.$$

(4)

is valid for all $|\lambda s^{-\alpha }|<1$. Here, ψ is the so-called digamma function $\psi \left(t\right)=(\ln \mathsf{\Gamma }(t){)}^{\prime }={\displaystyle \frac{{\mathsf{\Gamma }}^{\prime }\left(t\right)}{\mathsf{\Gamma }\left(t\right)}}$ (see, e.g., [15] [Chapter 5]).

Proof. 

Let us take arbitrary numbers $a,b$, $0<a<b<\infty$. Suppose that $\alpha \in [a,b]$.

First, we prove that the derivative in $\alpha$ of the function $t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)$ can be calculated term-wise, i.e.,

$${\displaystyle \frac{\partial }{\partial \alpha }}\left[t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right]=\sum _{k=0}^{\infty }{\displaystyle \frac{\partial }{\partial \alpha }}{\displaystyle \frac{t^{\alpha k+\beta -1}{\lambda }^k}{\mathsf{\Gamma }(\alpha k+\beta )}}=\sum _{k=0}^{\infty }k{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}(lnt-\psi (\alpha k+\beta ))}{\mathsf{\Gamma }(\alpha k+\beta )}}.$$

(5)

To see this, it is sufficient to show that the last series is converging uniformly with respect to $\alpha \in [a,b]$.

Let us consider the following series:

$$lnt·t^{\beta -1}\sum _{k=0}^{\infty }k{\lambda }^k{\displaystyle \frac{t^{\alpha k}}{\mathsf{\Gamma }(\alpha k+\beta )}}.$$

We can find a positive integer number $k_0\in \mathbb{N}$ such that $ak>4$ for all $k\ge k_0$. Since $\alpha \in [a,b]$, and for integer parts of numbers $[·]$ we have $\left[ak\right]\ge ak-1>1,\left[\beta \right]>0$, then the following inequalities can be derived:

$$\sum _{k=k_0}^{\infty }k{\lambda }^k{\displaystyle \frac{t^{\alpha k}}{\mathsf{\Gamma }(\alpha k+\beta )}}\le \sum _{k=k_0}^{\infty }k{\lambda }^k{\displaystyle \frac{t^{bk}}{\mathsf{\Gamma }(ak+\beta )}}\le \sum _{k=k_0}^{\infty }k{\displaystyle \frac{{\left(\lambda t^b\right)}^k}{\mathsf{\Gamma }\left(\right[ak]+[\beta \left]\right)}}\le$$

$$\sum _{k=k_0}^{\infty }{\displaystyle \frac{k}{\left[ak\right]+\left[\beta \right]}}{\displaystyle \frac{{\left(\lambda t^b\right)}^k}{\mathsf{\Gamma }\left(\right[ak]+[\beta ]-1)}}\le \sum _{k=k_0}^{\infty }{\displaystyle \frac{k}{ak+\beta -2}}{\displaystyle \frac{{\left(\lambda t^b\right)}^k}{\mathsf{\Gamma }(ak+\beta -3)}}.$$

The series

$$\sum _{k=k_0}^{\infty }{\displaystyle \frac{{\left(\lambda t^b\right)}^k}{\mathsf{\Gamma }(ak+\beta -3)}}={\left(\lambda t^b\right)}^{k_0}\sum _{k=k_0}^{\infty }{\displaystyle \frac{{\left(\lambda t^b\right)}^{k-k_0}}{\mathsf{\Gamma }(a(k-k_0)+a{k}_0+\beta -3)}}=$$

$${\left(\lambda t^b\right)}^{k_0}\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(\lambda t^b\right)}^m}{\mathsf{\Gamma }(am+a{k}_0+\beta -3)}}={\left(\lambda t^b\right)}^{k_0}{E}_{a,a{k}_0+\beta -3}\left(\lambda t^b\right),$$

converges for all $\lambda \in \mathbb{R}$, $t\in \mathbb{R}$ and does not depend on $\alpha$. The sequence ${\displaystyle \frac{k}{ak+\beta -2}}$ is bounded and monotonically decreasing. It gives the uniform convergence with respect to $\alpha \in [a,b]$ of the series ${\displaystyle \sum _{k=0}^{\infty }}k{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}lnt}{\mathsf{\Gamma }(\alpha k+\beta )}}$.

In order to show the uniform convergence with respect to $\alpha \in [a,b]$ of the series ${\displaystyle \sum _{k=0}^{\infty }}k{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}$, we use the following inequality:

$$ln(x+{\displaystyle \frac{1}{2}})\le \psi (x+1)\le ln(x+e^{-{\gamma }_0}),$$

(6)

valid for all $x>-1$ (see [16], cf. [17] for the best possible inequalities of this type). Here, ${\gamma }_0:=\underset{n\to \infty }{lim}\left(-lnn+{\displaystyle \sum _{k=1}^n}{\displaystyle \frac{1}{k}}\right)$, which is the Euler–Mascheroni constant. We also apply a Stirling-like asymptotic formula (see, e.g., Appendix A in [7]):

$$\mathsf{\Gamma }(\alpha k+\beta )=\sqrt{2\pi }{\left(\alpha k\right)}^{\alpha k+\beta -1/2}{e}^{-\alpha k}[1+o\left(1\right)],k\to \infty .$$

(7)

Thus, the following relations hold:

$$\sum _{k=0}^{\infty }k{\lambda }^k{\displaystyle \frac{t^{\alpha k}\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}\le \sum _{k=0}^{\infty }k{\lambda }^k{\displaystyle \frac{t^{bk}\psi (bk+\beta )}{\mathsf{\Gamma }(ak+\beta )}}\le \sum _{k=0}^{\infty }{\displaystyle \frac{kln(bk+\beta -1+e^{-{\gamma }_0})}{\sqrt{2\pi }{\left(\frac{ak}{e}\right)}^{ak}{\left(ak\right)}^{\beta -1/2}[1+o\left(1\right)]}}{\left(\lambda t^b\right)}^k.$$

Direct calculation shows that this power series has an infinite radius of convergence. It gives the uniform convergence with respect to $\alpha \in [a,b]$ of the series ${\displaystyle \sum _{k=0}^{\infty }}k{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}$.

The next step is to justify the validity of the formula

$$\mathcal{L}\left(\sum _{k=0}^{\infty }{\displaystyle \frac{\partial }{\partial \alpha }}{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}}{\mathsf{\Gamma }(\alpha k+\beta )}}\right)=\sum _{k=0}^{\infty }\underset{0}{\overset{\infty }{\int }}{e}^{-st}{\displaystyle \frac{\partial }{\partial \alpha }}\left({\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}}{\mathsf{\Gamma }(\alpha k+\beta )}}\right)dt.$$

(8)

For this it is sufficient to prove that the series

$$\sum _{k=0}^{\infty }\underset{0}{\overset{\infty }{\int }}{e}^{-st}{\displaystyle \frac{\partial }{\partial \alpha }}\left({\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}}{\mathsf{\Gamma }(\alpha k+\beta )}}\right)dt=\sum _{k=0}^{\infty }\underset{0}{\overset{\infty }{\int }}{e}^{-st}k{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}(lnt-\psi (\alpha k+\beta ))}{\mathsf{\Gamma }(\alpha k+\beta )}}dt,$$

(9)

converges uniformly with respect to $\alpha \in [a,b]$ in a proper domain of the variable s. Let us perform the change of variable $st=u$ in the following integral:

$$\underset{0}{\overset{\infty }{\int }}{e}^{-st}k{\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}lnt}{\mathsf{\Gamma }(\alpha k+\beta )}}dt={\displaystyle \frac{k{\left(\lambda s^{-\alpha }\right)}^k}{s^{\beta }}}\left[\underset{0}{\overset{\infty }{\int }}{e}^{-u}{\displaystyle \frac{u^{\alpha k+\beta -1}lnu}{\mathsf{\Gamma }(\alpha k+\beta )}}du-\underset{0}{\overset{\infty }{\int }}{e}^{-u}{\displaystyle \frac{u^{\alpha k+\beta -1}lns}{\mathsf{\Gamma }(\alpha k+\beta )}}du\right].$$

(10)

Note that $\underset{0}{\overset{\infty }{\int }}{e}^{-u}{u}^{\gamma }lnudu={\left(\underset{0}{\overset{\infty }{\int }}{e}^{-u}{u}^{\gamma }du\right)}_{\gamma }^{\prime }$. Hence,

$$\begin{array}{ccc}& & \underset{0}{\overset{\infty }{\int }}{e}^{-u}{\displaystyle \frac{u^{\alpha k+\beta -1}lnu}{\mathsf{\Gamma }(\alpha k+\beta )}}du={\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha k+\beta )}}{\left(\underset{0}{\overset{\infty }{\int }}{e}^{-u}{u}^{\alpha k+\beta }du\right)}_{\gamma =\alpha k+\beta }^{\prime }={\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha k+\beta )}}{\mathsf{\Gamma }}^{\prime }(\alpha k+\beta )\hfill \\ & & =\psi (\alpha k+\beta ).\hfill \end{array}$$

(11)

For the second term in the right hand side of (10) we have

$$-\underset{0}{\overset{\infty }{\int }}{e}^{-u}{\displaystyle \frac{u^{\alpha k+\beta -1}lns}{\mathsf{\Gamma }(\alpha k+\beta )}}du=-lns\underset{0}{\overset{\infty }{\int }}{e}^{-u}{\displaystyle \frac{u^{\alpha k+\beta -1}}{\mathsf{\Gamma }(\alpha k+\beta )}}du=-lns.$$

(12)

Substituting (11) and (12) into (9), we obtain

$$\sum _{k=0}^{\infty }\underset{0}{\overset{\infty }{\int }}{e}^{-st}{\displaystyle \frac{\partial }{\partial \alpha }}\left({\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}}{\mathsf{\Gamma }(\alpha k+\beta )}}\right)dt=-\sum _{k=0}^{\infty }{\displaystyle \frac{k{\left(\lambda s^{-\alpha }\right)}^k}{s^{\beta }}}lns.$$

(13)

for the series ${\displaystyle \sum _{k=0}^{\infty }}k{\left(\lambda s^{-\alpha }\right)}^k\le {\displaystyle \sum _{k=0}^{\infty }}k{\left(\lambda s^{-a}\right)}^k$. The last series does not depend on $\alpha$ and converges in the domain $\left|\lambda s^{-a}\right|<1$. Hence, we can conclude that the series (13) converges uniformly with respect to $\alpha \in [a,b]$ in the domain $\left|\lambda s^{-\alpha }\right|<1$. Let us denote $\lambda s^{-\alpha }=:v$ and note that ${\displaystyle \sum _{k=0}^{\infty }}k{v}^k=vs.{\displaystyle \sum _{k=0}^{\infty }}{\left(v^k\right)}_v^{\prime }={\displaystyle \frac{v}{{(1-v)}^2}}$. Therefore,

$${\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \underset{0}{\overset{\infty }{\int }}}{e}^{-st}{\displaystyle \frac{\partial }{\partial \alpha }}\left({\lambda }^k{\displaystyle \frac{t^{\alpha k+\beta -1}}{\mathsf{\Gamma }(\alpha k+\beta )}}\right)dt=-{\displaystyle \frac{lns·\lambda s^{-\alpha }}{s^{\beta }{(1-\lambda s^{-\alpha })}^2}}=-{\displaystyle \frac{lns·\lambda s^{\alpha -\beta }}{{(s^{-\alpha }-\lambda )}^2}}.$$

This completes the proof. □

Remark 1.

The obtained result can be formally derived by the direct differentiation of the relation (3) with respect to α. In Theorem 1, we justify the validity of this formal result.

Below, we present a number of other formulas for the Laplace transform of the derivatives with respect to parameters of different special functions. These formulas are justified in a way similar to that demonstrated in Theorem 1.

Theorem 2.

Let $\alpha >0$, $\beta >0$, $\lambda \in \mathbb{R}$. Then, the following formula

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}\left[t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)=\underset{0}{\overset{\infty }{\int }}{e}^{-st}\sum _{k=0}^{\infty }{\lambda }^k\left[{\displaystyle \frac{t^{\alpha k+\beta -1}\left(lnt-\psi (\alpha k+\beta )\right)}{\mathsf{\Gamma }(\alpha k+\beta )}}\right]dt=-{\displaystyle \frac{s^{\alpha -\beta }lns}{s^{\alpha }-\lambda }},$$

(14)

is valid for all $|\lambda s^{-\alpha }|<1$.

The direct application of the Laplace transforms to the Mittag–Leffler function leads to a slightly more cumbersome formula:

$$\mathcal{L}\left(E_{\alpha ,\beta }\left(t\right)\right)\left(s\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{k!}{\mathsf{\Gamma }(\alpha k+\beta )}}{s}^{-k-1}.$$

(15)

It follows from the asymptotic Formula (7) that the series on the right-hand side of (15) converge for all $s\in \mathbb{C}\setminus \left\{0\right\}$ whenever $\alpha >1$ and for all $s\in \mathbb{C},\left|s\right|>1,$ whenever $\alpha =1$. If $0<\alpha <1$, then this series diverges everywhere. The corresponding formulas of the Laplace transform of the derivatives of $E_{\alpha ,\beta }\left(t\right)$ are given in the next theorem.

Theorem 3.

Let $\alpha >1,\beta \in \mathbb{R}$. Then the following formulas of the Laplace transform of the derivatives of the function $E_{\alpha ,\beta }\left(t\right)$,

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}\left[E_{\alpha ,\beta }\left(t\right)\right]\right)\left(s\right)=-\sum _{k=0}^{\infty }{\displaystyle \frac{k!\times k\times \psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}{s}^{-k},$$

(16)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}\left[E_{\alpha ,\beta }\left(t\right)\right]\right)\left(s\right)=-\sum _{k=0}^{\infty }{\displaystyle \frac{k!\times \psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}{s}^{-k}.$$

(17)

are valid for all $s\in \mathbb{C}\setminus \left\{0\right\}$.

3. Laplace Transforms of the Derivatives of the Wright Function

In this section, we provide formulas of the Laplace transforms of the derivatives with respect to parameters of the Wright function (which can be considered for the following real values of parameters $\alpha >-1,\beta \in \mathbb{R}$),

$$\varphi (\alpha ,\beta ;z)=W_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{k!\mathsf{\Gamma }(\alpha k+\beta )}}.$$

(18)

Two formulas of the Laplace transforms of this function are known.The first one represents the Laplace transforms in terms of the Mittag–Leffler function (1) in the case $\alpha >0$,

$$\mathcal{L}\left[W_{\alpha ,\beta }(\pm t)\right]\left(s\right)={\displaystyle \frac{1}{s}}{E}_{\alpha ,\beta }(\pm s^{-1}).$$

(19)

As in Theorem 1, we can justify the following formulas of the Laplace transforms of the derivatives with respect to parameters:

Theorem 4.

Let $\alpha >0$, $\beta >0$. Then, the following formulas for the Laplace transforms of the derivatives of $W_{\alpha ,\beta }\left(t\right)$,

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}\left[W_{\alpha ,\beta }(\pm t)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s}}\sum _{k=0}^{\infty }{\displaystyle \frac{\pm s^{-k}k\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}},$$

(20)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}\left[W_{\alpha ,\beta }(\pm t)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s}}\sum _{k=0}^{\infty }{\displaystyle \frac{\pm s^{-k}\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}},$$

(21)

are valid for all $s\in \mathbb{C}\setminus \left\{0\right\}$.

The second variant of the Laplace transform of the Wright function is similar to Formulas (4) and (14),

$$\mathcal{L}\left(t^{\beta -1}\left[W_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)={\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\lambda s^{-\alpha }\right)}^k}{k!}}={\displaystyle \frac{1}{s^{\beta }}}{e}^{\lambda s^{-\alpha }}.$$

(22)

Corollary 1.

Let $\alpha >0,\beta >0$. Then, the following formulas for the Laplace transforms of the derivatives,

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}{t}^{\beta -1}\left[W_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)={\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\lambda s^{-\alpha }\right)}^k}{k!}}={\displaystyle \frac{-\lambda lns}{s^{\alpha +\beta }}}{e}^{\lambda s^{-\alpha }},$$

(23)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}{t}^{\beta -1}\left[W_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)={\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\lambda s^{-\alpha }\right)}^k}{k!}}={\displaystyle \frac{-lns}{s^{\beta }}}{e}^{\lambda s^{-\alpha }}.$$

(24)

are valid for all $s\in \mathbb{C},\mathrm{Re}s>0$.

4. Laplace Transforms of the Derivatives of the Three- and Four-Parametric Mittag–Leffler Functions

The so-called Prabhakar function is the most useful for applications among the multi-parametric functions (see, e.g., Section 5.1 in [7]),

$$E_{\alpha ,\beta }^{\gamma }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\gamma \right)}_k{z}^k}{\mathsf{\Gamma }(\alpha k+\beta )}}=\sum _{k=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma \right)z^k}{\mathsf{\Gamma }(\gamma +k)\mathsf{\Gamma }(\alpha k+\beta )}}.$$

(25)

The Laplace transforms of the Prabhakar function are given by the following formula:

$$\mathcal{L}\left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right)\left(s\right)={\displaystyle \frac{s^{\alpha \gamma -\beta }}{{(s^{\alpha }-\lambda )}^{\gamma }}};\lambda \in \mathbb{R};\mathrm{Re}s>0;\left|\lambda s^{-\alpha }\right|<1.$$

(26)

In a way similar to that of Theorem 1, we can justify the following formulas of the Laplace transforms of the derivatives of this function with respect to parameters.

Theorem 5.

Let $\alpha >0$, $\beta >0$, $\gamma >0$. Then, the following formulas of the Laplace transform of the derivatives with respect to parameters are satisfied:

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}\left[t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)={\displaystyle \frac{-\gamma \lambda s^{\alpha \gamma -\beta }lns}{{(s^{\alpha }-\lambda )}^{\gamma +1}}};\lambda \in \mathbb{R};\mathrm{Re}s>0;\left|\lambda s^{-\alpha }\right|<1.$$

(27)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}\left[t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)={\displaystyle \frac{-s^{\alpha \gamma -\beta }lns}{{(s^{\alpha }-\lambda )}^{\gamma }}};\lambda \in \mathbb{R};\mathrm{Re}s>0;\left|\lambda s^{-\alpha }\right|<1.$$

(28)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \gamma }}\left[t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)={\displaystyle \frac{s^{\alpha \gamma -\beta }}{{(s^{\alpha }-\lambda )}^{\gamma }}}\left[-\alpha lns+ln(s^{\alpha }-\lambda )\right];$$

(29)

$$\lambda \in \mathbb{R};\mathrm{Re}s>0;|\lambda s^{-\alpha }|<1.$$

The second variant of the Laplace transform has the following form:

$$\mathcal{L}\left(t^{\gamma -1}{E}_{\alpha ,\beta }^{\gamma }\left(t\right)\right)\left(s\right)=\mathsf{\Gamma }\left(\gamma \right)\sum _{k=0}^{\infty }{\displaystyle \frac{s^{-\gamma -k}}{\mathsf{\Gamma }(\alpha k+\beta )}}={\displaystyle \frac{\mathsf{\Gamma }\left(\gamma \right)}{s^{\gamma }}}{E}_{\alpha ,\beta }\left({\displaystyle \frac{1}{s}}\right);\mathrm{Re}s>0.$$

(30)

Corollary 2.

Let $\alpha >0$, $\beta >0$, $\gamma >0$. Then, the following formulas of the Laplace transforms of the derivatives with respect to parameters of the function $t^{\gamma -1}{E}_{\alpha ,\beta }^{\gamma }\left(t\right)$ are satisfied:

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}\left[t^{\gamma -1}{E}_{\alpha ,\beta }^{\gamma }\left(t\right)\right]\right)\left(s\right)=-{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma \right)}{s^{\gamma }}}\sum _{k=0}^{\infty }{\displaystyle \frac{k\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}{\displaystyle \frac{1}{s^k}};\mathrm{Re}s>0.$$

(31)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}\left[t^{\gamma -1}{E}_{\alpha ,\beta }^{\gamma }\left(t\right)\right]\right)\left(s\right)=-{\displaystyle \frac{\mathsf{\Gamma }\left(\gamma \right)}{s^{\gamma }}}\sum _{k=0}^{\infty }{\displaystyle \frac{\psi (\alpha k+\beta )}{\mathsf{\Gamma }(\alpha k+\beta )}}{\displaystyle \frac{1}{s^k}};\mathrm{Re}s>0.$$

(32)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \gamma }}\left[t^{\gamma -1}{E}_{\alpha ,\beta }^{\gamma }\left(t\right)\right]\right)\left(s\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\gamma \right)\left[\psi \right(\gamma )-lns]}{s^{\gamma }}}{E}_{\alpha ,\beta }\left({\displaystyle \frac{1}{s}}\right);\mathrm{Re}s>0.$$

(33)

The results for the four-parametric Mittag–Leffler function,

$$E_{({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)}\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }({\alpha }_{1}k+{\beta }_1)\mathsf{\Gamma }({\alpha }_{2}k+{\beta }_2)}},$$

(34)

are similar to those obtained above. For instance, we can use the following formula for the Laplace transforms ($\lambda \in \mathbb{R};\mathrm{Re}s>0$):

$$\mathcal{L}\left(t^{{\beta }_1-1}{E}_{({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)}\left(\lambda t^{{\alpha }_1}\right)\right)\left(s\right)={\displaystyle \frac{1}{s^{{\beta }_1}}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\lambda s^{-{\alpha }_1}\right)}^k}{\mathsf{\Gamma }({\alpha }_{2}k+{\beta }_2)}}={\displaystyle \frac{1}{s^{{\beta }_1}}}{E}_{{\alpha }_2,{\beta }_2}\left(\lambda s^{-{\alpha }_1}\right).$$

(35)

Theorem 6.

Let ${\alpha }_1>0,{\alpha }_2>0,{\beta }_1>0,{\beta }_2>0$. Then, the following formulas of the Laplace transforms of the derivatives are satisfied:

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial {\alpha }_1}}\left[t^{{\beta }_1-1}{E}_{({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)}\left(\lambda t^{{\alpha }_1}\right)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s^{{\beta }_1}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k{\left(\lambda s^{-{\alpha }_1}\right)}^k}{\mathsf{\Gamma }({\alpha }_{2}k+{\beta }_2)}},\mathrm{Re}s>0,$$

(36)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial {\beta }_1}}\left[t^{{\beta }_1-1}{E}_{({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)}\left(\lambda t^{{\alpha }_1}\right)\right]\right)\left(s\right)=-{\displaystyle \frac{lns}{s^{{\beta }_1}}}{E}_{{\alpha }_2,{\beta }_2}\left(\lambda s^{-{\alpha }_1}\right),\mathrm{Re}s>0,$$

(37)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial {\alpha }_2}}\left[t^{{\beta }_1-1}{E}_{({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)}\left(\lambda t^{{\alpha }_1}\right)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s^{{\beta }_1}}}\sum _{k=0}^{\infty }{\displaystyle \frac{k\psi ({\alpha }_{2}k+{\beta }_2)}{\mathsf{\Gamma }({\alpha }_{2}k+{\beta }_2)}}{\left(\lambda s^{-{\alpha }_1}\right)}^k,\mathrm{Re}s>0,$$

(38)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial {\beta }_2}}\left[t^{{\beta }_1-1}{E}_{({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)}\left(\lambda t^{{\alpha }_1}\right)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s^{{\beta }_1}}}\sum _{k=0}^{\infty }{\displaystyle \frac{\psi ({\alpha }_{2}k+{\beta }_2)}{\mathsf{\Gamma }({\alpha }_{2}k+{\beta }_2)}}{\left(\lambda s^{-{\alpha }_1}\right)}^k,\mathrm{Re}s>0.$$

(39)

5. Laplace Transforms of the Derivatives of the Le Roy-Type Function

Here, we deal with the Laplace transforms of the derivatives of the Le Roy-type function (see, e.g., Section 5.3 in [7] cf. [18,19,20]),

$$F_{\alpha ,\beta }^{\left(\gamma \right)}\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{\left[\mathsf{\Gamma }(\alpha k+\beta )\right]}^{\gamma }}}$$

(40)

which generalizes the function

$$R_{\gamma }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{{[k!]}^{\gamma }}},$$

(41)

introduced by E. Le Roy at the end of XIX century for the study of the analytic continuation of power series.

Let us calculate the Laplace transforms of the Le Roy-type function taken in the form similar to (3), (20), (30),

$$\mathcal{L}\left(t^{\gamma -1}{F}_{\alpha ,\beta }^{\left(\gamma \right)}\left(\lambda t^{\alpha }\right)\right)\left(s\right)={\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(\lambda s^{-\alpha }\right)}^k}{{\left[\mathsf{\Gamma }(\alpha k+\beta )\right]}^{\gamma -1}}}={\displaystyle \frac{1}{s^{\beta }}}{F}_{\alpha ,\beta }^{\left(\gamma \right)}\left({\displaystyle \frac{\lambda }{s^{\alpha }}}\right);$$

(42)

$$\lambda \in \mathbb{R},\mathrm{Re}s>0;\gamma >1.$$

In particular, for $\gamma =2$, the Laplace transform image is related to the Mittag–Leffler function,

$$\mathcal{L}\left(t^{\gamma -1}{F}_{\alpha ,\beta }^{\left(2\right)}\left(\lambda t^{\alpha }\right)\right)\left(s\right)={\displaystyle \frac{1}{s^{\beta }}}{E}_{\alpha ,\beta }\left({\displaystyle \frac{\lambda }{s^{\alpha }}}\right).$$

(43)

Theorem 7.

Let $\alpha >0,\beta >0,\gamma >1$. Then, the formulas of the Laplace transforms of the derivatives have the following forms ($\lambda \in \mathbb{R},\mathrm{Re}s>0$):

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \alpha }}\left[t^{\gamma -1}{F}_{\alpha ,\beta }^{\left(\gamma \right)}\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }k{\left({\displaystyle \frac{\lambda }{s^{\alpha }}}\right)}^k{\displaystyle \frac{lns+(\gamma -1)\psi (\alpha k+\beta )}{{\left[\mathsf{\Gamma }(\alpha k+\beta )\right]}^{\gamma -1}}},$$

(44)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \beta }}\left[t^{\gamma -1}{F}_{\alpha ,\beta }^{\left(\gamma \right)}\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }{\left({\displaystyle \frac{\lambda }{s^{\alpha }}}\right)}^k{\displaystyle \frac{lns+(\gamma -1)\psi (\alpha k+\beta )}{{\left[\mathsf{\Gamma }(\alpha k+\beta )\right]}^{\gamma -1}}},$$

(45)

$$\mathcal{L}\left({\displaystyle \frac{\partial }{\partial \gamma }}\left[t^{\gamma -1}{F}_{\alpha ,\beta }^{\left(\gamma \right)}\left(\lambda t^{\alpha }\right)\right]\right)\left(s\right)=-{\displaystyle \frac{1}{s^{\beta }}}\sum _{k=0}^{\infty }{\left({\displaystyle \frac{\lambda }{s^{\alpha }}}\right)}^k{\displaystyle \frac{ln\mathsf{\Gamma }(\alpha k+\beta )}{{\left[\mathsf{\Gamma }(\alpha k+\beta )\right]}^{\gamma -1}}}.$$

(46)

6. Convoluted Forms of the Derivatives of Mittag–Leffler and Wright-Type Functions with Respect to Parameters

The above-reported Laplace transforms of the partial derivatives of Mittag–Leffler and Wright-type functions with respect to parameters allow us to find new representations of the partial derivatives of these special functions in terms of convoluted forms. These forms are obtained via the convolution theorem and the Efros theorem, which are reported below for the sake of clarity.

As far as the convolution theorem is concerned, let ${\phi }_1$ and ${\phi }_2$ be functions, locally integrable on $0,+\infty$, that exhibit exponential growth $u_1$ and $u_2$, respectively. Thus, positive parameters $m_j$, $u_j$, and $T_j$ exist such that $\left|{\phi }_j\left(t\right)\right|\le m_{j}exp\left(u_{j}t\right)$, for all $t\ge T_j$, and $j=1,2$. The convolution product $\left({\phi }_1\ast {\phi }_2\right)\left(t\right)$, given by

$$\begin{array}{ccc}& & \left({\phi }_1\ast {\phi }_2\right)\left(t\right)={\int }_0^t{\phi }_1\left(\tau \right){\phi }_2\left(t-\tau \right)d\tau ,\hfill \end{array}$$

(47)

is properly defined on $[0,+\infty )$ and exhibits exponential growth $max\left\{u_1,u_2\right\}$. Additionally, the following Laplace transform,

$$\begin{array}{ccc}& & \mathcal{L}\left(\left({\phi }_1\ast {\phi }_2\right)\left(t\right)\right)\left(s\right)=\mathcal{L}\left({\phi }_1\left(t\right)\right)\left(s\right)\mathcal{L}\left({\phi }_2\left(t\right)\right)\left(s\right),\hfill \end{array}$$

(48)

holds for $Res>max\left\{u_1,u_2\right\}$.

At this stage, we are equipped to enunciate the various theorems concerning the representations of the special functions under study mentioned above.

Theorem 8.

The partial derivative $\partial /\partial \alpha \left(t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)$, involving the Mittag–Leffler function, results in a convoluted form with a kernel of the logarithmic type:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial \alpha }}{t}^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)=\lambda {\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)t_1^{\alpha +\beta -2}{E}_{\alpha ,\alpha +\beta -1}^2\left(\lambda t_1^{\alpha }\right)d{t}_1,\end{array}$$

(49)

for every $t\ge 0$, $\alpha >0$, $\beta >0$, $\lambda \in \mathbb{R}$.

For $\alpha >0$ and $\left(\alpha +\beta \right)>1$, the first term of the right side of Equation (49) is given by the expression below:

$$\begin{array}{ccc}& & {\int }_0^{t}ln\left(t-t_1\right)t_1^{\alpha +\beta -2}{E}_{\alpha ,\alpha +\beta -1}^2\left(\lambda t_1^{\alpha }\right)d{t}_1\hfill \\ & & =t^{\alpha +\beta -1}\left(lnt-{\gamma }_0\right)\left(\left(\alpha +\beta -1\right)E_{\alpha ,\alpha +\beta -1}^2\left(\lambda t^{\alpha }\right)+2\alpha \lambda t^{\alpha }{E}_{\alpha ,2\alpha +\beta -1}^3\left(\lambda t^{\alpha }\right)\right)\hfill \\ & & -t^{\alpha +\beta -1}\left(\left(\alpha +\beta -1\right)\sum _{n=0}^{\infty }{\displaystyle \frac{\left(n+1\right)\psi \left(\alpha n+\alpha +\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\alpha n+\alpha +\beta -1\right)}}\right.\hfill \\ & & \left.+\alpha \lambda t^{\alpha }\sum _{n=0}^{\infty }{\displaystyle \frac{\left(n+2\right)\left(n+1\right)\psi \left(\alpha n+2\alpha +\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\alpha n+2\alpha +\beta -1\right)}}\right).\hfill \end{array}$$

(50)

For $\alpha >0$ and $\left(\alpha +\beta \right)>1$, the second term on the right side of Equation (49) is given by the form below:

$$\begin{array}{ccc}& & {\int }_0^t{t}_1^{\alpha +\beta -2}{E}_{\alpha ,\alpha +\beta -1}^2\left(\lambda t_1^{\alpha }\right)d{t}_1=t^{\alpha +\beta -1}{E}_{\alpha ,\alpha +\beta }^2\left(\lambda t^{\alpha }\right).\hfill \end{array}$$

(51)

Proof of Theorem 8. 

Form (49) is obtained by expressing the right side of Equation (4) as follows:

$$\lambda {\displaystyle \frac{-lns}{s}}{\displaystyle \frac{s^{1+\alpha -\beta }}{{\left(s^{\alpha }-\lambda \right)}^2}}.$$

The Laplace inversion of the above-reported expression is obtained by performing the Laplace inversion of each of the two fractions and adopting the convolution theorem [21].

The term-by-term integration of the power series representation of the involved Mittag–Leffler function provides Equation (50) via the following integral [21]:

$$\begin{array}{ccc}& & {\int }_0^t{t}_1^{r}ln\left(t-t_1\right)d{t}_1=\left(1+r\right)t^{1+r}\left(lnt-{\gamma }_0-\psi \left(2+r\right)\right),\hfill \end{array}$$

(52)

holding for every $t\ge 0$ and $r>-1$. Instead, Equation (51) is obtained in a straightforward way via the term-by-term integration of the power series representation of the involved Mittag–Leffler function. □

Theorem 9.

The partial derivative $\partial /\partial \beta \left(t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)$, involving the Mittag–Leffler function, results in a convoluted form with a kernel of the logarithmic type:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial \beta }}\left(t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)={\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)t_1^{\beta -2}{E}_{\alpha ,\beta -1}\left(\lambda t_1^{\alpha }\right)d{t}_1,\end{array}$$

(53)

for every $t\ge 0$, $\alpha >0$, $\beta >0$, $\lambda \in \mathbb{R}$.

For $\alpha >0$ and $\beta >1$, the first term on the right side of Equation (53) is given by

$$\begin{array}{ccc}& & {\int }_0^t\left(ln\left(t-t_1\right)\right)t_1^{\beta -2}{E}_{\alpha ,\beta -1}\left(\lambda t_1^{\alpha }\right)d{t}_1\hfill \\ & & =t^{\beta -1}\left(lnt-{\gamma }_0\right)\left(\left(\beta -1\right)E_{\alpha ,\beta -1}\left(\lambda t^{\alpha }\right)+\alpha \lambda t^{\alpha }{E}_{\alpha ,\alpha +\beta -1}^2\left(\lambda t^{\alpha }\right)\right)\hfill \\ & & -t^{\beta -1}\left(\left(\beta -1\right)\sum _{n=0}^{\infty }{\displaystyle \frac{\psi \left(\alpha n+\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\alpha n+\beta -1\right)}}\right.\hfill \\ & & \left.+\alpha \sum _{n=1}^{\infty }{\displaystyle \frac{n\psi \left(\alpha n+\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\alpha n+\beta -1\right)}}\right).\hfill \end{array}$$

(54)

For $\alpha >0$ and $\beta >1$, the second term of the right side of Equation (53) is given by

$$\begin{array}{ccc}& & {\int }_0^t{t}_1^{\beta -2}{E}_{\alpha ,\beta -1}\left(\lambda t_1^{\alpha }\right)d{t}_1=t^{\beta -1}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right).\hfill \end{array}$$

(55)

The proof of Theorem 9 is similar to the proof of Theorem 8.

The Efros theorem is a generalization of the convolution theorem that represents in the present scenario a valuable tool for evaluating the involved Laplace inversions [12]. Let $F\left(s\right)$ be the Laplace transform of the function $f\left(t\right)$,

$$F\left(s\right)=\mathcal{L}\left(f\left(t\right)\right)\left(s\right).$$

Let $G\left(s\right)$ and $q\left(s\right)$ be two analytic functions of the complex variable s such that $G\left(s\right)exp\left(-\tau q\left(s\right)\right)$ is the Laplace transform of the function $g\left(t,\tau \right)$,

$$G\left(s\right)exp\left(-\tau q\left(s\right)\right)=\mathcal{L}\left(g\left(t,\tau \right)\right)\left(s\right).$$

The Efros theorem states that the following Laplace transform holds [12]:

$$G\left(s\right)F\left(q\left(s\right)\right)=\mathcal{L}\left({\int }_0^{+\infty }f\left(\tau \right)g\left(t,\tau \right)d\tau \right)\left(s\right).$$

The Efros theorem provides the following Laplace inversion [22]:

$$\begin{array}{c}\hfill s^{-b}F\left(s^a\right)=\mathcal{L}\left({\int }_0^{+\infty }{\mathsf{\Phi }}_{a,b}\left(t,t^{\prime }\right)f\left(t^{\prime }\right)d{t}^{\prime }\right)\left(s\right).\end{array}$$

(56)

The function ${\mathsf{\Phi }}_{a,b}\left(t,t^{\prime }\right)$ can be defined as follows [22]:

$$\begin{array}{ccc}& & {\mathsf{\Phi }}_{a,b}\left(t,t^{\prime }\right)={\pi }^{-1}{\int }_0^t{\displaystyle \frac{sin\left(t^{\prime }{u}^{a}sin\left(\pi a\right)+\pi b\right)}{u^{b}exp\left(ut+t^{\prime }{u}^{a}cos\left(\pi a\right)\right)}}du+{\delta }_{b,1},\hfill \end{array}$$

(57)

where ${\delta }_{b,1}$ is the Kronecker symbol. Particularly, the function ${\mathsf{\Phi }}_{a,b}\left(t,t^{\prime }\right)$ reproduces the Heaviside function for $a=b=1$ or the delta function for $a=1$ and $b=0$, respectively,

$${\mathsf{\Phi }}_{1,1}\left(t,t^{\prime }\right)=\mathsf{\Theta }\left(t-t^{\prime }\right),{\mathsf{\Phi }}_{1,0}\left(t,t^{\prime }\right)=\delta \left(t-t^{\prime }\right).$$

The Efros theorem allows us to obtain further integral forms of the special functions under study.

Theorem 10.

The Wright function $W_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)$ results in an integral form involving the modified Bessel function of the first kind $I_0\left(t\right)$:

$$\begin{array}{c}\hfill t^{\beta -1}{W}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)={\int }_0^{+\infty }{\mathsf{\Phi }}_{\alpha ,\beta -\alpha }\left(t,t_1\right)I_0\left(2\sqrt{\lambda t_1}\right)d{t}_1,\end{array}$$

(58)

for every $t\ge 0$, $\alpha >0$, $\beta >0$.

Proof of Theorem 10. 

Form (58) is obtained by expressing the right side of Equation (22) as follows:

$$s^{\alpha -\beta }{\displaystyle \frac{exp\left(\lambda /s^{\alpha }\right)}{s^{\alpha }}}.$$

The Laplace inversion of the above-reported expression is obtained in a straightforward way from Equations (56) and (57) [21]. □

Theorem 11.

The partial derivative $\partial /\partial \alpha \left(t^{\beta -1}{W}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)$, involving the Wright function, results in a convoluted form with logarithmic and other types of kernels:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial \alpha }}{t}^{\beta -1}{W}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)=\lambda {\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)\left({\int }_0^{+\infty }{\mathsf{\Phi }}_{\alpha ,\beta -1}\left(t_1,t_2\right)I_0\left(2\sqrt{\lambda t_2}\right)d{t}_2\right)d{t}_1,\end{array}$$

(59)

for every $t\ge 0$, $\alpha >0$, $\beta >0$.

Proof of Theorem 11. 

Form (59) is obtained by expressing the right side of Equation (23) as follows:

$$-\lambda {\displaystyle \frac{lns}{s}}{s}^{1-\beta }{\displaystyle \frac{exp\left(\lambda /s^{\alpha }\right)}{s^{\alpha }}}.$$

The Laplace inversion of the above-reported expression is obtained in straightforward way from Equations (56) and (57) and the convolution theorem [21]. □

Theorem 12.

The partial derivative $\partial /\partial \beta \left(\left(t^{\beta -1}{W}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)\right)$, involving the Wright function, results to be a convoluted form with logarithmic and other types of kernels:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial \beta }}\left(t^{\beta -1}{W}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)\right)={\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)\left({\int }_0^{+\infty }{\mathsf{\Phi }}_{\alpha ,\beta -\alpha -1}\left(t_1,t_2\right)I_0\left(2\sqrt{\lambda t_2}\right)d{t}_2\right)d{t}_1,\end{array}$$

(60)

for every $t\ge 0$, $\alpha >0$, $\beta >0$.

Proof of Theorem 12. 

Form (60) is obtained by expressing the right side of Equation (24) as follows:

$$-{\displaystyle \frac{lns}{s}}{s}^{\alpha -\beta +1}{\displaystyle \frac{exp\left(\lambda /s^{\alpha }\right)}{s^{\alpha }}}.$$

The Laplace inversion of the above-reported expression is obtained in straightforward way from Equations (56) and (57) and the convolution theorem [21]. □

Theorem 13.

The partial derivative $\partial /\partial \alpha \left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right)$, involving the Prabhakar function, results in a convoluted form with a kernel of the logarithmic type:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial \alpha }}{t}^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)=\gamma \lambda {\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)t_1^{\alpha +\beta -2}{E}_{\alpha ,\alpha +\beta -1}^{\gamma +1}\left(\lambda t_1^{\alpha }\right)d{t}_1,\end{array}$$

(61)

for every $t\ge 0$, $\alpha >0$, $\beta >0$, $\gamma >0$, $\lambda \in \mathbb{R}$.

For $\alpha >0$, $\left(\alpha +\beta \right)>1$, the first term of the right side of Equation (61) results in

$$\begin{array}{ccc}& & {\int }_0^t\left(ln\left(t-t_1\right)\right)t_1^{\alpha +\beta -2}{E}_{\alpha ,\alpha +\beta -1}^{\gamma +1}\left(\lambda t_1^{\alpha }\right)d{t}_1\hfill \\ & & =t^{\alpha +\beta -1}\left(lnt-{\gamma }_0\right)\left(\left(\alpha +\beta -1\right)E_{\alpha ,\alpha +\beta -1}^{\gamma +1}\left(\lambda t^{\alpha }\right)\right.\hfill \\ & & \left.+\alpha \left(\gamma +1\right)\lambda t^{\alpha }{E}_{\alpha ,2\alpha +\beta -1}^{\gamma +2}\left(\lambda t^{\alpha }\right)\right)\hfill \\ & & -t^{\alpha +\beta -1}\left(\left(\alpha +\beta -1\right)\sum _{n=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(n+\gamma +1\right)\psi \left(\alpha n+\alpha +\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\gamma +1\right)n!\mathsf{\Gamma }\left(\alpha n+\alpha +\beta -1\right)}}\right.\hfill \\ & & \left.+\alpha \left(\gamma +1\right)\lambda t^{\alpha }\sum _{n=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(n+\gamma +2\right)\psi \left(\alpha n+2\alpha +\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\gamma +2\right)n!\mathsf{\Gamma }\left(\alpha n+2\alpha +\beta -1\right)}}\right).\hfill \end{array}$$

(62)

For $\alpha >0$, $\left(\alpha +\beta \right)>1$, $\gamma >-1$, the second term of the right side of Equation (61) results in

$$\begin{array}{ccc}& & {\int }_0^t{t}_1^{\alpha +\beta -2}{E}_{\alpha ,\alpha +\beta -1}^{\gamma +1}\left(\lambda t_1^{\alpha }\right)d{t}_1=t^{\alpha +\beta -1}{E}_{\alpha ,\alpha +\beta }^{\gamma +1}\left(\lambda t^{\alpha }\right).\hfill \end{array}$$

(63)

Proof of Theorem 13. 

Form (61) is obtained by expressing the right side of Equation (27) as follows:

$$-\gamma \lambda {\displaystyle \frac{lns}{s}}{\displaystyle \frac{s^{\alpha \gamma -\beta +1}}{{\left(s^{\alpha }-\lambda \right)}^{\gamma +1}}}.$$

The Laplace inversion of the above-reported expression is obtained by performing the Laplace inversion of each of the two fractions and applying the convolution theorem [21].

The term-by-term integration of the series representation of the involved Prabhakar function provides Equation (62) via Equation (52), while Equation (63) is obtained in a straightforward way via the term-by-term integration [21]. □

Theorem 14.

The partial derivative $\partial /\partial \beta \left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right)$, involving the Prabhakar function, results in a convoluted form with a kernel of the logarithmic type:

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial }{\partial \beta }}\left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right)={\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)t_1^{\beta -2}{E}_{\alpha ,\beta -1}^{\gamma }\left(\lambda t_1^{\alpha }\right)d{t}_1,\hfill \end{array}$$

(64)

for every $t\ge 0$, $\alpha >0$, $\beta >0$, $\gamma >0$, $\lambda \in \mathbb{R}$.

For $\alpha >0$ and $\beta >1$, the first term of the right side of Equation (64) results in

$$\begin{array}{ccc}& & {\int }_0^t\left(ln\left(t-t_1\right)\right)t_1^{\beta -2}{E}_{\alpha ,\beta -1}^{\gamma }\left(\lambda t_1^{\alpha }\right)d{t}_1\hfill \\ & & =t^{\beta -1}\left(lnt-{\gamma }_0\right)\left(\left(\beta -1\right)E_{\alpha ,\beta -1}^{\gamma }\left(\lambda t^{\alpha }\right)+\alpha \gamma \lambda t^{\alpha }{E}_{\alpha ,\alpha +\beta -1}^{\gamma +1}\left(\lambda t^{\alpha }\right)\right)\hfill \\ & & -t^{\beta -1}\left(\left(\beta -1\right)\sum _{n=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(n+\gamma \right)\psi \left(\alpha n+\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\gamma \right)n!\mathsf{\Gamma }\left(\alpha n+\beta -1\right)}}\right.\hfill \\ & & \left.+\alpha \gamma \lambda t^{\alpha }\sum _{n=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(n+\gamma +1\right)\psi \left(\alpha n+\alpha +\beta \right){\left(\lambda t^{\alpha }\right)}^n}{\mathsf{\Gamma }\left(\gamma +1\right)n!\mathsf{\Gamma }\left(\alpha n+\alpha +\beta -1\right)}}\right).\hfill \end{array}$$

(65)

For $\alpha >0$, $\gamma >0$, and $\beta >1$, the second term of the right side of Equation (64) results in

$$\begin{array}{ccc}& & {\int }_0^t{t}_1^{\beta -2}{E}_{\alpha ,\beta -1}^{\gamma }\left(\lambda t_1^{\alpha }\right)d{t}_1=t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right).\hfill \end{array}$$

(66)

Proof of Theorem 14. 

Form (64) is obtained by expressing the right side of Equation (28) as follows:

$$-{\displaystyle \frac{lns}{s}}{\displaystyle \frac{s^{\alpha \gamma -\beta +1}}{{\left(s^{\alpha }-\lambda \right)}^{\gamma }}}.$$

The Laplace inversion of the above-reported expression is obtained by performing the Laplace inversion of each of the two fractions and applying the convolution theorem [21]. □

Theorem 15.

The partial derivative $\partial /\partial \gamma \left(t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)\right)$, involving the Prabhakar function, results in the sum of convoluted forms with logarithmic and other types of kernels:

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial }{\partial \gamma }}{t}^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }\left(\lambda t^{\alpha }\right)=\alpha {\int }_0^t\left(ln\left(t-t_1\right)+{\gamma }_0\right)t_1^{\beta -2}{E}_{\alpha ,\beta -1}^{\gamma }\left(\lambda t_1^{\alpha }\right)d{t}_1\hfill \\ & & -{\int }_0^t{\left(t-t_1\right)}^{\beta -\alpha -1}{E}_{\alpha ,\beta -\alpha }^{\gamma -1}\left(\lambda {\left(t-t_1\right)}^{\alpha }\right)\left({\int }_0^{+\infty }{\mathsf{\Phi }}_{\alpha ,0}\left(t_1,t_2\right)exp\left(\lambda t_2\right)\left(ln{t}_2+{\gamma }_0\right)d{t}_2\right)d{t}_1,\hfill \end{array}$$

(67)

for every $t\ge 0$, $\alpha >0$, $\beta >0$, $\gamma >0$, $\lambda \in \mathbb{R}$.

Proof of Theorem 15. 

Form (67) is obtained by expressing the right side of Equation (29) as follows:

$$-\alpha {\displaystyle \frac{lns}{s}}{\displaystyle \frac{s^{\alpha \gamma -\beta +1}}{{\left(s^{\alpha }-\lambda \right)}^{\gamma }}}+{\displaystyle \frac{s^{\alpha \gamma -\beta }}{{\left(s^{\alpha }-\lambda \right)}^{\gamma -1}}}{\displaystyle \frac{ln\left(s^{\alpha }-\lambda \right)}{\left(s^{\alpha }-\lambda \right)}}.$$

The Laplace inversion of the above-reported expression is obtained by performing the Laplace inversion of each of the two terms of the sum via the convolution theorem and Equations (56) and (57) [21]. □

7. Discussion and Outlook

This paper provides several formulas of the Laplace transforms of the derivatives with respect to parameters of a number of special functions. Some of these formulas are represented via elementary or known special functions, but others are related to new types of special functions. These results could serve to improve our understanding about the local and global behaviors of the considered special functions. They can be useful for solving differential equations.

The formulas of the Laplace transforms of the derivatives are reconstructed using the Efros theorem. Using a special case of this theorem and operational calculus, it will be possible to derive new formulas for special functions of the Mittag–Leffler type and their generalizations. These formulas allow new ways of considering different models and of solving new classes of differential and integral equations. The obtained formulas contain series which can be considered as a new type of special function. The coefficients in these series contain not only products of $\mathsf{\Gamma }$ functions but also their powers, their logarithms, and their derivatives. The characteristic for these new functions is that they can be represented via the Mellin–Barnes integrals. It paves the way to study asymptotics of the Laplace transforms of the derivatives of special functions with respect to parameters and to solve the corresponding differential equations.