On Symmetrical Sonin Kernels in Terms of Hypergeometric-Type Functions

Abstract

In this paper, a new class of kernels of integral transforms of the Laplace convolution type that we named symmetrical Sonin kernels is introduced and investigated. For a symmetrical Sonin kernel given in terms of elementary or special functions, its associated kernel has the same form with possibly different parameter values. In the paper, several new kernels of this type are derived by means of the Sonin method in the time domain and using the Laplace integral transform in the frequency domain. Moreover, for the first time in the literature, a class of Sonin kernels in terms of the convolution series, which are a far-reaching generalization of the power series, is constructed. The new symmetrical Sonin kernels derived in the paper are represented in terms of the Wright function and the new special functions of the hypergeometric type that are extensions of the corresponding Horn functions in two variables.

1. Introduction

In the theory of the integral transforms of the Mellin convolution type, the functions p and q are said to form a pair of the Fourier kernels if the relations

$$F(x)={\int }_0^{+\infty }p(x\xi )f(\xi )d\xi ,x>0,$$

(1)

$$f(\xi )={\int }_0^{+\infty }q(\xi x)F(x)dx,\xi >0$$

(2)

are simultaneously valid on some functional spaces that the functions f and F belong to; see, e.g., [1] for details and particular cases. In the case $p(x)=q(x),x\in {\mathbb{R}}_{+}$, the kernels p and q are called symmetrical Fourier kernels.

Under the condition that the Mellin integral transforms

$$P(s)={\int }_0^{+\infty }p(x)x^{s-1}dx,Q(s)={\int }_0^{+\infty }q(x)x^{s-1}dx$$

of the Fourier kernels p and q exist in the domain $D=\{s\in \mathbb{C}:C_1<\Re (s)<C_2$}, with $C_1$ and $C_2$ being some constants from $\mathbb{R}$, the Mellin convolution theorem applied to relations (1) and (2) leads to the functional equation

$$P(s)·Q(1-s)=1,C_1<\Re (s)<C_2$$

(3)

in the frequency domain.

Because the majority of elementary and special functions are particular cases of the Meijer G- or Fox H-functions, their Mellin integral transforms can be represented in terms of quotients of products of Gamma functions. Thus, Formula (3) lets a straightforward derivation of both Fourier kernels and symmetrical Fourier kernels in terms of the Meijer G and Fox H functions [1].

In this paper, we focus on kernels of so-called general fractional integrals (GFIs),

$$F(x)=({\mathbb{I}}_{(\kappa )}f)(x):=(\kappa \ast f)(x)={\int }_0^x\kappa (x-\xi )f(\xi )d\xi$$

(4)

and general fractional derivatives (GFDs),

$$f(\xi )=({\mathbb{D}}_{(k)}F)(\xi ):={\displaystyle \frac{d}{d\xi }}({\mathbb{I}}_{(k)}F)(\xi )={\displaystyle \frac{d}{d\xi }}(k\ast F)(\xi )={\displaystyle \frac{d}{d\xi }}{\int }_0^{\xi }k(\xi -x)F(x)dx$$

(5)

of the Laplace convolution type with Sonin kernels $\kappa$ and k ([2,3,4,5,6]).

In [7], Sonin extended the Abel method for solving the integral equation of type (4) with the power law kernel $\kappa (x)=x^{\alpha -1},\alpha \in (0,1)$ to the case of a general kernel, $\kappa$, that satisfies the condition

$$(\kappa \ast k)(x)={\int }_0^x\kappa (x-\xi )k(\xi )d\xi =1,x>0.$$

(6)

Nowadays, condition (6) is referred to as the Sonin condition, and the function k is called the Sonin kernel associated with the kernel $\kappa$.

It is worth mentioning that, in the mathematical literature, the famous Russian mathematician Sonin is referred to either as Sonin (English spelling) or as Sonine (French spelling). In this paper, we employ the English spelling of his surname.

Under certain conditions, the GFD (5) with the Sonin kernel k is a left-inverse operator to the GFI (4) with its associated kernel $\kappa$; see, e.g., [2,4,5,6,8,9,10] for details and examples of Sonin kernels in terms of elementary and special functions.

If the Laplace integral transforms

$$\tilde{\kappa }(p)={\int }_0^{+\infty }\kappa (x)e^{-px}dx,\tilde{k}(p)={\int }_0^{+\infty }k(x)e^{-px}dx$$

of the Sonin kernels $\kappa$ and k exist in the domain $D=\{p\in \mathbb{C}:\Re (p)>C\}$, the relation (6) is reduced to a simple formula in the Laplace domain:

$$\tilde{\kappa }(p)·\tilde{k}(p)={\displaystyle \frac{1}{p}},\Re (p)>C.$$

(7)

This formula can be used for the derivation of some particular cases of Sonin kernels using the tables of the Laplace integral transforms of elementary and special functions. However, in contrast to the case of the Fourier kernels, we cannot directly use the technique of the Mellin integral transform and provide a general description of the Sonin kernels in terms of the Meijer G and Fox H functions. In this paper, we suggest some alternative approaches for the derivation of Sonin kernels and, in particular, of so-called symmetrical Sonin kernels that will be defined in the next section.

It is worth mentioning that, recently, GFIs and GFDs with Sonin kernels became a subject of active research in both fractional calculus (FC) (see, e.g., [2,3,4,5,6,11,12,13,14]) and its applications (see [15,16,17,18,19] for the models of the general fractional dynamics, the general non-Markovian quantum dynamics, the general non-local electrodynamics, the non-local classical theory of gravity, and non-local statistical mechanics, respectively, and [20,21,22,23,24] for the mathematical models of anomalous diffusion and linear viscoelasticity in terms of GFIs and GFDs with Sonin kernels). Thus, the investigation of the general properties and particular cases of the Sonin kernels is an important topic both for the theory of FC and for its applications. This paper is devoted to these matters, and it aims to extend the set of known Sonin kernels in terms of the special functions of the hypergeometric type.

The rest of the paper is organized as follows. In Section 2, we provide some preliminary information regarding Sonin kernels and their examples. In particular, for the first time in the literature, we introduce a class of symmetrical Sonin kernels. Section 3 is devoted to the construction of Sonin kernels, both symmetrical and non-symmetrical, in the time domain. Following Sonin, we provide a procedure for the derivation of the Sonin kernels in the form of hypergeometric-type functions, and we present several examples of Sonin kernels, both known and new ones, obtained by means of this procedure. In particular, a new pair of symmetrical Sonin kernels in terms of the Wright function is deduced. Then, we introduce a new class of Sonin kernels in the form of the convolution series, which constitutes a far-reaching generalization of the power series, and we present some examples of this kind. In Section 4, the Laplace integral transform technique is employed for the construction of symmetrical Sonin kernels in the frequency domain. Using the inverse Laplace transform, these kernels are then represented in the time domain in terms of the hypergeometric functions in one and two variables. In particular, we present several new Sonin kernels expressed in the form of the new special functions of the hypergeometric type that are extensions of the corresponding Horn functions in two variables.

2. Definitions and Examples

For the first time, a pair of Sonin kernels appeared in the publications [25,26] by Abel devoted to the so-called tautochrone problem. The tautochrone is a curve for which the time taken for an object that slides without friction in uniform gravity to its lowest point is independent of its starting point on the curve. By the time of Abel’s publications, Christiaan Huygens could already prove, through some advanced geometrical methods, that the tautochrone is a suitable part of the cycloid. In [25,26], Abel provided an analytical solution to a somewhat more general mechanical problem that is nowadays referred to as the generalized tautochrone problem, which is formulated as follows: for a given function, $F=F(x)$, find a curve for which the sliding time taken for an object sliding without friction in uniform gravity depends on the position x of its starting point on the curve as $F=F(x)$. In [25,26], Abel first derived a mathematical model of the generalized tautochrone problem in the form of the following integro-differential equation (with slightly different notations):

$$F(x)={\int }_0^x{\displaystyle \frac{f^{\prime }(\xi )}{\sqrt{(x-\xi )}}}d\xi .$$

(8)

Then Abel considered a more general equation:

$$F(x)={\int }_0^x{\displaystyle \frac{f^{\prime }(\xi )}{{(x-\xi )}^{\alpha }}}d\xi ,0<\alpha <1.$$

(9)

The solution to the integro-differential Equation (9) derived by Abel has the following form (according to his tautochrone model, the condition $f(0)=0$ is valid):

$$f(x)={\displaystyle \frac{sin(\alpha \pi )}{\pi }}{\int }_0^x{(x-\xi )}^{\alpha -1}F(\xi )d\xi ,x>0.$$

(10)

It is worth mentioning that the known formula

$${\displaystyle \frac{sin(\alpha \pi )}{\pi }}={\displaystyle \frac{1}{\Gamma (\alpha )\Gamma (1-\alpha )}}$$

immediately leads to a representation of Equations (9) and (10) in terms of the so-called Caputo fractional derivative, ${}_{\ast }D_{0+}^{\alpha }$, and the Riemann–Liouville fractional integral, $I_{0+}^{\alpha }$, of the order $\alpha$:

$${\displaystyle \frac{F(x)}{\Gamma (1-\alpha )}}=({}_{\ast }D_{0+}^{\alpha }f)(x),\Gamma (1-\alpha )f(x)=(I_{0+}^{\alpha }F)(x),x>0,0<\alpha <1.$$

The main ingredient of the solution method for the integro-differential Equation (9) invented by Abel was a simple formula for the kernels of the operators at the right-hand sides of Equations (9) and (10), i.e., of the kernels of the Riemann–Liouville fractional integral and the Riemann–Liouville or Caputo fractional derivatives (in modern notations):

$$(h_{\alpha }\ast h_{1-\alpha })(x)=1,0<\alpha <1,x>0,$$

(11)

where the power law function $h_{\alpha }$ is defined as follows:

$$h_{\alpha }(x)={\displaystyle \frac{x^{\alpha -1}}{\Gamma (\alpha )}},\alpha >0.$$

(12)

Thus, in [25,26], Abel derived the first and a very important pair of Sonin kernels:

$$\kappa (x)=h_{\alpha }(x),k(x)=h_{1-\alpha }(x),0<\alpha <1,x>0.$$

(13)

In particular, the tautochrone problem (8) corresponds to kernels (13) with $\alpha ={\displaystyle \frac{1}{2}}$. In this case, the kernels $\kappa$ and k have exactly the same form:

$$\kappa (x)=k(x)=h_{1/2}(x)={\displaystyle \frac{1}{\sqrt{\pi }}}{\displaystyle \frac{1}{\sqrt{x}}},x>0.$$

(14)

It is natural to call such kernels symmetrical Sonin kernels. However, up to a constant multiplier, this is the only pair of Sonin kernels that satisfies the relation $\kappa (x)=k(x),x\in {\mathbb{R}}_{+}$ if we suppose that the Laplace integral transforms of the kernels exist. Indeed, in this case, we get the following chain of implications from the Sonin condition (7) in the frequency domain:

$$\tilde{\kappa }(p)·\tilde{k}(p)={\tilde{\kappa }}^2(p)={\displaystyle \frac{1}{p}}\Rightarrow \tilde{\kappa }(p)=k(p)=\pm {\displaystyle \frac{1}{\sqrt{p}}}\Rightarrow \kappa (x)=k(x)=\pm h_{1/2}(x).$$

Even the power-law Sonin kernels (13) derived by Abel in [25,26] do not coincide unless $\alpha =1/2$. However, they are expressed in terms of the same power law function, $h_{\beta }$, with different values of the parameter $\beta$, and they generate the Riemann–Liouville and Caputo fractional derivatives and the Riemann–Liouville fractional integral, which are probably the most used FC operators.

Motivated by this example, we define the symmetrical Sonin kernels as follows:

Definition 1. 

A pair $(\kappa ,k)$ of Sonin kernels is called symmetrical if both the kernels are of the same form in terms of certain elementary or special functions with possibly different parameter values.

According to this definition, the power-law Sonin kernels (13) can be called symmetrical. However, not all pairs of Sonin kernels are symmetrical. In particular, we mention here the following known pairs of non-symmetrical Sonin kernels (see [9,11,27]):

$$\kappa (x)=h_{\alpha ,\rho }(x),k(x)=h_{1-\alpha ,\rho }(x)+\rho {\int }_0^x{h}_{1-\alpha ,\rho }(\xi )d\xi ,0<\alpha <1,\rho >0,$$

where

$$h_{\alpha ,\rho }(x)={\displaystyle \frac{x^{\alpha -1}}{\Gamma (\alpha )}}{e}^{-\rho x}$$

and

$$\kappa (x)=h_{1-\beta +\alpha }(x)+h_{1-\beta }(x),k(x)=x^{\beta -1}{E}_{\alpha ,\beta }(-x^{\alpha }),0<\alpha <\beta <1,$$

(15)

where $E_{\alpha ,\beta }$ stands for the two-parameter Mittag–Leffler function defined by the following convergent series:

$$E_{\alpha ,\beta }(z)=\sum _{n=0}^{+\infty }{\displaystyle \frac{z^n}{\Gamma (\alpha n+\beta )}},\alpha >0,\beta ,z\in \mathbb{C}.$$

(16)

For further examples of symmetrical and non-symmetrical Sonin kernels, we refer to [4,5,7,8,9]; see also the references therein.

It is worth mentioning that any pair, $(\kappa ,k)$, of the Sonin kernels generates a pair of the general FC operators: the GFI (4) with the kernel $\kappa$ and the GFD (5) with the kernel k. As an example, the Sonin kernels (15) generate the GFI in the form of a sum of two Riemann–Liouville fractional integrals,

$$({\mathbb{I}}_{(\kappa )}f)(x)=(I_{0+}^{1-\beta +\alpha }f)(x)+(I_{0+}^{1-\beta }f)(x),x>0$$

(17)

and the GFD with the Mittag–Leffler function in the kernel

$$({\mathbb{D}}_{(k)}f)(x)={\displaystyle \frac{d}{dx}}{\int }_0^x{(x-\xi )}^{\beta -1}{E}_{\alpha ,\beta }(-{(x-\xi )}^{\alpha })f(\xi )d\xi ,0<\alpha <\beta <1,x>0.$$

(18)

In suitable spaces of functions, the GFD (18) is a left-inverse operator to the GFI (17) (see, e.g., [4,5] for details):

$$({\mathbb{D}}_{(k)}{\mathbb{I}}_{(\kappa )}f)(x)=f(x).$$

As we see, in the case of non-symmetrical Sonin kernels, the formulas for the GFIs and the GFDs look very different. In the rest of this paper, we mainly deal with the symmetrical Sonin kernels and discuss the derivation of such kernels in the time and frequency domains, as well as several known and new kernels of this kind.

3. Sonin Kernels in the Time Domain

In [7], Sonin introduced an important class of Sonin kernels in the form of the products of power law functions and analytic functions:

$$\kappa (x)=x^{\beta -1}·{\kappa }_1(x),{\kappa }_1(x)=\sum _{n=0}^{+\infty }{a}_n{x}^n,a_0\ne 0,0<\beta <1,$$

(19)

$$k(x)=x^{-\beta }·k_1(x),k_1(x)=\sum _{n=0}^{+\infty }{b}_n{x}^n,0<\beta <1.$$

(20)

Whereas the coefficients $a_n,n\in {\mathbb{N}}_0$ (${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$) of the analytic function ${\kappa }_1$ can be arbitrarily chosen, the coefficients $b_n,n\in {\mathbb{N}}_0$ of the function $k_1$ have to satisfy the following triangular system of the linear equations:

$$\Gamma (\beta )\Gamma (1-\beta )a_0{b}_0=1,\sum _{n=0}^N\Gamma (n+\beta )\Gamma (N-n+1-\beta )a_n{b}_{N-n}=0,N\in \mathbb{N}.$$

(21)

Equations (21) are obtained via the substitution of the functions $\kappa$ and k defined as in (19) and (20) into the Sonin condition (6), interchanging the order of the integration and summation operations (which is justified by the uniform convergence of the series defining the analytic functions ${\kappa }_1$ and $k_1$ at any finite interval) and by applying the formula

$$(h_{\alpha }\ast h_{\beta })(x)=h_{\alpha +\beta }(x),\alpha ,\beta >0,x>0,$$

(22)

that immediately follows from the well-known representation of the Euler beta function in terms of the Gamma function:

$$B(\alpha ,\beta )={\int }_0^1{u}^{\alpha -1}{(1-u)}^{\beta -1}du={\displaystyle \frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}},\alpha ,\beta >0.$$

In the following theorem, a more general construction of the Sonin kernels compared to the one suggested by Sonin in forms (19) and (20) is provided.

Theorem 1. 

Let ${\kappa }_1$ and $k_1$ be analytic functions, and let the conditions

$$0<\beta <1,\alpha >0,\lambda \in \mathbb{R}$$

(23)

be satisfied.

The functions

$$\kappa (x)=x^{\beta -1}·{\kappa }_1(\lambda x^{\alpha }),{\kappa }_1(x)=\sum _{n=0}^{+\infty }{a}_n{x}^n,a_0\ne 0$$

(24)

and

$$k(x)=x^{-\beta }·k_1(\lambda x^{\alpha }),k_1(x)=\sum _{n=0}^{+\infty }{b}_n{x}^n$$

(25)

build a pair of Sonin kernels iff the coefficients $a_n,n\in {\mathbb{N}}_0$ and $b_n,n\in {\mathbb{N}}_0$ of the functions ${\kappa }_1$ and $k_1$ satisfy the following triangular system of equations:

$$\Gamma (\beta )\Gamma (1-\beta )a_0{b}_0=1,\sum _{n=0}^N\Gamma (\alpha n+\beta )\Gamma (\alpha (N-n)+1-\beta )a_n{b}_{N-n}=0,N\in \mathbb{N}.$$

(26)

The statement formulated in Theorem 1 is proved through the same direct calculations as the ones employed for the Sonin kernels in forms (19) and (20) (see, e.g., [9]), i.e., by substitution of the functions $\kappa$ and k defined as in (24) and (25) into the Sonin condition (6), interchanging the order of the integration and summation operations, and applying the Formula (22) for the calculation of convolution integrals.

Remark 1. 

It is worth mentioning that Theorem 1 also remains valid in the case of the convergence radii of the series that define the functions ${\kappa }_1$ and $k_1$ not being infinite. The essential condition for the validity of Theorem 1 is that the convergence radius of the series that defines ${\kappa }_1$ is greater than zero. Then, the conditions (26) implicate that the series for $k_1$ has the same convergence radius, and the functions κ and k defined by Formulas (24) and (25), respectively, are Sonin kernels (see [28] for the proof of this statement in the case of $\alpha =1$). The proof for an arbitrary $\alpha >0$ is completely analogous to the one for $\alpha =1$, and we omit it here.

Now, we follow another idea suggested by Sonin in his paper [7] concerning kernels in forms (19) and (20) to derive some particular cases of Sonin kernels in forms (24) and (25), and in particular, of symmetrical Sonin kernels.

First, we mention that Equation (26) can be interpreted as just one functional equation:

$$\left(\sum _{n=0}^{+\infty }{c}_n{x}^n\right)·\left(\sum _{n=0}^{+\infty }{d}_n{x}^n\right)\equiv 1,x>0,$$

(27)

where

$$c_n:=\Gamma (\alpha n+\beta )a_n\mathrm{and}{d}_n:=\Gamma (\alpha n+1-\beta )b_n,n\in {\mathbb{N}}_0.$$

(28)

Moreover, without any loss of generality, we can set

$$c_0=d_0=1.$$

(29)

Remark 2. 

When Equation (27) is employed, the Sonin kernels in the form (24) and (25) can be derived using the following simple procedure:

• Step 1: Construct an arbitrary power series with a convergence radius greater than zero:

  $$\varphi (x)=\sum _{n=0}^{+\infty }{c}_n{x}^n,c_0=1.$$

  (30)
• Step 2: Build the function ${\displaystyle \frac{1}{\varphi (x)}}$ and determine its Taylor series:

  $${\displaystyle \frac{1}{\varphi (x)}}=\sum _{n=0}^{+\infty }{d}_n{x}^n,d_0=1.$$

  (31)
• Step 3: The functions (24) and (25) with the coefficients

  $$a_n={\displaystyle \frac{c_n}{\Gamma (\alpha n+\beta )}}and{b}_n={\displaystyle \frac{d_n}{\Gamma (\alpha n+1-\beta )}},n\in {\mathbb{N}}_0,$$

  (32)

  respectively, are Sonin kernels.

Remark 3. 

In the case when the functions $\varphi (x)$ and ${\displaystyle \frac{1}{\varphi (x)}}$ defined by the series at the right-hand sides of (30) and (31), respectively, have the same form in terms of certain elementary or special functions with possibly different parameter values, the procedure presented in Remark 2 leads to symmetrical Sonin kernels.

Now, we demonstrate the application of Theorem 1, Remarks 2 and 3 for the derivation of some symmetrical Sonin kernels, both already known and new ones.

Example 1. 

In Formula (30), we set $\varphi (x)=exp(x)$. Then,

$$\varphi (x)=exp(x)=\sum _{n=0}^{+\infty }{\displaystyle \frac{1}{n!}}{x}^n,{\displaystyle \frac{1}{\varphi (x)}}=exp(-x)=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n}{n!}}{x}^n.$$

Thus,

$$c_n={\displaystyle \frac{1}{n!}},d_n={\displaystyle \frac{{(-1)}^n}{n!}},n\in {\mathbb{N}}_0$$

and (see the relations (32))

$$a_n={\displaystyle \frac{c_n}{\Gamma (\alpha n+\beta )}}={\displaystyle \frac{1}{n!\Gamma (\alpha n+\beta )}},n\in {\mathbb{N}}_0,$$

(33)

$$b_n={\displaystyle \frac{d_n}{\Gamma (\alpha n+1-\beta )}}={\displaystyle \frac{{(-1)}^n}{n!\Gamma (\alpha n+1-\beta )}},n\in {\mathbb{N}}_0.$$

(34)

The Sonin kernels (24) and (25) from Theorem 1 with the coefficients as in (33) and (34), respectively, take the form ($\beta \in (0,1),\alpha >0,\lambda \in \mathbb{R}$)

$$\kappa (x)=x^{\beta -1}\sum _{n=0}^{+\infty }{\displaystyle \frac{{(\lambda x^{\alpha })}^n}{n!\Gamma (\alpha n+\beta )}}=x^{\beta -1}{W}_{\alpha ,\beta }(\lambda x^{\alpha })$$

(35)

and

$$k(x)=x^{-\beta }\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-\lambda x^{\alpha })}^n}{n!\Gamma (\alpha n+1-\beta )}}=x^{-\beta }{W}_{\alpha ,1-\beta }(-\lambda x^{\alpha }),$$

(36)

where the Wright function $W_{\alpha ,\beta }(z)$ is defined by the convergent series (see [29] for its properties and applications):

$$W_{\alpha ,\beta }(z):=\sum _{n=0}^{+\infty }{\displaystyle \frac{z^n}{n!\Gamma (\alpha n+\beta )}},z,\beta \in \mathbb{C},\alpha >-1.$$

(37)

To the best knowledge of the author, the pair of (35) and (36) of symmetrical (in the sense of Definition 1) Sonin kernels has not yet been reported in the literature. However, in his paper [7], Sonin mentioned a particular case of the kernels (35) and (36) with $\alpha =1$. In this case, Sonin kernels (35) and (36) can be expressed in terms of the cylinder or Bessel functions; see [7,9] for details.

Example 2. 

In this example, we derive another pair of symmetrical Sonin kernels that is generated via the function $\varphi (x)={(1+x)}^{-\gamma }$. The Taylor series for $\varphi (x)$ and its reciprocal are well known:

$$\varphi (x)={(1+x)}^{-\gamma }=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(\gamma )}_n}{n!}}{x}^n,{\displaystyle \frac{1}{\varphi (x)}}={(1+x)}^{\gamma }=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(-\gamma )}_n}{n!}}{x}^n,$$

where the Pochhammer symbol ${(\gamma )}_n$ is defined as follows:

$${(\gamma )}_n:={\displaystyle \frac{\Gamma (\gamma +n)}{\Gamma (\gamma )}}=\gamma ·(\gamma +1)·\dots ·(\gamma +n-1).$$

As we see, the functions $\varphi (x)={(1+x)}^{-\gamma }$ and ${\displaystyle \frac{1}{\varphi (x)}}={(1+x)}^{\gamma }$ generate the coefficients

$$c_n={\displaystyle \frac{{(-1)}^n{(\gamma )}_n}{n!}},d_n={\displaystyle \frac{{(-1)}^n{(-\gamma )}_n}{n!}},n\in {\mathbb{N}}_0,$$

and

$$a_n={\displaystyle \frac{c_n}{\Gamma (\alpha n+\beta )}}={\displaystyle \frac{{(-1)}^n{(\gamma )}_n}{n!\Gamma (\alpha n+\beta )}},b_n={\displaystyle \frac{d_n}{\Gamma (\alpha n+1-\beta )}}={\displaystyle \frac{{(-1)}^n{(-\gamma )}_n}{n!\Gamma (\alpha n+1-\beta )}},n\in {\mathbb{N}}_0.$$

The Sonin kernels (24) and (25) take then the form ($\beta \in (0,1),\alpha >0,\lambda \in \mathbb{R}$)

$$\kappa (x)=x^{\beta -1}\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(\gamma )}_n}{n!\Gamma (\alpha n+\beta )}}{(\lambda x^{\alpha })}^n=x^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }(-\lambda x^{\alpha })$$

(38)

and

$$k(x)=x^{-\beta }\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(-\gamma )}_n}{n!\Gamma (\alpha n+1-\beta )}}{(\lambda x^{\alpha })}^n=x^{-\beta }{E}_{\alpha ,1-\beta }^{-\gamma }(-\lambda x^{\alpha }),$$

(39)

where the three-parameter Mittag–Leffler function or the Prabhakar function $E_{\alpha ,\beta }^{\gamma }(z)$ is defined by the convergent series (see, e.g., [30,31] for its properties and applications)

$$E_{\alpha ,\beta }^{\gamma }(z):=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(\gamma )}_n}{n!\Gamma (\alpha n+\beta )}},z,\beta ,\gamma \in \mathbb{C},\alpha >0.$$

(40)

Please note that the function (40) can also be interpreted as a particular case of the so-called generalized Wright or Fox–Wright function; see [29] for details.

In [7], Sonin derived a particular case of the kernels (38) and (39) with $\alpha =1$ in the form of the power law series. As mentioned in [9], the kernels (38) and (39) with $\alpha =1$ can be represented in terms of the confluent hypergeometric function or the Kummer function ${}_{1}F^1$:

$$\kappa (x)=x^{\beta -1}\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(\gamma )}_n}{n!\Gamma (n+\beta )}}{(\lambda x)}^n={\displaystyle \frac{x^{\beta -1}}{\Gamma (\beta )}}{}_{1}F^1(\gamma ;\beta ;-\lambda x),$$

(41)

$$k(x)=x^{-\beta }\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(-\gamma )}_n}{n!\Gamma (n+1-\beta )}}{(\lambda x)}^n={\displaystyle \frac{x^{-\beta }}{\Gamma (1-\beta )}}{}_{1}F^1(-\gamma ;1-\beta ;-\lambda x),$$

(42)

where the Kummer function ${}_{1}F^1$ is defined by the convergent series

$${}_{1}F^1(\gamma ;\beta ;z):=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(\gamma )}_n}{{(\beta )}_n}}{\displaystyle \frac{z^n}{n!}},z,\beta ,\gamma \in \mathbb{C}.$$

(43)

The integral operator in the form of (4) (which we called GFI with kernel $\kappa$) and the integro-differential operator (5) (which we called GFD with kernel k) with the Sonin kernels (41) and (42) were considered by Prabhakar in his paper [32] (see also [10], formulas (37.1) and (37.31)). In particular, in [32], the GFD (5) with the kernel (42) was shown to be a left-inverse operator to the GFI (5) with the kernel (41).

Two years after the publication of his paper [32], in [31], Prabhakar also studied the general case of the GFI (4) and the GFD (5) with the Sonin kernels (38) and (39), respectively, in terms of the three-parameter Mittag–Leffler function. For this reason, nowadays, this function is often referred to as the Prabhakar function. For a detailed presentation of the theory and applications of these operators, nowadays called the Prabhakar fractional integral and derivative, we refer the interested reader to a recent survey [30].

As mentioned in the introduction, the investigation of GFIs and GFDs with Sonin kernels is a very recent topic initialized by Kochubei in the paper [2] published in 2011. In the earlier publications by Prabhakar and other authors who investigated the operators of type (4) and (5) with kernels in terms of the Bessel function, Kummer function, three-parameter Mittag–Leffler function, etc., these operators were introduced and studied independently of the theory of GFIs and GFDs with Sonin kernels. However, recently, one started to interpret these earlier results from the viewpoint of the GFIs and the GFDs; see, e.g., [33] for a discussion of a connection between general fractional calculus (GFC) and the Prabhakar fractional integral and derivative.

Example 3. 

In this example, we consider a pair of Sonin kernels generated via the function $\varphi (x)=exp(x){(1+x)}^{-\gamma }$. The Taylor series for the function $\varphi (x)$ and its reciprocal can be obtained through the multiplication of the Taylor series from Examples 1 and 2:

$$\varphi (x)=exp(x){(1+x)}^{-\gamma }=\left(\sum _{n=0}^{+\infty }{\displaystyle \frac{1}{n!}}{x}^n\right)·\left(\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(\gamma )}_n}{n!}}{x}^n\right)=\sum _{n=0}^{+\infty }{c}_{n,\gamma }{x}^n,$$

where

$$c_{n,\gamma }=\sum _{m=0}^n{\displaystyle \frac{{(-1)}^m{(\gamma )}_m}{m!(n-m)!}},n\in {\mathbb{N}}_0,$$

(44)

$${\displaystyle \frac{1}{\varphi (x)}}=exp(-x){(1+x)}^{\gamma }=\left(\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n}{n!}}{x}^n\right)·\left(\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(-\gamma )}_n}{n!}}{x}^n\right)=\sum _{n=0}^{+\infty }{d}_{n,\gamma }{(-x)}^n,$$

where

$$d_{n,\gamma }=\sum _{m=0}^n{\displaystyle \frac{{(-\gamma )}_m}{m!(n-m)!}},n\in {\mathbb{N}}_0.$$

(45)

Taking into account the relations (28) and applying Theorem 1, we arrive at the following new pair of Sonin kernels ($\beta \in (0,1),\alpha >0,\lambda \in \mathbb{R}$):

$$\kappa (x)=x^{\beta -1}\sum _{n=0}^{+\infty }{\displaystyle \frac{c_{n,\gamma }}{\Gamma (\alpha n+\beta )}}{(\lambda x^{\alpha })}^n,k(x)=x^{-\beta }\sum _{n=0}^{+\infty }{\displaystyle \frac{d_{n,\gamma }}{\Gamma (\alpha n+1-\beta )}}{(-\lambda x^{\alpha })}^n,$$

(46)

where the coefficients $c_{n,\gamma }$ and $d_{n,\gamma }$ are given by Formulas (44) and (45), respectively.

The series in (46) are convergent for all $x\ge 0$ because of the known asymptotic formulas for the Gamma function:

$$\Gamma (\alpha x+\beta )\sim \sqrt{2\pi }exp(-\alpha x){(\alpha x)}^{\alpha x+\beta -1/2}asx\to +\infty ,\alpha >0,\beta \in \mathbb{C},$$

(47)

$${\displaystyle \frac{\Gamma (x+a)}{\Gamma (x+b)}}\sim x^{a-b}asx\to +\infty ,a,b\in \mathbb{C}.$$

(48)

Indeed, we can estimate the coefficients $c_{n,\gamma }$ as follows:

$$|c_{n,\gamma }|\le \sum _{m=0}^n{\displaystyle \frac{{|(\gamma )}_m|}{m!(n-m)!}}\le {\displaystyle \frac{{(|\gamma |)}_n}{n!}}\sum _{m=0}^n{\displaystyle \frac{n!}{m!(n-m)!}}={\displaystyle \frac{{(|\gamma |)}_n}{n!}}·2^n.$$

(49)

Taking into account the asymptotic Formulas (47) and (48), we get the estimate

$${\left(\left|{\displaystyle \frac{c_{n,\gamma }}{\Gamma (\alpha n+\beta )}}\right|\right)}^{1/n}\le C_1{\left({\displaystyle \frac{2^n{(|\gamma |)}_n}{n!}}exp(\alpha n){(\alpha n)}^{-\alpha n-\beta +1/2}\right)}^{1/n}\le C_2{n}^{-\alpha }\to 0asn\to +\infty$$

and, thus, the first series in (46) is convergent for any $x\ge 0$. For the second series, similar estimates are evidently valid, and thus this series is also convergent for $x\ge 0$.

In the rest of this section, we discuss an important extension of Sonin kernels (19) and (20) in the form of products of the power-law functions and analytic functions to the so-called convolution series. The convolution series are a far-reaching generalization of the power-law series that was recently introduced in the framework of the general FC (see, e.g., [5,34]). First, we remind readers of their definition and properties that we need for further discussions.

Let a function, f, belong to the space $C_{-1}(0,+\infty )=\{\varphi :\varphi (x)=x^p{\varphi }_1(x),x>0,$ $p>-1,{\varphi }_1\in C[0,+\infty )\}$ and be represented in the form

$$f(x)=x^p{f}_1(x),x>0,p>0,f_1\in C[0,+\infty )$$

and let the convergence radius of the power series

$$\mathsf{\Sigma }(x)=\sum _{n=0}^{+\infty }{a}_n{x}^n,a_n,x\in \mathbb{R}$$

be non-zero.

The convolution series generated via the function f has the form

$${\mathsf{\Sigma }}_f(x)=\sum _{n=0}^{+\infty }{a}_n{f}^{<n+1>}(x),$$

(50)

where $f^{<n>},n\in \mathbb{N}$ stands for the convolution powers

$$f^{<n>}(x):=\left\{\begin{array}{cc}f(x),\hfill & n=1,\hfill \\ (\underset{n\mathrm{times}}{\underbrace{f\ast \dots \ast f}})(x),\hfill & n=2,3,\dots .\hfill \end{array}\right.$$

(51)

As was shown in [5,34], the convolution series (50) is convergent for all $x>0$, and it defines a function from the space $C_{-1}(0,+\infty )$. Moreover, the series

$$x^{1-\alpha }{\mathsf{\Sigma }}_f(x)=\sum _{n=0}^{+\infty }{a}_n{x}^{1-\alpha }{f}^{<n+1>}(x),\alpha =min\{p,1\}$$

(52)

is uniformly convergent for $x\in [0,X]$ for any $X>0$.

In particular, any power series can be represented in the form of the convolution series (50) generated via the function $f(x)=h_1(x)={\displaystyle \frac{x^0}{\Gamma (1)}}\equiv 1,x\ge 0$:

$${\mathsf{\Sigma }}_{h_1}(x)=\sum _{n=0}^{+\infty }{a}_n{h}_1^{<n+1>}(x)=\sum _{n=0}^{+\infty }{a}_n{h}_{n+1}(x)=\sum _{n=0}^{+\infty }{a}_n{\displaystyle \frac{x^n}{n!}}.$$

For examples and applications of the convolution series in the form (50), we refer to [5,34]. In the following theorem, we present a construction of the Sonin kernels in terms of the convolution series.

Theorem 2. 

Let $({\kappa }_1,k_1)$ be a pair of Sonin kernels, let f be any function from the space $C_{-1}(0,+\infty )$ that is not identically equal to zero, and let the convergence radii of the power series

$${\mathsf{\Sigma }}_a(x)=\sum _{n=0}^{+\infty }{a}_n{x}^n,a_n,x\in \mathbb{R}and{\mathsf{\Sigma }}_b(x)=\sum _{n=0}^{+\infty }{b}_n{x}^n,b_n,x\in \mathbb{R}$$

be non-zero.

Then, the functions

$$\kappa (x)=a_0{\kappa }_1(x)+({\kappa }_1\ast \sum _{n=1}^{+\infty }{a}_n{f}^{<n>})(x),a_0\ne 0,$$

(53)

$$k(x)=b_0{k}_1(x)+(k_1\ast \sum _{n=1}^{+\infty }{b}_n{f}^{<n>})(x)$$

(54)

build a pair of Sonin kernels iff the conditions

$$a_0{b}_0=1,\sum _{m=0}^n{a}_m{b}_{n-m}=0,n\in \mathbb{N}$$

(55)

hold true.

Proof. 

According to the Sonin condition (6) and the definition of the convolution powers, we immediately get the relations

$$(a_0{\kappa }_1\ast b_0{k}_1)(x)=a_0{b}_0,(a_0{\kappa }_1\ast (b_m{k}_1\ast f^{<m>}))(x)=a_0{b}_m(1\ast f^{<m>})(x),$$

$$(({\kappa }_1\ast a_i{f}^{<i>})\ast b_0{k}_1)(x)=a_i{b}_0(1\ast f^{<i>})(x),$$

$$(({\kappa }_1\ast a_i{f}^{<i>})\ast (k_1\ast b_m{f}^{<m>}))(x)=a_i{b}_m(1\ast f^{<i+m>})(x).$$

To calculate the convolution $(\kappa \ast k)(x)$ of the functions $\kappa$ and k defined by (53) and (54), respectively, we interchange the orders of integration and summation (which is allowed because of the uniform convergence of the convolution series (52)) and use the relations from above. Thus, we arrive at the representation

$$(\kappa \ast k)(x)=a_0{b}_0+\sum _{n=1}^{+\infty }\left(\sum _{m=0}^n{a}_m{b}_{n-m}\right)(1\ast f^{<n>})(x).$$

This formula ensures that the pair of functions $\kappa$ and k given by the relations (53) and (54) are Sonin kernels iff the conditions (55) are satisfied. □

Remark 4. 

The Sonin kernels (19) and (20) in the form of the products of the power law functions $x^{\beta -1}$ and $x^{-\beta }$ with $0<\beta <1$ and the analytic functions are a particular case of the general convolution kernels (53) and (54) with ${\kappa }_1(x)=h_{\beta }(x)$, $k_1(x)=h_{1-\beta }(x)$ and $f(x)=h_1(x)\equiv 1$. Other forms of Sonin kernels can be constructed by employing any other Sonin kernels ${\kappa }_1,k_1$ and any function $f\in C_{-1}(0,+\infty )$ that generates the convolution series in the Formulas (53) and (54). In particular, the Sonin kernels (24) and (25) presented in Theorem 1 correspond to the convolution kernels (53) and (54) with ${\kappa }_1(x)=h_{\beta }(x)$, $k_1(x)=h_{1-\beta }(x)$ ($0<\beta <1$) and $f(x)=\lambda h_{\alpha }(x),\alpha >0,\lambda \in \mathbb{R}$.

4. Sonin Kernels in the Laplace Domain

As already mentioned in the introduction, the Laplace integral transforms of any pair of Sonin kernels (if they exist) are connected to each other via the relation (7) that can be easily solved for, say, $\tilde{k}$:

$$\tilde{k}(p)={\displaystyle \frac{1}{p\tilde{\kappa }(p)}},\Re (p)>C.$$

(56)

Using the Formula (56), several important pairs of the Sonin kernels have already been derived (see, e.g., [2,9,11,33]). In this section, we apply this formula to obtain some new symmetrical Sonin kernels in terms of the hypergeometric type functions.

Let us suppose that the Laplace integral transforms of the Sonin kernels $\kappa$ and k exist. Then, the relation (56) implies that these kernels are symmetrical iff their Laplace transforms $\tilde{\kappa }(p)$ and $\tilde{k}(p)={\displaystyle \frac{1}{p\tilde{\kappa }(p)}}$ are represented in terms of the same elementary or special function with possibly different parameter values. In this section, we consider some examples of the known and new symmetrical Sonin kernels that possess this property.

Example 4. 

We start with the well-known case of the Laplace transform $\tilde{\kappa }(p)$ in the form of a power function,

$$\tilde{\kappa }(p)=p^{-\alpha },\alpha >0,\Re (p)>0.$$

Then, the Laplace transform of the associated kernel, k, is given by the relation

$$\tilde{k}(p)={\displaystyle \frac{1}{p\tilde{\kappa }(p)}}=p^{\alpha -1},\alpha <1,\Re (p)>0.$$

The basic formula

$${\tilde{h}}_{\gamma }(p)=p^{-\gamma },\gamma >0,\Re (p)>0$$

(57)

for the Laplace transform of the power function $h_{\gamma }(t)=t^{\gamma -1}/\Gamma (\gamma )$ leads to the well-known pair of symmetrical Sonin kernels in the time domain:

$$\kappa (t)=h_{\alpha }(t),k(t)=h_{1-\alpha }(t),0<\alpha <1.$$

Example 5. 

For the kernel κ with the Laplace transform

$$\tilde{\kappa }(p)=p^{-\beta }exp(\lambda p^{-\alpha }),\alpha >0,\beta >0,\lambda \in \mathbb{R},\Re (p)>0,$$

(58)

the Laplace transform of its associated kernel k has a similar form,

$$\tilde{k}(p)={\displaystyle \frac{1}{p\kappa (p)}}=p^{\beta -1}exp(-\lambda p^{-\alpha }),\alpha >0,\beta <1,\lambda \in \mathbb{R},\Re (p)>0.$$

(59)

We note here that the functions (58) and (59) with $\lambda =0$ have already been considered in Example 5.

To represent the function κ in the time domain, we employ the series representation of its Laplace transform

$$\tilde{\kappa }(p)=p^{-\beta }exp(\lambda p^{-\alpha })=\sum _{n=0}^{+\infty }{\displaystyle \frac{{\lambda }^n}{n!}}{p}^{-\alpha n-\beta }$$

(60)

and the Formula (57), and thus, we arrive at the expression

$$\kappa (x)=\sum _{n=0}^{+\infty }{\displaystyle \frac{{\lambda }^n}{n!}}{h}_{\alpha n+\beta }(x)=\sum _{n=0}^{+\infty }{\displaystyle \frac{{\lambda }^n}{n!}}{\displaystyle \frac{x^{\alpha n+\beta -1}}{\Gamma (\alpha n+\beta )}}=x^{\beta -1}{W}_{\alpha ,\beta }(\lambda x^{\alpha }),\alpha >0,\beta >0,\lambda \in \mathbb{R},$$

(61)

where $W_{\alpha ,\beta }$ is the Wright function (37). Using the same method for the Laplace transform $\tilde{k}(p)$ given by (59), we get a representation of the associated kernel k in the time domain:

$$k(x)=x^{-\beta }{W}_{\alpha ,1-\beta }(-\lambda x^{\alpha }),\alpha >0,\beta <1,$$

(62)

where $W_{\alpha ,\beta }$ is the Wright function (37).

To the best knowledge of the author, the pair of Sonin kernels in forms (61) and (62) was not yet reported in the literature. However, we already derived the same Sonin kernels in Section 3; see Example 1.

Example 6. 

In this example, we set

$$\tilde{\kappa }(p)={\displaystyle \frac{p^{\alpha \gamma -\beta }}{{(p^{\alpha }+\lambda )}^{\gamma }}},\alpha >0,\beta >0,\lambda \in \mathbb{R},\left|\lambda p^{-\alpha }\right|<1.$$

(63)

The Laplace transform of the associated kernel k has the same form with different parameter values:

$$\tilde{k}(p)={\displaystyle \frac{1}{p\kappa (p)}}={\displaystyle \frac{{(p^{\alpha }+\lambda )}^{\gamma }}{p^{\alpha \gamma -\beta +1}}}={\displaystyle \frac{p^{-\alpha \gamma +\beta -1}}{{(p^{\alpha }+\lambda )}^{-\gamma }}},\beta <1,\alpha >0,\lambda \in \mathbb{R},\left|\lambda p^{-\alpha }\right|<1.$$

(64)

For $\lambda =0$, (63) and (64) are reduced to the functions that have been discussed in Example 4.

Using the power series representation

$$\tilde{\kappa }(p)={\displaystyle \frac{p^{\alpha \gamma -\beta }}{{(p^{\alpha }+\lambda )}^{\gamma }}}=p^{-\beta }{(1+\lambda p^{-\alpha })}^{-\gamma }=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(\gamma )}_n{\lambda }^n}{n!}}{p}^{-\alpha n-\beta }$$

(65)

and the Formula (57), we get the following result in the time domain:

$$\kappa (x)=\sum _{n=0}^{+\infty }{\displaystyle \frac{{(-1)}^n{(\gamma )}_n{\lambda }^n}{n!}}{h}_{\alpha n+\beta }(x)=x^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }(-\lambda x^{\alpha }),\alpha >0,\beta >0,\lambda \in \mathbb{R},$$

(66)

where $E_{\alpha ,\beta }^{\gamma }$ is the three-parameter Mittag–Leffler function defined as in (40).

The same method applied to the Laplace transform (64) easily leads to the formula

$$k(x)=x^{-\beta }{E}_{\alpha ,1-\beta }^{-\gamma }(-\lambda x^{\alpha }),\alpha >0,\beta <1,\lambda \in \mathbb{R},$$

(67)

where $E_{\alpha ,\beta }^{\gamma }$ is the three-parameter Mittag–Leffler function defined as in (40).

The pair of the symmetrical Sonin kernels (66) and (67) has already been discussed in Section 3; see Example 2.

Example 7. 

In this example, we consider the Sonin kernel κ with the Laplace integral transform $\tilde{\kappa }$ in the form

$$\tilde{\kappa }(p)=p^{-\beta }{\displaystyle \frac{p^{{\alpha }_1\gamma }}{{(p^{{\alpha }_1}+{\lambda }_1)}^{\gamma }}}exp({\lambda }_2{p}^{-{\alpha }_2})=p^{{\alpha }_1\gamma -\beta }{(p^{{\alpha }_1}+{\lambda }_1)}^{-\gamma }exp({\lambda }_2{p}^{-{\alpha }_2}),$$

(68)

where the parameters and the Laplace variable satisfy the conditions ${\alpha }_1>0,{\alpha }_2>0,\beta >0,$${\lambda }_1,{\lambda }_2\in \mathbb{R},\Re (p)>0,\left|{\lambda }_1{p}^{-{\alpha }_1}\right|<1$.

Please note that we already considered two important particular cases of the kernels in form (68), namely the case of ${\lambda }_1=0$ in Example 5 and the case of ${\lambda }_2=0$ in Example 6.

The Laplace transform of the associated kernel k has, then, a similar form:

$$\tilde{k}(p)={\displaystyle \frac{1}{p\kappa (p)}}=p^{-{\alpha }_1\gamma +\beta -1}{(p^{{\alpha }_1}+{\lambda }_1)}^{\gamma }exp(-{\lambda }_2{p}^{-{\alpha }_2}),$$

(69)

where the parameters and the Laplace variable satisfy the conditions ${\alpha }_1>0,{\alpha }_2>0,\beta <1,$${\lambda }_1,{\lambda }_2\in \mathbb{R},\Re (p)>0,\left|{\lambda }_1{p}^{-{\alpha }_1}\right|<1$.

The series representations of the Functions (68) and (69) immediately follow from the Formulas (60) and (65):

$$\tilde{\kappa }(p)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{(\gamma )}_m{\lambda }_2^n}{m!n!}}{p}^{-{\alpha }_{1}m-{\alpha }_{2}n-\beta },$$

(70)

$$\tilde{k}(p)={\displaystyle \frac{1}{p\kappa (p)}}=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{(-\gamma )}_m{(-{\lambda }_2)}^n}{m!n!}}{p}^{-{\alpha }_{1}m-{\alpha }_{2}n+\beta -1}.$$

(71)

The last two formulas are valid under the conditions ${\alpha }_1>0,{\alpha }_2>0,0<\beta <1,$${\lambda }_1,{\lambda }_2\in \mathbb{R},\left|{\lambda }_1{p}^{-{\alpha }_1}\right|<1,\Re (p)>0$.

Now, we apply the Formula (57) and get the following representations of Sonin kernels κ and k in the time domain:

$$\kappa (x)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{(\gamma )}_m{\lambda }_2^n}{m!n!}}{h}_{{\alpha }_{1}m+{\alpha }_{2}n+\beta }(x)=x^{\beta -1}{\varphi }_3(\gamma ;({\alpha }_1,{\alpha }_2;\beta );-{\lambda }_1{x}^{{\alpha }_1},{\lambda }_2{x}^{{\alpha }_2}),$$

(72)

$$k(x)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{(-\gamma )}_m{(-{\lambda }_2)}^n}{m!n!}}{h}_{{\alpha }_{1}m+{\alpha }_{2}n-\beta +1}(x)=$$

$$x^{-\beta }{\varphi }_3(-\gamma ;({\alpha }_1,{\alpha }_2;1-\beta );-{\lambda }_1{x}^{{\alpha }_1},-{\lambda }_2{x}^{{\alpha }_2}),$$

(73)

where ${\alpha }_1>0,{\alpha }_2>0,0<\beta <1,{\lambda }_1,{\lambda }_2\in \mathbb{R}$ and ${\varphi }_3$ is a new special function of the hypergeometric type in two variables defined by the convergent series

$${\varphi }_3(\gamma ;({\alpha }_1,{\alpha }_2;\beta );y,z):=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(\gamma )}_m}{\Gamma ({\alpha }_{1}m+{\alpha }_{2}n+\beta )}}{\displaystyle \frac{y^m{z}^n}{m!n!}},{\alpha }_1>0,{\alpha }_2>0,\gamma ,\beta ,y,z\in \mathbb{C}.$$

(74)

The asymptotical Formulas (47) and (48) for the gamma function ensure convergence of the series in (74) for all $y,z\in \mathbb{C}$. The proof of this fact closely follows the lines of the derivation of the convergence conditions for the Horn function ${\mathsf{\Phi }}_3$ ([35]), and we omit it here.

The denotation ${\varphi }_3$ is motivated by a particular case of this function for ${\alpha }_1={\alpha }_2=1$ that is reduced to the known Horn function ${\mathsf{\Phi }}_3$:

$${\varphi }_3(\gamma ;(1,1;\beta );y,z)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(\gamma )}_m}{\Gamma ((m+n)+\beta )}}{\displaystyle \frac{y^m{z}^n}{m!n!}}={\displaystyle \frac{1}{\Gamma (\beta )}}{\mathsf{\Phi }}_3(\gamma ;\beta ;y,z),$$

(75)

where ${\mathsf{\Phi }}_3$ is one of the Horn functions defined by the double-confluent series of the hypergeometric type (see Formula (22) in Section 5.7.1 in [35]):

$${\mathsf{\Phi }}_3(\gamma ;\beta ;y,z):=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(\gamma )}_m}{{(\beta )}_{m+n}}}{\displaystyle \frac{y^m{z}^n}{m!n!}}.$$

(76)

Thus, for ${\alpha }_1={\alpha }_2=1$, the pair (72) and (73) of the symmetrical Sonin kernels is expressed in terms of the Horn function ${\mathsf{\Phi }}_3$:

$$\kappa (x)=h_{\beta }(x){\mathsf{\Phi }}_3(\gamma ;\beta ;-{\lambda }_{1}x,{\lambda }_{2}x),\beta >0,$$

(77)

$$k(x)=h_{1-\beta }(x){\mathsf{\Phi }}_3(-\gamma ;1-\beta ;-{\lambda }_{1}x,-{\lambda }_{2}x),\beta <1.$$

(78)

The representations (77) and (78) can also be obtained by employing the Formula (2.2.3.16) from [36] that is valid under the conditions $\Re (\mu +\nu )<0,\Re (p)>max\{0,-\Re (b)\}$:

$$\{{\mathcal{L}}^{-1}{p}^{\mu }{(p+b)}^{\nu }exp(a/p)\}(x)=h_{-\mu -\nu }(x){\mathsf{\Phi }}_3(-\nu ;-\mu -\nu ;-bx,ax),$$

(79)

where ${\mathsf{\Phi }}_3$ is the Horn function defined by (76), and $\{{\mathcal{L}}^{-1}f(p)\}(x)$ stands for the inverse Laplace transform of the function f at point $x>0$.

To the best knowledge of the author, both functions (72) and (73) and functions (77) and (78) are new symmetrical Sonin kernels not yet mentioned in the literature.

Example 8. 

In this last example, the Laplace integral transform $\tilde{\kappa }$ of the Sonin kernel κ is a product of three different power-law functions:

$$\tilde{\kappa }(p)=p^{-\beta }{\displaystyle \frac{p^{{\alpha }_1{\gamma }_1}}{{(p^{{\alpha }_1}+{\lambda }_1)}^{{\gamma }_1}}}{\displaystyle \frac{p^{{\alpha }_2{\gamma }_2}}{{(p^{{\alpha }_2}+{\lambda }_2)}^{{\gamma }_2}}}=p^{{\alpha }_1{\gamma }_1+{\alpha }_2{\gamma }_2-\beta }{(p^{{\alpha }_1}+{\lambda }_1)}^{-{\gamma }_1}{(p^{{\alpha }_2}+{\lambda }_2)}^{-{\gamma }_2},$$

(80)

where the parameters and the Laplace variable satisfy the conditions ${\alpha }_1>0,{\alpha }_2>0,\beta >0,$${\lambda }_1,{\lambda }_2\in \mathbb{R},\Re (p)>0,|{\lambda }_1{p}^{-{\alpha }_1}|<1,|{\lambda }_2{p}^{-{\alpha }_2}|<1$.

In the case of ${\lambda }_1=0$ or ${\lambda }_2=0$, the kernel (80) is reduced to the kernel already considered in Example 6.

The Laplace transform of the associated kernel k takes a similar form:

$$\tilde{k}(p)={\displaystyle \frac{1}{p\kappa (p)}}=p^{-{\alpha }_1{\gamma }_1-{\alpha }_2{\gamma }_2+\beta -1}{(p^{{\alpha }_1}+{\lambda }_1)}^{{\gamma }_1}{(p^{{\alpha }_2}+{\lambda }_2)}^{{\gamma }_2},$$

(81)

where the parameters and the Laplace variable satisfy the conditions ${\alpha }_1>0,{\alpha }_2>0,\beta <1,$${\lambda }_1,{\lambda }_2\in \mathbb{R},\Re (p)>0,|{\lambda }_1{p}^{-{\alpha }_1}|<1,|{\lambda }_2{p}^{-{\alpha }_2}|<1$.

Applying the Formula (65), we get the following series representations of the functions as in the Formulas (80) and (81):

$$\tilde{\kappa }(p)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{({\gamma }_1)}_m{(-{\lambda }_2)}^n{({\gamma }_2)}_n}{m!n!}}{p}^{-{\alpha }_{1}m-{\alpha }_{2}n-\beta },$$

(82)

$$\tilde{k}(p)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{(-{\gamma }_1)}_m{(-{\lambda }_2)}^n{(-{\gamma }_2)}_n}{m!n!}}{p}^{-{\alpha }_{1}m-{\alpha }_{2}n+\beta -1},$$

(83)

that are valid under the conditions ${\alpha }_1>0,{\alpha }_2>0,0<\beta <1,{\lambda }_1,{\lambda }_2\in \mathbb{R},\Re (p)>0,$$|{\lambda }_1{p}^{-{\alpha }_1}|<1,|{\lambda }_2{p}^{-{\alpha }_2}|<1$.

To transform the functions $\tilde{\kappa }$ and $\tilde{k}$ into the time domain, we again apply Formula (57) and arrive at the following representations:

$$\kappa (x)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{({\gamma }_1)}_m{(-{\lambda }_2)}^n{({\gamma }_2)}_n}{m!n!}}{h}_{{\alpha }_{1}m+{\alpha }_{2}n+\beta }(x)=$$

$$x^{\beta -1}{\xi }_2({\gamma }_1;{\gamma }_2;({\alpha }_1,{\alpha }_2;\beta );-{\lambda }_1{x}^{{\alpha }_1},-{\lambda }_2{x}^{{\alpha }_2}),$$

(84)

$$k(x)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{(-{\lambda }_1)}^m{(-{\gamma }_1)}_m{(-{\lambda }_2)}^n{(-{\gamma }_2)}_n}{m!n!}}{h}_{{\alpha }_{1}m+{\alpha }_{2}n+1-\beta }(x)=$$

$$x^{-\beta }{\xi }_2(-{\gamma }_1;-{\gamma }_2;({\alpha }_1,{\alpha }_2;1-\beta );-{\lambda }_1{x}^{{\alpha }_1},-{\lambda }_2{x}^{{\alpha }_2}),$$

(85)

where ${\alpha }_1>0,{\alpha }_2>0,0<\beta <1,{\lambda }_1,{\lambda }_2\in \mathbb{R}$, and ${\xi }_2$ is a new special function of the hypergeometric type in two variables defined by the convergent series

$${\xi }_2({\gamma }_1;{\gamma }_2;({\alpha }_1,{\alpha }_2;\beta );y,z):=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{({\gamma }_1)}_m{({\gamma }_2)}_n}{\Gamma ({\alpha }_{1}m+{\alpha }_{2}n+\beta )}}{\displaystyle \frac{y^m{z}^n}{m!n!}},$$

(86)

$${\alpha }_1>0,{\alpha }_2>0,\left|y\right|<1,{\gamma }_1,{\gamma }_2,\beta ,z\in \mathbb{C}.$$

The convergence conditions for the series in (86) follow from the asymptotical Formulas (47) and (48) for the Gamma function by applying the same arguments as in the derivation of the convergence conditions for the Horn function ${\mathsf{\Xi }}_2$ ([35]).

Similar to the case considered in Example 7, we denoted the function defined by Equation (86) by ${\xi }_2$ because, for ${\alpha }_1={\alpha }_2=1$, it is reduced to the known Horn function ${\mathsf{\Xi }}_2$:

$${\xi }_2({\gamma }_1;{\gamma }_2;(1,1;\beta );y,z)=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{({\gamma }_1)}_m{({\gamma }_2)}_n}{\Gamma (m+n+\beta )}}{\displaystyle \frac{y^m{z}^n}{m!n!}}={\displaystyle \frac{1}{\Gamma (\beta )}}{\mathsf{\Xi }}_2({\gamma }_1;{\gamma }_2;\beta ;y,z),$$

(87)

where the Horn function ${\mathsf{\Xi }}_2$ is defined by the double-confluent series of the hypergeometric type (see Formula (26) in Section 5.7.1 in [35]):

$${\mathsf{\Xi }}_2({\gamma }_1;{\gamma }_2;\beta ;y,z):=\sum _{m,n=0}^{+\infty }{\displaystyle \frac{{({\gamma }_1)}_m{({\gamma }_2)}_n}{{(\beta )}_{m+n}}}{\displaystyle \frac{y^m{z}^n}{m!n!}}.$$

(88)

Thus, for ${\alpha }_1={\alpha }_2=1$ and under the conditions $0<\beta <1,{\lambda }_1,{\lambda }_2\in \mathbb{R}$, the pair (84) and (85) of the symmetrical Sonin kernels is expressed in terms of the Horn function ${\mathsf{\Xi }}_2$:

$$\kappa (x)=h_{\beta }(x){\mathsf{\Xi }}_2({\gamma }_1;{\gamma }_2;\beta ;-{\lambda }_{1}x,-{\lambda }_{2}x),$$

(89)

$$k(x)=h_{1-\beta }(x){\mathsf{\Xi }}_2(-{\gamma }_1;-{\gamma }_2;1-\beta ;-{\lambda }_{1}x,-{\lambda }_{2}x).$$

(90)

To the best knowledge of the author, both the pair of functions (84) and (85) and its particular case in forms (89) and (90) are new symmetrical Sonin kernels not yet reported in the literature.

5. Conclusions and Some Open Problems

In this paper, we have introduced a new class of Sonin kernels referred to as symmetrical Sonin kernels, and we have discussed several methods for the derivation of Sonin kernels in terms of elementary and special functions. Even if these methods can be applied to the construction of arbitrary Sonin kernels, the focus of this paper was the case of symmetrical kernels that are expressed in terms of the same elementary or special functions with possibly different values for their parameters.

The first method discussed in this paper can be traced back to the publication [7] by Sonin, who applied this technique for the construction of several kernels, including the famous pair of kernels in terms of the Bessel and modified Bessel functions. The procedure presented in Remark 2 is a generalization of the method suggested by Sonin. By applying this technique, a new pair of symmetrical Sonin kernels in terms of the Wright function was derived in Example 1. In Example 3, we constructed another new pair of symmetrical Sonin kernels in the form of the hypergeometric-type series.

The second method for the derivation of Sonin kernels presented in this paper was not yet reported in the literature until now. In Theorem 2, we introduced a new class of Sonin kernels in the form of the convolution series that are a far-reaching generalization of the power series. The new Sonin kernels in the form of the convolution series can be constructed for any known pair of the Sonin kernels and any function from the space $C_{-1}(0,+\infty )$. In this way, we have achieved a very general representation of Sonin kernels that includes an arbitrary function from the space $C_{-1}(0,+\infty )$. In some cases, the convolution series can be represented in terms of the power series, and one obtains Sonin kernels in the conventional form; see Remark 4.

The third method discussed in this paper is based on the Laplace integral transform technique. Even if this method is known, with its help, some new and important pairs of symmetrical Sonin kernels were constructed in this paper. In Example 7, a pair of the symmetrical Sonin kernels was derived in terms of a generalization of the Horn function ${\mathsf{\Phi }}_3$, whereas, in Example 8, the symmetrical Sonin kernels were expressed by means of a generalization of the Horn function ${\mathsf{\Xi }}_2$.

Because only very few symmetrical Sonin kernels are known in the literature (symmetrical Sonin kernels in terms of the power function, the Bessel function, the Kummer function, and the Prabhakar function), the construction of four new symmetrical Sonin kernels (Examples 1, 3, 7 and 8) can be considered an essential contribution to this subject.

In the rest of this section, we mention some open problems and directions for further research related to the topic of this paper. First of all, searching for the new Sonin kernels, and in particular for new symmetrical Sonin kernels, is a very important subject for further research. Another topic worth consideration is an investigation of the properties of Sonin kernels. In particular, Sonin kernels in terms of the power function and those in terms of the Prabhakar function are known to be completely monotonic. This property of the kernels is very essential for many applications, in particular, for applications in linear viscoelasticity. The question of whether there exist some other completely monotonic symmetrical Sonin kernels is still open.

Regarding applications of Sonin kernels, let us mention a very recent and actively developing branch of FC in form of GFIs and GFDs with Sonin kernels; see, e.g., [2,3,4,5,6,11,12,13,14] for their theory and [15,16,17,18,19,20,21,22,23,24] for applications in models for fractional dynamics, general non-Markovian quantum dynamics, general non-local electrodynamics, anomalous diffusion, linear viscoelasticity, and other non-local physical theories. Thus, the investigation of the general properties and particular cases of Sonin kernels is an important topic both for the theory of FC and for its applications.

It is worth mentioning that any pair of Sonin kernels, and especially any pair of symmetrical Sonin kernels, generates a kind of new FC. For instance, one can consider the GFI and the corresponding GFD with the kernels $\kappa$ and k in terms of the Horn function ${\mathsf{\Xi }}_2$, as in (89) and (90), respectively:

$$({\mathbb{I}}_{(\kappa )}f)(x)={\int }_0^x{h}_{\beta }(x-\xi ){\mathsf{\Xi }}_2({\gamma }_1;{\gamma }_2;\beta ;-{\lambda }_1(x-\xi ),-{\lambda }_2(x-\xi ))f(\xi )d\xi ,$$

(91)

$$({\mathbb{D}}_{(k)}f)(x)={\displaystyle \frac{d}{dx}}{\int }_0^x{h}_{1-\beta }(x-\xi ){\mathsf{\Xi }}_2(-{\gamma }_1;-{\gamma }_2;1-\beta ;-{\lambda }_1(x-\xi ),-{\lambda }_2(x-\xi ))f(\xi )d\xi .$$

(92)

It is easy to see that all the Sonin kernels that we considered in this paper, both the known and new ones, belong to the space of functions $C_{-1}(0,+\infty )$. The GFIs and the GFDs with the Sonin kernels from this space were introduced and investigated in [4,5,6] and other related publications. In particular, the general theory of the GFIs and the GFDs developed in [4,5,6] ensures that the GFD defined by (92) is a left-inverse operator to the GFI defined by (91) on the suitable spaces of functions (see, e.g., [4] for details). However, developing a theory of these operators on other spaces of functions, the investigation of their mapping properties, the derivation of their norm estimates, etc. are among further important directions for research devoted to Sonin kernels and the FC operators with these kernels.