Revisiting Ramanujan’s Master Theorem

Abstract

Ramanujan’s master theorem is considered to express particular values of the Liouville fractional derivatives. This fact is used to immediately obtain generalizations. Firstly, it is shown that the original formulation is suitable only for anti-causal functions. Consequently, an expression for causal functions is introduced. Secondly, the two formulations are extended to be valid for any real order and related to the Laplace transform, corresponding to transforms with left- or right-sided regions of convergence. This leads to introducing the concept of the unilateral MacLaurin series and the substitution of the Liouville derivative by the (Liouville–)Grünwald–Letnikov derivative, which essentially expresses a discrete formulation that is suitable for numerical implementations. The way in which this procedure arose and its relationship with the Liouville derivatives suggested other approaches, which are briefly discussed, such as replacing the MacLaurin series with the Mittag–Leffler series or the Liouville derivative with the Riesz, Feller, or Hadamard derivatives.

1. Introduction

Among the myriad of results that S. Ramanujan introduced, the so-called Ramanujan’s master theorem (RMT) is probably one of the most curious and interesting. One of its main features is the ability to provide an analytic continuation of the Taylor coefficients of a given function [1]. It has been used in many interesting mathematical applications [1,2,3,4,5,6,7]. One such application was conducted by Ramanujan himself. He obtained the causal fractional derivative proposed by Liouville in his first paper on fractional calculus [2,8]. Another very useful application concerns providing a mathematical base for “the method of brackets” [1] that is used for the evaluation of integrals associated with Feynman diagrams [9].

The importance of RMT lies in the fact that it provides a tool for extrapolating successive integer derivative orders at the origin into real (or even complex) derivative orders. This is a new interpretation of the theorem that stems from its relationship with Liouville’s causal fractional derivative [8,10]. Therefore, we reverse the reasoning: RMT becomes a consequence rather than a cause. This verification allows us to open doors to new formulations and generalizations.

Our approach begins by verifying two facts: (a) the original RMT is expressed in terms of the part of the function defined in ${\mathbb{R}}^{-}$, although it is assumed to have a MacLaurin series expansion; (b) this requirement is equivalent to ensuring that the successive derivatives of integer order at the origin exist and are bounded. The verification of these facts leads us to reformulate RMT so that it can be applied to discontinuous functions at the origin but with finite lateral derivatives. Therefore, we arrive at two different formulae that can be applied to left or right functions.

Previous studies and proofs of RMT refer to its relationship with the Mellin transform. This would be required if we were basing our approach on scale-invariant derivatives like Hadamard’s [11]. In contrast, our approach is based on shift-invariant Liouville derivatives. This makes the Laplace transform (bilateral) the suitable operator.

Therefore, we intend to:

1.

Enlarge the applicability of RMT for functions that are not analytical at the origin;

2.

Allow its application to piecewise continuous functions;

3.

Define RMT for values on the half-positive real axis.

• Bearing in mind these ideas, we established the following novelties:

1.

Reinterpretation of RMT as a particular case of the causal Liouville derivative;

2.

Introduction of the anti-causal version;

3.

Generalizations for negative orders;

4.

Formulations that are suitable for discontinuous functions at the origin by introducing the concept of the “unilateral MacLaurin series”;

5.

Suggestion of a version based on the (Liouville–)Grünwald–Letnikov derivatives.

The paper is outlined as follows. In Section 2, the classic RMT is presented and its reinterpretation conducted. The main features of the paper are discussed in Section 3. Here, the Liouville derivatives are reiterated, and, from them, we obtain the classic RMT and a first reformulation. This supports an extension for the whole real line. All the results are defined for bilateral functions. In Section 4 the case of one-sided functions is addressed. Other generalizations are suggested in Section 5.

2. Ramanujan’s Master Theorem

In his first quarterly report, Ramanujan presented what is known as Ramanujan’s master theorem (RMT) [2].

Theorem 1.  

Let $f\left(t\right)$ be defined and continuous in ${\mathbb{R}}^{+}$ such that $t^{n}f\left(t\right)\to 0$ as $t\to \infty$ for $n\in \mathbb{N}.$ Suppose that, in some neighborhood of $t=0$,

$$f\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{\psi \left(k\right)}{k!}}{(-t)}^k.$$

(1)

Then,

$$I_n\equiv {\int }_0^{\infty }{t}^{n-1}f\left(t\right)dt=\Gamma \left(n\right)\psi (-n).$$

(2)

Conversely, if the value of I is given by (2), then the MacLaurin coefficients of $f\left(t\right)$ can be determined.

The more rigorous Hardy’s version is stated in the following.

Theorem 2.  

Let $\phi \left(z\right)$ be an analytic (single-valued) function, defined on a half-plane

$$H\left(ϵ\right)=\{z\in \mathbb{C}:Re(z)>-ϵ\}$$

for some $0<ϵ<1$. Suppose that, for some $A<\pi ,\phi$ satisfies the growth condition

$$\left|\phi (v+iw)\right|<C{e}^{Pv+A\left|w\right|},$$

for all $z=v+iw\in H\left(ϵ\right)$. Then, for $0<Re\left(s\right)<ϵ$, it follows that

$${\int }_0^{\infty }{t}^{s-1}{\displaystyle \sum _{n=0}^{\infty }}{(-1)}^n\phi \left(n\right)t^{n}dt={\displaystyle \frac{\pi }{sins\pi }}\phi (-s)$$

For the proof, see [1,2,7].

Setting $\phi \left(u\right)=\psi \left(u\right)/\Gamma (u+1)$ and using ${\displaystyle \frac{sins\pi }{\pi }}={\displaystyle \frac{1}{\Gamma (1-s)\Gamma \left(s\right)}},$ we obtain Theorem 1.

Example 1.  

Let $f\left(t\right)=e^{2t},t\in \mathbb{R}$. As

$$e^{2t}={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{2^m}{m!}}{t}^m={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(-2)}^m}{m!}}{(-t)}^m,\psi \left(m\right)={(-2)}^m,m\in {\mathbb{N}}_0,$$

Then

$${\int }_0^{\infty }{\displaystyle \frac{t^{n-1}}{\Gamma \left(n\right)}}{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{(-2)}^m}{m!}}{(-t)}^{m}dt={\int }_0^{\infty }{\displaystyle \frac{t^{n-1}}{\Gamma \left(n\right)}}{e}^{2t}dt=\infty .$$

But, if

$$f\left(t\right)=e^{-2t}={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{2^m}{m!}}{(-t)}^m,\psi \left(m\right)=2^m,$$

and

$${\int }_0^{\infty }{\displaystyle \frac{t^{n-1}}{\Gamma \left(n\right)}}{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{2^m}{m!}}{(-t)}^{m}dt={\int }_0^{\infty }{\displaystyle \frac{t^{n-1}}{\Gamma \left(n\right)}}{e}^{-2t}dt={\displaystyle \frac{1}{2^n}}.$$

This example shows that RMT exhibits unexpected limitations.

We must note that

1.

The function $f\left(t\right)$ must have a MacLaurin expansion. In such a case, $\psi \left(k\right)$ is related to the kth order derivative at $t=0$,

$$\psi \left(k\right)={(-1)}^k{f}^{\left(k\right)}\left(0\right).$$

(3)

2.

The MacLaurin expansion only serves to define the $\psi \left(k\right)$ function and does not play any role in the process since the series is not integrable term by term.

3.

The integral $I_n$ gives the anti-derivative of order n at the origin:

$$\psi (-n)={(-1)}^n{f}^{(-n)}\left(0\right).$$

(4)

4.

The integral $I_n$ is, in fact, the Mellin transform of $f\left(t\right)$.

5.

The function is really assumed to be of exponential order. This condition is more demanding than necessary for the existence of the Mellin transform, but it is required to guarantee the existence of the Laplace transform.

6.

It is not absolutely necessary that n is a positive integer. However, it is valid only for $n\in {\mathbb{R}}^{+}$.

7.

The MacLaurin series ${\sum }_{k=0}^{\infty }{\displaystyle \frac{\psi \left(k\right)}{k!}}{t}^k$ defines a bilateral function, but only its anti-causal part is used in (2). Nothing is said about the convergence of the series.

8.

The function $f\left(t\right)$ must be continuous at the origin.

For reasons that will become clear later, we rewrite Ramanujan’s theorem in the following form.

Theorem 3.  

Suppose that, in some neighborhood of $t=0$,

$$f\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k.$$

(5)

Then,

$$I_n\equiv {\int }_0^{\infty }{\displaystyle \frac{t^{n-1}}{\Gamma \left(n\right)}}f(-t)dt=\varphi (-n).$$

(6)

In this reformulation of RMT, $\varphi \left(k\right)=f^{\left(k\right)}\left(0\right)$, and, if $n\in \mathbb{N}$, the integral on the right-hand side represents the repeated integration of $f(-t)$.

This form of RMT allows an interesting interpretation: (6) states that the function $\varphi (-n)$ is a linear prediction from past values of the function. The predictor is nothing else than the fractional integrator. This leads us to require similar prediction using future values.

Example 2.  

Consider the sinc function $h\left(t\right)={\displaystyle \frac{sin\left(t\right)}{t}},t\in \mathbb{R}$. By using (6), we get

$$h\left(t\right)={\displaystyle \frac{sin\left(t\right)}{t}}={\displaystyle \frac{{\sum }_{n=0}^{\infty }{(-1)}^n\frac{t^{2n+1}}{(2n+1)!}}{t}}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{(-1)}^n}{2n+1}}{\displaystyle \frac{t^{2n}}{\left(2n\right)!}}={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{sin\left(\right(n+1)\pi /2)}{n+1}}{\displaystyle \frac{t^n}{n!}},$$

with $h\left(0\right)=1$. Therefore, $\varphi \left(n\right)={\displaystyle \frac{sin\left(\right(n+1)\pi /2)}{n+1}}$. Let us use (6) as [12]

$${\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}{e}^{-ivt}dt={\displaystyle \frac{1}{{\left(iv\right)}^{\beta }}},$$

and we get

$${\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}{\displaystyle \frac{sin\left(t\right)}{t}}dt={\int }_0^{\infty }{\displaystyle \frac{t^{\beta -2}}{\Gamma \left(\beta \right)}}sin\left(t\right)dt={\displaystyle \frac{1}{2i(\beta -1)}}\left[{\displaystyle \frac{1}{{(-i)}^{\beta -1}}}-{\displaystyle \frac{1}{{\left(i\right)}^{\beta -1}}}\right]={\displaystyle \frac{sin\left((-\beta +1)\pi /2\right)}{(-\beta +1)}}.$$

Remark 1.  

1. 

In most applications of RMT, the MacLaurin series is not used for anything. Furthermore, we do not require it to be convergent. In fact, it can be replaced by the requirement that $|f^{\left(k\right)}\left(0\right)|<\infty ,k=0,1,2,\cdots$.

2. 

In the whole paper, we will consider real derivative orders, $\alpha \in \mathbb{R}$, but almost all the results and mainly the generalizations of RMT remain valid for complex orders. In such cases involving the multivalued expression $t^z,z\in \mathbb{C}$, we assume the half-negative real axis as the branch cut line and work always on the principal Riemann surface.

3. Meeting Liouville Derivatives

3.1. The Liouville Framework

In the following, we shall be working with functions, $h\left(t\right)$, defined on $\mathbb{R}$ that [13]

• Are continuous almost everywhere;
• Have bounded variation;
• Are locally integrable (in the sense that the function is absolutely integrable in any real interval $[a,b]$ so that ${\int }_a^b\left|h\left(t\right)\right|dt<\infty$;
• Are of exponential order.

These conditions are sufficient for the existence of the (bilateral) Laplace transform (LT).

Definition 1.  

The direct Laplace transform is defined by the following [14],

$$\mathcal{L}\left[h\left(t\right)\right]=H\left(s\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}h\left(t\right)e^{-st}\mathrm{d}t,s\in \mathbb{C}$$

(7)

and the inverse reads

$$h\left(t\right)={\mathcal{L}}^{-1}H\left(s\right)={\displaystyle \frac{1}{2\pi i}}{\displaystyle \underset{c-i\infty }{\overset{c+i\infty }{\int }}}H\left(s\right)e^{st}\mathrm{d}s,t\in \mathbb{R},$$

(8)

where $c\in \mathbb{R},$ and right-hand side represents the Bromwich integral.

Under the above conditions, the integral in (7) converges absolutely and uniformly in a vertical strip in the complex plane defined by $b<Re\left(s\right)<a$, where $H\left(s\right)$ is analytic. This strip is called the region of convergence (ROC), and the values of the constants a and b are the abscissas of convergence. Eventually, such ROC may include or degenerate into the imaginary axis so that the function has Fourier transform (FT). Therefore, we must include in our framework absolutely ($L_1$) or square ($L_2$) integrable functions. Some particular domain features are important.

Definition 2.  

We define a right function by $f\left(t\right)=0,t<a\in \mathbb{R}.$ If $a=0$, we call it a causal function. A generalized right function verifies $f(-\infty )=0.$ Similarly, we define left function by $f\left(t\right)=0,t>a\in \mathbb{R}.$ If $a=0$, we call it an anti-causal function. A generalized left function verifies $f(\infty )=0.$

To illustrate: the function $e^{2t}\left(e^{-2t}\right)$ is a generalized right (left) function. The causal and anti-causal functions will be designated generically by unilateral (one-sided) functions.

The Heaviside function is represented by $u\left(t\right)$, and the signum function is defined by

$$sgn\left(t\right)=2u\left(t\right)-1.$$

3.2. Liouville Derivatives

The designation “fractional derivative” is assigned to any operator that generalizes the classical concept of derivative and that enjoys some properties [15]. We represent such operator by $D^{\alpha },\alpha \in \mathbb{R}$. Among the required properties, we impose here the composition rule

$$D^{\alpha }{D}^{\beta }f\left(t\right)=D^{\alpha +\beta }f\left(t\right),\alpha ,\beta \in \mathbb{R}.$$

Often, the use of the designation of derivative is reserved for positive orders, $D^{\alpha },\alpha \in {\mathbb{R}}^{+}$. The composition rule implies that there is an anti-derivative, $D^{-\alpha }$, such that

$$D^{\alpha }{D}^{-\alpha }f\left(t\right)=D^{-\alpha }{D}^{\alpha }f\left(t\right)=f\left(t\right).$$

This operator must be compared with the primitive. While the anti-derivative is a left and right inverse, the primitive is only right.

There are two main types of derivatives: shift-invariant and scale-invariant [16]. We will consider the former.

Definition 3.  

In his first paper on fractional calculus (FC) [8], Liouville introduced the first formula for the fractional anti-derivative that reads [10,17]

$$D_a^{-\beta }f\left(t\right)=e^{i\beta \pi }{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\displaystyle \underset{0}{\overset{\infty }{\int }}}{\tau }^{\beta -1}f(t+\tau )\mathrm{d}\tau ,$$

(9)

where $\beta >0$.

Example 3.  

Let $f\left(t\right)=e^{st},s\in \mathbb{C}$. We obtain

$$D_a^{-\beta }{e}^{st}={\displaystyle \frac{e^{st}{e}^{i\beta \pi }}{\Gamma \left(\beta \right)}}{\int }_0^{\infty }{e}^{s\tau }{\tau }^{\beta -1}\mathrm{d}\tau =s^{-\beta }{e}^{st},$$

(10)

provided that $Re\left(s\right)<0.$

Remark 2.  

• This example shows that (9) is suitable for (left) functions that decrease to zero with increasing argument, in particular those with Laplace transform having an ROC of the type $Re\left(s\right)<a\in \mathbb{R}$.
• The presence of the $e^{i\beta \pi }$ factor deserves comment. At first glance, it may seem useless and even harmful. However, it serves to ensure that there are two derivatives with the same expression of the Laplace transform, obviously with different regions of convergence. On the other hand, it ensures consistency and backward compatibility between cases corresponding to integer and fractional orders.

At the publication time (1832) [8], Liouville was not conscious that (9) is not a causal operator: only present and future values are used. It was Serret who noted this fact and proposed the causal version [18] that led Liouville to also adopt his expression [19].

Definition 4.  

The causal version of the Liouville anti-derivative can be stated as [10]

$$D_c^{-\beta }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^{\infty }f\left(t-\tau \right){\tau }^{\beta -1}\mathrm{d}\tau$$

(11)

where $\beta >0.$ In fact only present and past values are used.

Example 4.  

Proceeding as in the previous example, it is not very difficult to show that

$$D_c^{-\beta }{e}^{st}=s^{-\beta }{e}^{st},$$

(12)

for $Re\left(s\right)>0.$ It can be shown that this relation remains valid even if $Re\left(s\right)=0$ provided that $s\ne 0$ [12].

Similarly to the above, this derivative definition is valid for (right) functions that decrease to zero as the argument tends to $-\infty$, in particular those for which the ROC assumes the form $Re\left(s\right)>b\in \mathbb{R}$.

Remark 3.  

It is important to note that the (anti-)causality of the Liouville operators implies that:

1. 

The causal anti-derivative (9) of a causal function is causal;

2. 

The causal anti-derivative (9) of an anti-causal function is bilateral;

3. 

The anti-causal anti-derivative (11) of a causal function is bilateral;

4. 

The anti-causal anti-derivative (11) of an anti-causal function is anti-causal.

3.3. A First Visit

The previous examples and considerations show that

1.

Any function having Laplace (or Fourier) transform has fractional anti-derivative;

2.

Relations (9) and (11) define two variable functions. For a fixed value $t=t_0\in \mathbb{R}$ we obtain a function of $\beta$.

Ramanujan’s theorem results from (11) by setting $t=0$.

$$D_c^{-\beta }f\left(0\right)={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^{\infty }f\left(-\tau \right){\tau }^{\beta -1}\mathrm{d}\tau$$

(13)

where $\beta >0$. If $f\left(t\right)$ has an associated MacLaurin series

$$f\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{f}^{\left(k\right)}\left(0\right){\displaystyle \frac{t^k}{k!}},$$

(14)

we obtain (1). However, as mentioned above, only the anti-causal part of $f\left(t\right)$ is useful. Therefore, we can obtain another version of RMT by considering the causal part and the anti-causal anti-derivative (9) to obtain

$$D_a^{-\beta }f\left(0\right)={\displaystyle \frac{e^{i\beta \pi }}{\Gamma \left(\beta \right)}}{\int }_0^{\infty }f\left(\tau \right){\tau }^{\beta -1}\mathrm{d}\tau ,$$

(15)

for $\beta >0.$

Theorem 4.  

Let $\beta \in {\mathbb{R}}^{+}$. Suppose that

$$f\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k,t\in \mathbb{R}.$$

(16)

Then,

$${\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}f(-t)dt=\varphi (-\beta ),$$

(17)

or

$$e^{i\beta \pi }{\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}f\left(t\right)dt=\varphi (-\beta ).$$

(18)

Proof. 

For proof it is enough to set $t=0$ in (11) and (9) to obtain $\varphi (-\beta )=D_{c,a}^{-\beta }f\left(0\right).$ □

This theorem states the existence of two versions of RMT, one valid for right functions (17) and the other for left functions (18). The examples stated in (10) and (12) allow us to see the different applicability of the two formulations: we use one or the other in agreement with the region of convergence of the Laplace transform of the function (see Examples 3 and 4).

Remark 4.  

Expression (18) assumes another interesting form,

$${\int }_{-\infty }^0{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}f(-t)dt=\varphi (-\beta ).$$

However, we will continue with the other.

Example 5.  

Let $f\left(t\right)=t{e}^t$ be a right-exponential-type function. As $f\left(t\right)={\sum }_{n=0}^{\infty }(n+1){\displaystyle \frac{t^{n+1}}{(n+1)!}}={\sum }_{n=1}n{\displaystyle \frac{t^n}{n!}},$ we have $\varphi \left(n\right)=n,n\in {\mathbb{N}}_0$ and, successively,

$$-{\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}t{e}^{-t}dt=-{\displaystyle \frac{\Gamma (\beta +1)}{\Gamma \left(\beta \right)}}=-\beta ,$$

and

$$e^{i\beta \pi }{\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}t{e}^{t}dt=\infty .$$

Now, set $f\left(t\right)=t{e}^{-t}=-{\sum }_{n=1}^{\infty }{(-1)}^{n}n{\displaystyle \frac{t^n}{n!}}$ (left) with $\varphi \left(n\right)=-{(-1)}^{n}n=n{e}^{i(n-1)\pi },n\in {\mathbb{N}}_0$. We obtain

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

and

$$e^{i\beta \pi }{\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}t{e}^{-t}dt=e^{i\beta \pi }{\displaystyle \frac{\Gamma (\beta +1)}{\Gamma \left(\beta \right)}}=e^{i\beta \pi }\beta =-e^{i(\beta -1)\pi }\beta .$$

For $\beta =n\in \mathbb{N}$, we get $\varphi \left(n\right)=-e^{i(n-1)\pi }n={(-1)}^{n}n.$

Example 6.  

Consider the complex variable function:

$$F\left(s\right)={\displaystyle \frac{1}{s-a}},a\in \mathbb{R}.$$

If we adjoint a region of convergence, it may be considered as a Laplace transform. Let $Re\left(s\right)>a$. We have

$$F\left(s\right)={\displaystyle \frac{1}{s-a}}={\displaystyle \sum _{k=0}^{\infty }}{a}^k{s}^{-k-1},$$

whose inverse is

$$f\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{a^k}{k!}}{t}^{k}u\left(t\right)=e^{at}u\left(t\right).$$

So, $\varphi \left(k\right)=a^k.$ As the function is causal, the classic RMT should not be applied. However, let us use (18). We obtain

$$e^{i\beta \pi }{\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}{e}^{at}dt=e^{i\beta \pi }{\displaystyle \frac{1}{{(-a)}^{\beta }}}={\displaystyle \frac{1}{a^{\beta }}},$$

provided that $a<0$. So, we get a result that is in agreement with the RMT.

Now, let us assume that $Re\left(s\right)<a$. The inverse of $s^{-k-1},Re\left(s\right)<0$ is easily shown to be $-{\displaystyle \frac{t^k}{k!}}u(-t)$. Then,

$$f\left(t\right)=-{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{a^k}{k!}}{t}^{k}u(-t)=-e^{at}u(-t).$$

Now, $\varphi \left(k\right)=-a^k.$ With (17), we get

$$-{\int }_0^{\infty }{\displaystyle \frac{t^{\beta -1}}{\Gamma \left(\beta \right)}}{e}^{-at}dt=-{\displaystyle \frac{1}{a^{\beta }}},$$

if $a>0$. Again, we got a coherent result.

This example shows how RMT can be applied to a particular expression of an LT, yielding two different forms, depending, as one might expect, on the ROC. However, we should not apply RMT discussed above to these cases because we do not have a function with a MacLaurin series; rather, we have two distinct functions, and, if we add them together, the resulting function is not continuous at the origin. We must therefore seek another generalization of the RMT.

3.4. Second Visit

The above reformulation of RMT assumed negative real orders allowing the definition of Liouville anti-derivatives. The use of (9) and (11) in the derivative case leads to singular integrals. To avoid this problem, we can regularize them [20] or use the trick introduced by Liouville [8]. It consists of transferring the effect of the singularity to an integer-order derivative. He noted that $s^{\alpha }=s^{\alpha -N}{s}^N,N\in {\mathbb{N}}_0$ (the alternative $s^{\alpha }=s^N{s}^{\alpha -N}$ complicates the solution). We go on by using the Liouville derivative definitions. In the same conditions used above, we introduce the following theorems.

Definition 5.  

Let N be the smallest positive integer greater than or equal to α ($N-1<\alpha \le N$). The Liouville causal derivative can be stated as [8,10,21]

$$D_c^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (N-\alpha )}}{\int }_0^{\infty }{f}^{\left(N\right)}\left(t-\tau \right){\tau }^{N-\alpha -1}\mathrm{d}\tau .$$

(19)

Similarly, the anti-causal reads

$$D_a^{\alpha }f\left(t\right)={\displaystyle \frac{e^{i(N-\alpha )\pi }}{\Gamma (N-\alpha )}}{\int }_0^{\infty }{f}^{\left(N\right)}\left(t+\tau \right){\tau }^{N-\alpha -1}\mathrm{d}\tau .$$

(20)

If $\alpha <0$ (anti-derivative), we set $N=0$. These formulae follow directly from (9) and (11) by using $s^{\alpha }=s^{\alpha -N}{s}^N.$

Proceeding as above, we obtain the corresponding Ramanujan’s master theorem.

Theorem 5.  

Let $f\left(t\right)={\sum }_{k=0}^{\infty }{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k,t\in \mathbb{R}$. The general RMT can be formulated by

$${\varphi }_c\left(\alpha \right)={\displaystyle \frac{1}{\Gamma (N-\alpha )}}{\int }_0^{\infty }{f}^{\left(N\right)}\left(-\tau \right){\tau }^{N-\alpha -1}\mathrm{d}\tau ={\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{\varphi (k+N)}{\Gamma (k+1)}}{(-\tau )}^k\mathrm{d}\tau ,$$

(21)

and

$${\varphi }_a\left(\alpha \right)={\displaystyle \frac{e^{i(N-\alpha )\pi }}{\Gamma (N-\alpha )}}{\int }_0^{\infty }{f}^{\left(N\right)}\left(\tau \right){\tau }^{N-\alpha -1}\mathrm{d}\tau =e^{i(N-\alpha )\pi }{\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{\varphi (k+N)}{\Gamma (k+1)}}{\tau }^k\mathrm{d}\tau .$$

(22)

Proof. 

As above, we set $t=0$ in (19) and (20). Then attend to the fact that, from the MacLaurin series, we have

$$f^{\left(N\right)}\left(t\right)={\displaystyle \sum _{k=N}^{\infty }}k(k-1)\cdots (k-N-1){\displaystyle \frac{\varphi \left(k\right)}{k!}}{t}^{k-N}={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{\varphi (k+N)}{k!}}{t}^k,$$

(23)

which leads to (21) and (22) since ${\displaystyle \frac{k(k-1)\cdots (k-N-1)}{k!}}={\displaystyle \frac{k(k-1)\cdots (k-N-1)}{k(k-1)\cdots \left(1\right)}}={\displaystyle \frac{1}{(k-N)!}}$. □

Therefore, we can state some conclusions:

1.

The generalized RMT recovers the above RMT by setting $N=0$ and $\alpha \to -\alpha$;

2.

The formula corresponding to the causal case uses the left part of the function;

3.

The anti-causal uses the right part of the function;

4.

It is not a difficult task to enlarge the above formulation to include complex orders so that they allow us to extend the function $\varphi$ to the whole complex plane;

5.

If $\alpha =N\in {\mathbb{N}}_0$, we recover the $\varphi \left(k\right)$ values since [22]

$${\displaystyle \frac{t^{-n-1}}{\Gamma (-n)}}={\delta }^{\left(n\right)}\left(t\right)n\in {\mathbb{N}}_0,t\in \mathbb{R}.$$

Example 7.  

Consider the exponential case again, where $\varphi \left(k\right)=s^k$. Inserting the expansion in (21), we get

$${\varphi }_c\left(\alpha \right)={\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{s^{k+N}}{\Gamma (k+1)}}{(-\tau )}^k\mathrm{d}\tau =s^N{\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{e}^{-s\tau }\mathrm{d}\tau ,$$

giving

$${\varphi }_c\left(\alpha \right)=s^N{\displaystyle \frac{1}{s^{N-\alpha }}}=s^{\alpha },$$

provided that $Re\left(s\right)>0.$ According to what we told above, it is valid for $s=i\omega \ne 0,\omega \in \mathbb{R}$. In a similar way, we have

$${\varphi }_a\left(\alpha \right)=e^{i(N-\alpha )\pi }{\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{s^{k+N}}{\Gamma (k+1)}}{\tau }^k\mathrm{d}\tau =e^{i(N-\alpha )\pi }{s}^N{\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{e}^{s\tau }\mathrm{d}\tau .$$

Therefore,

$${\varphi }_a\left(\alpha \right)=e^{i(N-\alpha )\pi }{s}^N{\displaystyle \frac{1}{{(-s)}^{N-\alpha }}}=s^{\alpha }.$$

This relation is valid for $Re\left(s\right)<0$.

4. RMT for Unilateral Functions

4.1. One-Sided MacLaurin Series

RMT presented above has one major disadvantage: it can be applied to functions that are analytic near the origin. This fact is expressed in the requirement for the existence of the MacLaurin series. We will try to overcome this shortcoming.

Let $f\left(t\right)$ be any function defined on $\mathbb{R}$ that is expanded into a MacLaurin series

$$f\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{f^{\left(k\right)}\left(0\right)}{k!}}{t}^k.$$

(24)

Definition 6.  

We define its causal part as

$$f_c\left(t\right)=f\left(t\right)u\left(t\right)$$

(25)

and the anti-causal part by

$$f_a\left(t\right)=f\left(t\right)u(-t),$$

(26)

so that

$$f\left(t\right)=f_a\left(t\right)+f_c\left(t\right).$$

(27)

We can then write

$$f_c\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{f^{\left(k\right)}\left(0^{+}\right)}{k!}}{t}^{k}u\left(t\right)$$

(28)

and

$$f_a\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{f^{\left(k\right)}\left(0^{-}\right)}{k!}}{t}^{k}u(-t),$$

(29)

as the causal/anti-causal decomposition of $f\left(t\right).$ The presence of $0^{\pm }$ instead of 0 is due to the fact that both parts have a jump at the origin, motivating the appearance of impulses since the derivatives of the Heaviside functions are Dirac impulses. So, to broaden the applicability of these formulae, we consider the derivatives in classic sense and retain the finite components only.

Definition 7.  

We define the causal and anti-causal MacLaurin series by

$$f_c\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{+}^{\left(k\right)}\left(0^{+}\right)}{\Gamma (k+1)}}{t}^{k}u\left(t\right),$$

(30)

with ${\varphi }_{+}\left(k\right)=f^{\left(k\right)}\left(0^{+}\right)$, and

$$f_a\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{-}^{\left(k\right)}\left(0^{-}\right)}{\Gamma (k+1)}}{t}^{k}u(-t),$$

(31)

with ${\varphi }_{-}\left(k\right)=f^{\left(k\right)}\left(0^{-}\right)$. Therefore, the lateral derivatives are used. We should point out that, in general, they are different.

Example 8.  

Let

$$f\left(t\right)=\left\{\begin{array}{cc}{e}^{at}\hfill & t>0\hfill \\ -e^{-bt}\hfill & t<0,\hfill \end{array}\right.$$

with $a,b\in \mathbb{R}$. As it is easy to verify, it is discontinuous at the origin. However,

$$f_c\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{a^k}{k!}}{t}^{k}u\left(t\right)$$

and

$$f_a\left(t\right)=-e^{-bt}=-{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{(-b)}^k}{k!}}{t}^{k}u(-t)$$

For applying (17) to $f\left(t\right)$ we use $f_a\left(t\right)$ so that ${\varphi }_{+}\left(k\right)={(-a)}^k$ and

$${\varphi }_{-}\left(\alpha \right)={\int }_0^{\infty }{\displaystyle \frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}}{e}^{at}dt={\displaystyle \frac{1}{{(-a)}^{\alpha }}},$$

if $a<0$. The $a>0$ case is divergent. For (18), we use $f_c\left(t\right)$, giving ${\varphi }_{-}\left(k\right)=-b^k$ and

$${\varphi }_{+}\left(\alpha \right)=-e^{i\alpha \pi }{\int }_0^{\infty }{\displaystyle \frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}}{e}^{bt}dt=-e^{i\alpha \pi }{\displaystyle \frac{1}{{(-b)}^{\alpha }}}=-{\displaystyle \frac{1}{b^{\alpha }}},$$

if $b<0$.

4.2. One-Sided RMT

The previous example and Example 6 allow us to observe that Theorem 5 remains valid for unilateral functions with suitable evident modifications. We then set

Theorem 6.  

Let $f\left(t\right)$ be defined by (30) or (31) and $N-1<\alpha \le N\in \mathbb{N}$. The general RMT can be formulated by

$${\varphi }_{+}\left(\alpha \right)={\displaystyle \frac{1}{\Gamma (N-\alpha )}}{\int }_0^{\infty }{f}_{+}^{\left(N\right)}\left(-\tau \right){\tau }^{N-\alpha -1}\mathrm{d}\tau ={\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{+}(k+N)}{\Gamma (k+1)}}{(-\tau )}^k\mathrm{d}\tau ,$$

(32)

and

$${\varphi }_{-}\left(\alpha \right)={\displaystyle \frac{e^{i(N-\alpha )\pi }}{\Gamma (N-\alpha )}}{\int }_0^{\infty }{f}_{-}^{\left(N\right)}\left(\tau \right){\tau }^{N-\alpha -1}\mathrm{d}\tau =e^{i(N-\alpha )\pi }{\int }_0^{\infty }{\displaystyle \frac{{\tau }^{N-\alpha -1}}{\Gamma (N-\alpha )}}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{-}(k+N)}{\Gamma (k+1)}}{\tau }^k\mathrm{d}\tau ,$$

(33)

with ${\varphi }_{+}\left(k\right)=f_{+}^{\left(k\right)}\left(0^{+}\right)$ and ${\varphi }_{-}\left(k\right)=f_{-}^{\left(k\right)}\left(0^{-}\right)$.

Proof. 

To show these results we begin by noting that, similarly to (23), we have

$$f_{\pm }^{\left(N\right)}\left(t\right)={\displaystyle \sum _{k=N}^{\infty }}k(k-1)\cdots (k-N-1){\displaystyle \frac{\varphi {\left(k\right)}_{\pm }}{k!}}{t}^{k-N}={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{\pm }(k+N)}{k!}}{t}^k,$$

(34)

since the domains are ${\mathbb{R}}^{\pm }$. The expressions (32) and (33) are easily obtained by inserting (34) into Liouvillle derivatives (19) and (20), respectively. □

Obviously, we should use (32) for anti-causal functions, while (33) must be used for causal functions. Example 6 shows how the generalized RMT can be applied to functions defined by their LT. With this new theorem, we showed how to extrapolate the discrete functions ${\varphi }_{+}\left(k\right)=f_{+}^{\left(k\right)}\left(0^{+}\right)$ and ${\varphi }_{-}\left(k\right)=f_{-}^{\left(k\right)}\left(0^{-}\right)$ into the whole real line (and the complex plane). These functions define unilateral MacLaurin series.

Example 9.  

Consider the exponential-order function $g\left(t\right)=\left[e^{-t}+e^{-2t}\right]u\left(t\right)$. Now, we have

$${\varphi }_{-}\left(k\right)={(-1)}^k+2^{-k}=e^{ik\pi }+2^{-k}.$$

Then

$${\varphi }_{-}(-\beta )={\displaystyle \frac{e^{\beta \pi }}{\Gamma \left(\beta \right)}}{\int }_0^{\infty }\left[e^{-\tau }+e^{-2\tau }\right]{\tau }^{\beta -1}\mathrm{d}\tau =e^{-\beta \pi }+{\displaystyle \frac{1}{{(-2)}^{\beta }}}.$$

If $1<\beta \le 2$, we need to compute the second-order derivative of $g\left(t\right)$ but retain the finite terms only. Therefore, $g^{\prime \prime }\left(t\right)=\left[e^{-t}+4e^{-2t}\right]u\left(t\right).$ Then,

$${\varphi }_{-}\left(k\right)={(-1)}^k+2^{-k+2}=e^{ik\pi }+4·2^{-k}$$

and

$${\varphi }_{-}(-\beta )={\displaystyle \frac{e^{\beta \pi }}{\Gamma \left(\beta \right)}}{\int }_0^{\infty }\left[e^{-\tau }+4e^{-2\tau }\right]{\tau }^{\beta -1}\mathrm{d}\tau =e^{-\beta \pi }+{\displaystyle \frac{4}{{(-2)}^{\beta }}}.$$

Remark 5.  

The results expressed by Theorem 6 can be used to solve an interesting problem. Consider a causal ARMA-type system [13] with input $x\left(t\right)$ and output $y\left(t\right)$. Suppose that, at a given moment, the system is switched off, halting the dynamics but thereby preserving the state values since the system has memory. If we switch it again at a later instant (t=0) we have to take as initial conditions the values $x_{-}^{\left(k\right)}\left(0^{-}\right)$ and $y_{-}^{\left(k\right)}\left(0^{-}\right)$. However, we observed $x\left(t\right)$ and $y\left(t\right)$, not their derivatives. Therefore, we have to use (33) to obtain the required initial conditions [23].

4.3. The (Liouville–)Grünwald–Letnikov Derivative-Based RMT Generalizations

As

$${(1-e^{-sh})}^{\alpha }={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{(-\alpha )}_n}{n!}}{e}^{-nsh}$$

it is a simple matter to obtain the (Liouville–)Grünwald–Letnikov (GL) [10,24] derivatives using the translation property of the LT.

Definition 8.  

Let $\alpha \in \mathbb{R}$. We define causal GL derivative by [10,17]

$$D_c^{\alpha }f\left(t\right)=\underset{h\to 0^{+}}{lim}{h}^{-\alpha }{\displaystyle \sum _{n=0}^{+\infty }}{\displaystyle \frac{{\left(-\alpha \right)}_n}{n!}}f(t-nh),$$

(35)

while, for the anti-causal GL, set

$$D_a^{\alpha }f\left(t\right)=e^{-i\alpha \pi }\underset{h\to 0^{+}}{lim}{h}^{-\alpha }{\displaystyle \sum _{n=0}^{+\infty }}{\displaystyle \frac{{\left(-\alpha \right)}_n}{n!}}f(t+nh).$$

(36)

There is an intimate connection between these derivatives and Liouville’s [10,20]. With these derivatives, we can introduce a summation-based RMT as follows.

Theorem 7.  

Let $f\left(t\right)$ be defined by (30) or (31) and $\alpha \in \mathbb{R}$. The GL-type RMT can be formulated by

$${\varphi }_{+}\left(\alpha \right)=\underset{h\to 0^{+}}{lim}{h}^{-\alpha }{\displaystyle \sum _{n=0}^{+\infty }}{\displaystyle \frac{{\left(-\alpha \right)}_n}{n!}}f(-nh),$$

(37)

or

$${\varphi }_{-}\left(\alpha \right)=e^{-i\alpha \pi }\underset{h\to 0^{+}}{lim}{h}^{-\alpha }{\displaystyle \sum _{n=0}^{+\infty }}{\displaystyle \frac{{\left(-\alpha \right)}_n}{n!}}f\left(nh\right).$$

(38)

with ${\varphi }_{+}\left(k\right)=f_{+}^{\left(k\right)}\left(0^{+}\right)$ and ${\varphi }_{-}\left(k\right)=f_{-}^{\left(k\right)}\left(0^{-}\right)$.

Proof. 

Although these relations come directly from the previous definition, we can show that they are equivalent to the above-introduced generalized RMT. We therefore set $t=nh$ and $dt=h$. Consider the anti-derivative case ($\alpha \to -\alpha$) and let h decrease to 0. Using the quotient property of the gamma function [17], we have

$$\begin{array}{c}\hfill h^{\alpha }{\displaystyle \frac{{\left(\alpha \right)}_n}{n!}}=h^{\alpha }{\displaystyle \frac{\Gamma \left(\alpha +n\right)}{\Gamma (n+1)}}=h^{\alpha }{\displaystyle \frac{\Gamma \left(\alpha +\frac{t}{h}\right)}{\Gamma (\frac{t}{h}+1)}}\approx h^{\alpha }{\left({\displaystyle \frac{t}{h}}\right)}^{\alpha -1}=t^{\alpha -1}h.\end{array}$$

Therefore, (37) leads to

$${\varphi }_{+}\left(\alpha \right)=\underset{h\to 0^{+}}{lim}{\displaystyle \sum _{n=0}^{+\infty }}f(-nh)t^{\alpha -1}h,$$

and to (21), taking into account the traditional definition of Riemann integral. □

It is important to highlight the fact that (37) and (38) are valid, without any modification, for any real or complex values of $\alpha$.

Example 10.  

Consider $f\left(t\right)=e^{st}$ again. We have ${\varphi }_{+}\left(n\right)=s^n,n\in {\mathbb{N}}_0$ and get

$${\varphi }_{+}\left(\alpha \right)=\underset{h\to 0^{+}}{lim}{h}^{-\alpha }{\displaystyle \sum _{n=0}^{+\infty }}{\displaystyle \frac{{\left(-\alpha \right)}_n}{n!}}{e}^{-nsh}=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{{(1-e^{-sh})}^{\alpha }}{h^{-\alpha }}}=s^{\alpha },$$

if $Re\left(s\right)>0$.

• Similarly, we obtain 

$${\varphi }_{-}\left(\alpha \right)=e^{-i\alpha \pi }\underset{h\to 0^{+}}{lim}{h}^{-\alpha }{\displaystyle \sum _{n=0}^{+\infty }}{\displaystyle \frac{{\left(-\alpha \right)}_n}{n!}}{e}^{nsh}=\underset{h\to 0^{+}}{lim}{\displaystyle \frac{{(1-e^{sh})}^{\alpha }}{h^{-\alpha }}}=e^{-i\alpha \pi }{(-s)}^{\alpha }=s^{\alpha },$$

if $Re\left(s\right)<0$.

This example shows that the GL RMT is valid for functions with LT.

5. Forward

The relationship between RMT and the Liouville derivatives presented in the previous sections leads us to take things a step further and attempt to apply the same ideas to other derivatives or functions. Below, we present three examples that are worth exploring in the future.

• The above-introduced RMT versions assumed the existence of MacLaurin-type series. These are associated with integer-order Laplace transforms. However, fractional-order derivatives and transforms are very important and have interesting applications. With the Liouville derivatives, we can introduce a generalization of the unilateral MacLaurin series constituted by the Mittag–Leffler-type series. These read [25]:
  $$f_c\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{f_{+}^{\left(k\theta \right)}\left(0^{+}\right)}{\Gamma (k\theta +1)}}{t}^{k\theta }u\left(t\right)$$

  (39)

  and

  $$f_a\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{f_{-}^{\left(k\theta \right)}\left(0^{-}\right)}{\Gamma (k\theta +1)}}{t}^{k\theta }u(-t),$$

  (40)

  where $\theta \in {\mathbb{R}}^{+}$.
  We are led to ask for the corresponding RMT. To have some insight let us see what happens with a function that has a Mittag–Leffler-type series.

Example 11. 

Let $f\left(t\right)=e^{{a\left|t\right|}^{\theta }},\theta \in {\mathbb{R}}^{+}$. As is easy to verify,

$$f_c\left(t\right)=e^{a{t}^{\theta }}={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{a^k}{k!}}{t}^{k\theta }u\left(t\right)$$

and

$$f_a\left(t\right)=e^{a{(-t)}^{\theta }}={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{a^k{e}^{ik\theta \pi }}{k!}}{t}^{k\theta }u(-t).$$

For applying (17) to $f\left(t\right)$ we use $f_a\left(t\right)$ so that

$$\varphi \left(k\right)={\displaystyle \frac{a^k{e}^{ik\theta \pi }\Gamma (k\theta +1)}{k!}}={\displaystyle \frac{\theta {(-a^{1/\theta })}^{k\theta }\Gamma \left(k\theta \right)}{\Gamma \left(k\right)}}$$

and

$${\int }_0^{\infty }{\displaystyle \frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}}{e}^{a{t}^{\theta }}dt={\int }_0^{\infty }{\displaystyle \frac{t^{\frac{\alpha }{\theta }-1}}{\theta \Gamma \left(\alpha \right)}}{e}^{at}dt={\displaystyle \frac{\Gamma \left(\frac{\alpha }{\theta }\right)}{\theta \Gamma \left(\alpha \right)}}{\displaystyle \frac{1}{{(-a)}^{\frac{\alpha }{\theta }}}},$$

if $a<0$.

This example shows that we need to introduce a modification to the above RMT to be suitable for this type of series. We have to realize that we made the substitution ${\displaystyle \frac{t^{k\theta }}{\Gamma (k\theta +1)}}$ for ${\displaystyle \frac{t^k}{k!}}$ or $s^{\theta }$ for s in the transform domain. This is an open problem.

• In the previous sections, we dealt mainly with one-sided functions. When the functions were bilateral, we decomposed into their causal and anti-causal parts, which were treated separately. However, there is an alternative decomposition: even/odd. In such a situation, both components are bilateral.
  Generalizations of RMT can be obtained for a bilateral function using the two-sided derivatives [20]. The two most important particular cases are Riesz and Feller. If $\beta >0$, the Riesz anti-derivative is defined by [17,20]
  $${}^{R}D^{-\beta }f\left(x\right)={\int }_{-\infty }^{\infty }f(x-y){\displaystyle \frac{{\left|y\right|}^{\beta -1}}{2cos\left(\beta \frac{\pi }{2}\right)\Gamma \left(\beta \right)}}dy.$$

  (41)

  while Feller’s anti-derivative is given by

  $${}^{F}D^{-\beta }f\left(x\right)={\int }_{-\infty }^{\infty }f(x-y){\displaystyle \frac{{\left|y\right|}^{\beta -1}}{2sin\left(\beta \frac{\pi }{2}\right)\Gamma \left(\beta \right)}}sgn\left(y\right)dy.$$

  (42)

  As we can observe they correspond to anti-derivatives of the even and odd parts of the function.
  Therefore, if the function at hand has MacLaurin series, we can introduce two new forms of the Ramanujan master theorem. Before, we need obtain the even/odd decompositions of the MacLaurin series. As above, we set $\varphi \left(k\right)=f^{\left(k\right)}\left(0\right),k\in {\mathbb{N}}_0.$ The even, $f_e\left(t\right),$ and odd, $f_o\left(t\right),$ components are given by
  $$\begin{array}{cc}\hfill f_e\left(t\right)& ={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{1+{(-1)}^k}{2}}{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k\hfill \\ & ={\displaystyle \sum _{k=0}^{\infty }}cos\left(k{\displaystyle \frac{\pi }{2}}\right){(-1)}^{{\displaystyle \frac{k}{2}}}{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k={\displaystyle \sum _{k=0}^{\infty }}cos\left(k{\displaystyle \frac{\pi }{2}}\right){\displaystyle \frac{{\overline{\varphi }}_e\left(k\right)}{\Gamma (k+1)}}{t}^k\hfill \end{array}$$

  (43)

  and

  $$\begin{array}{cc}\hfill f_o\left(t\right)& ={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{1-{(-1)}^k}{2}}{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k\hfill \\ & ={\displaystyle \sum _{k=0}^{\infty }}sin\left(k{\displaystyle \frac{\pi }{2}}\right){(-1)}^{{\displaystyle \frac{k-1}{2}}}{\displaystyle \frac{\varphi \left(k\right)}{\Gamma (k+1)}}{t}^k={\displaystyle \sum _{k=0}^{\infty }}sin\left(k{\displaystyle \frac{\pi }{2}}\right){\displaystyle \frac{{\overline{\varphi }}_o\left(k\right)}{\Gamma (k+1)}}{t}^k.\hfill \end{array}$$

  (44)

  The new variables ${\overline{\varphi }}_e\left(k\right)={(-1)}^{{\displaystyle \frac{k}{2}}}\varphi \left(k\right),$ for even k, and ${\overline{\varphi }}_o\left(k\right)={(-1)}^{{\displaystyle \frac{k-1}{2}}}\varphi \left(k\right),$ for odd k, represent the corresponding two-sided derivatives. The exponential factors express the relations between both kinds of derivatives [26].
  Attending to the procedure used in the previous sections we can introduce two new generalizations of RMT given by
  $$\begin{array}{cc}\hfill {}^R\varphi \left(\beta \right)& ={}^{R}D^{-\beta }f\left(0\right)={\int }_{-\infty }^{\infty }f\left(y\right){\displaystyle \frac{{\left|y\right|}^{\beta -1}}{2cos\left(\beta \frac{\pi }{2}\right)\Gamma \left(\beta \right)}}dy\hfill \\ & ={\int }_0^{\infty }\left[{\displaystyle \frac{f\left(y\right)+f(-y)}{2}}\right]{\displaystyle \frac{{\left|y\right|}^{\beta -1}}{cos\left(\beta \frac{\pi }{2}\right)\Gamma \left(\beta \right)}}dy,\hfill \end{array}$$

  (45)

  and

  $$\begin{array}{cc}\hfill {}^F\varphi \left(\beta \right)& ={}^{F}D^{-\beta }f\left(0\right)={\int }_{-\infty }^{\infty }f\left(y\right){\displaystyle \frac{{\left|y\right|}^{\beta -1}}{2sin\left(\beta \frac{\pi }{2}\right)\Gamma \left(\beta \right)}}sgn\left(y\right)dy\hfill \\ & ={\int }_0^{\infty }\left[{\displaystyle \frac{f\left(y\right)-f(-y)}{2}}\right]{\displaystyle \frac{{\left|y\right|}^{\beta -1}}{sin\left(\beta \frac{\pi }{2}\right)\Gamma \left(\beta \right)}}dy.\hfill \end{array}$$

  (46)

  This new approach challenges us to move forward, but it raises a number of issues that remain to be resolved.
• So far, we have considered various versions of RMT derived from shift-invariant derivatives. However, there are other important derivatives. Of these, perhaps the most interesting are the Hadamard scale-invariant derivatives [16]. They can lead to logarithmic MacLaurin series and left/right derivatives, from which we can obtain several versions of a log-RMT. In the following we will open the door to a new version of RMT. However, it requires more work.
  Consider the Hadamard stretching derivative [16], which is defined by

$${\mathfrak{D}}_{s+}x\left(\tau \right)=\underset{q\to 1^{+}}{lim}{\displaystyle \frac{x\left(\tau \right)-x\left(\tau q^{-1}\right)}{lnq}}=\tau x^{\prime }\left(\tau \right),$$

(47)

This derivative allows us to define a log-MacLaurin series for indefinitely scale-differentiable bounded functions as follows

$$f\left(\tau \right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\mathfrak{D}}_{s+}^{k}f\left(1^{+}\right)}{\Gamma (k+1)}}{ln}^k\left(\tau \right),\tau \ge 1.$$

(48)

Let $\alpha <0$ and $\tau \in {\mathbb{R}}^{+}$. We define a stretching Hadamard anti-derivative by [16]

$${\mathfrak{D}}_{s+}^{-\alpha }f\left(\tau \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_1^{\infty }f(\tau /\eta ){ln}^{\alpha -1}\left(\eta \right){\displaystyle \frac{\mathrm{d}\eta }{\eta }}.$$

(49)

Similarly to the above, we set ${\varphi }_{+}\left(k\right)={\mathfrak{D}}_{s+}^{k}f\left(1^{+}\right)$ so that

$$f\left(\tau \right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{+}\left(k\right)}{\Gamma (k+1)}}{ln}^k\left(\tau \right),\tau \ge 1.$$

(50)

for $\tau$ sufficiently close to $1^{+}$. Then, the (stretching) log-RMT reads

$$I_n\equiv {\int }_1^{\infty }{\displaystyle \frac{{ln}^{n-1}\left(\tau \right)}{\Gamma \left(n\right)}}f\left({\tau }^{-1}\right){\displaystyle \frac{d\tau }{\tau }}={\varphi }_{+}(-n).$$

(51)

Example 12. 

Consider $f\left(\tau \right)={\tau }^{\nu }$ with $Re\left(\nu \right)>0$. Then

$${\tau }^{\nu }={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\varphi }_{+}\left(k\right)}{k!}}{ln}^k\left(\tau \right),$$

where $\varphi {\left(k\right)}_{+}={\nu }^k$ [16]. With a suitable variable change, it is not hard to verify that

$$I_n={\int }_1^{\infty }{\displaystyle \frac{{ln}^{n-1}\left(\tau \right)}{(n-1)!}}{\tau }^{-\nu }{\displaystyle \frac{d\tau }{\tau }}={\displaystyle \frac{1}{{\nu }^n}}$$

(52)

Therefore $I_n={\varphi }_{+}(-n)$.

• As in the Liouville case, we define a shrinking-scale derivative by

$${\mathfrak{D}}_{s-}x\left(\tau \right)=\underset{q\to 1^{+}}{lim}{\displaystyle \frac{x\left(\tau q\right)-x\left(\tau \right)}{lnq}},$$

(53)

which allows us to introduce another log-MacLaurin series for indefinitely scale-differentiable bounded functions similar to (48), which reads

$$f\left(\tau \right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{{\mathfrak{D}}_{s-}^{k}f\left(1^{-}\right)}{\Gamma (k+1)}}{ln}^k\left(\tau \right),0<\tau \le 1.$$

(54)

Here, we set ${\varphi }_{-}\left(k\right)={\mathfrak{D}}_{s-}^{k}f\left(1^{-}\right)$.

• As in (49), we define a shrinking Hadamard anti-derivative by [16]

$${\mathfrak{D}}_{s-}^{-\alpha }f\left(\tau \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{1}f(\tau /\eta ){ln}^{\alpha -1}\left(\eta \right){\displaystyle \frac{\mathrm{d}\eta }{\eta }},$$

(55)

for $0<\tau \le 1$. We are led to define a (shrinking) log-RMT by

$$I_n\equiv {\int }_0^1{\displaystyle \frac{{ln}^{n-1}\left(\tau \right)}{\Gamma \left(n\right)}}f\left({\tau }^{-1}\right){\displaystyle \frac{d\tau }{\tau }}={\int }_1^{\infty }{\displaystyle \frac{{ln}^{n-1}\left({\tau }^{-1}\right)}{\Gamma \left(n\right)}}f\left(\tau \right){\displaystyle \frac{d\tau }{\tau }}={\varphi }_{-}(-n)$$

(56)

As is easy to see, we have just opened up a new and interesting avenue for other formulations, particularly for positive orders. However, this requires a great deal more work.

6. Conclusions

In this paper, we studied Ramanujan’s master theorem from the point of view of Liouville fractional derivatives, of which it was considered as expressing particular values. This fact was used to obtain two generalizations. Firstly, we realized that the original formulation is suitable for anti-causal functions only, so expressions for causal functions were introduced. Both formulations were related to the regions of convergence of the Laplace transform. In passing, the concept of unilateral MacLaurin series was introduced and used. Other generalizations were suggested.