Umbral Methods, Function Factorisation and Mittag–Leffler Fourier-Type Integral Transform

Abstract

We propose a systematic way to construct trigonometric-like functions beyond the classical sine–cosine pair by factorising rational expressions in the umbral operator and then evaluating their action on the vacuum. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an n-fold decomposition of rational umbral expressions. For each integer $n\ge 2$, the construction produces n functions which play the role of higher-order trigonometric-like components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case $n=2$ gives the Gaussian-trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case $n=3$ yields a three-component cyclic system and shows how the same idea extends beyond the usual even–odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin–Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations and local expansions for the functions obtained formally, while also fixing the branch and residue conventions needed for their analytic continuation. Finally, we indicate how the same cyclic kernels act as deformations of the Fourier transform. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators.

1. Introduction

The theory of trigonometric functions beyond the elementary circular case has appeared in many forms throughout the development of mathematics over the last three centuries.

Several classical references, recalled in [1], provide a broad historical perspective on this subject. They investigate the structure of trigonometric functions in depth and provide the technical background needed to understand how the elementary circular functions are related to hypergeometric and elliptic functions, and how sine- and cosine-like functions may arise from higher transcendental functions, including functions of Bessel type [2].

Both prominent and less widely known mathematicians contributed to this development. Some of these contributions became part of the standard university curriculum, whereas others remained confined to a more specialised literature.

Euler introduced the so-called quasi-trigonometric functions [3] in connection with the summation of series extending beyond the Basel problem, namely the evaluation of the infinite sum of the reciprocal squares of the positive integers. We shall return to this point in the Section 5. Legendre and Jacobi [4] laid the foundations of the theory of elliptic functions, which arose from the inversion of elliptic integrals. Other authors, including Ferrari and Lindqvist [1,2], discussed trigonometric functions associated with elliptic curves.

More recently, the emergence of new mathematical languages has led to renewed interest in this circle of ideas; see, for example, [5,6]. In particular, indicial umbral calculus [7] provides a useful framework for constructing trigonometric-like functions from a different perspective. The motivations for this approach are manifold and will be discussed below.

In Ref. [8], Di Palma, Licciardi and one of the present authors (G.D.) introduced Gaussian trigonometry, using the umbral theory of Gaussian functions as a benchmark. A different umbral representation of the same functions has recently been proposed by another of the present authors (R.R.) in [9].

We briefly recall the relevant formalism and its conceptual content.

(a)

The umbral definition of the Gaussian-cosine is

$$C_G\left(x\right)={\displaystyle \frac{1}{1+\widehat{c}{x}^2}}{\phi }_0,{\widehat{c}}^{\alpha }{\phi }_0={\displaystyle \frac{1}{\Gamma (1+\alpha )}}.$$

(1)

In this formula, $\alpha$ is a complex parameter, $\widehat{c}$ denotes the formal umbral operator and ${\phi }_0$ the corresponding vacuum.

(b)

This Lorentzian umbral expression evaluates to the Gaussian function. This is easily verified by expanding $C_G\left(x\right)$ in series and using the action of $\widehat{c}$ on the vacuum:

$$C_G\left(x\right)={\displaystyle \sum _{r=0}^{\infty }}{(-1)}^r{x}^{2r}{\widehat{c}}^r{\phi }_0={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{(-1)}^r{x}^{2r}}{r!}}={\mathrm{e}}^{-x^2}.$$

The umbral formalism therefore establishes a correspondence between rational functions and transcendental functions. Applying the partial fraction decomposition to the Lorentzian factor, we obtain

$$C_G\left(x\right)={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{1-i{\widehat{c}}^{\frac{1}{2}}x}}+{\displaystyle \frac{1}{1+i{\widehat{c}}^{\frac{1}{2}}x}}\right]{\phi }_0.$$

(2)

This suggests the introduction of the Gaussian-exponential

$${\mathrm{e}}_G\left(ix\right)={\displaystyle \frac{1}{1-i{\widehat{c}}^{\frac{1}{2}}x}}{\phi }_0={\displaystyle \sum _{r=0}^{\infty }}{\left(ix\right)}^r{\widehat{c}}^{{\displaystyle \frac{r}{2}}}{\phi }_0={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{\left(ix\right)}^r}{\Gamma \left(\frac{r}{2}+1\right)}}$$

(3)

and of the associated Gaussian sine and cosine functions

$$S_G\left(x\right)={\displaystyle \frac{1}{2i}}\left[{\mathrm{e}}_G\left(ix\right)-{\mathrm{e}}_G(-ix)\right]={\mathrm{e}}^{-x^2}\mathrm{erfi}\left(x\right),$$

(4)

$$C_G\left(x\right)={\displaystyle \frac{1}{2}}\left[{\mathrm{e}}_G\left(ix\right)+{\mathrm{e}}_G(-ix)\right]={\mathrm{e}}^{-x^2}.$$

(5)

Here, erfi denotes the imaginary error function,

$$\mathrm{erfi}\left(x\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^x{\mathrm{e}}^{{\xi }^2}\mathrm{d}\xi =-i\mathrm{erf}\left(ix\right).$$

These functions are not trigonometric functions in the classical sense, since they do not possess the standard circular properties, such as periodicity or the ordinary derivative cyclic properties. Nevertheless, they share important structural analogies with the circular functions, and these analogies will be developed in the following sections.

The function ${\mathrm{e}}_G\left(ix\right)$ plays a central role in the partial fraction decomposition of the Lorentzian umbral image of the Gaussian. It is also recognised as a particular case of the two-parameter Mittag–Leffler function [10],

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

(6)

For $\alpha =1/2$ and $\beta =1$, one has [11]

$${\mathrm{e}}_G\left(ix\right)=E_{{\displaystyle \frac{1}{2}},1}\left(ix\right)={\mathrm{e}}^{-x^2}\left(1+i\mathrm{erfi}\left(x\right)\right).$$

(7)

The Mittag–Leffler function is a classical generalisation of the exponential function [12]. Moreover, the resemblance of the above construction with Euler’s formula justifies, at least at the formal level, the terminology of Gaussian trigonometric functions.

The Gaussian-exponential also appears in plasma physics through its close connection with the plasma dispersion function. The Gaussian sine and cosine satisfy Kramers–Kronig-type relations:

$$S_G\left(x\right)=-{\displaystyle \frac{1}{\pi }}\mathcal{P}{\int }_{-\infty }^{\infty }{\displaystyle \frac{C_G\left(\xi \right)}{\xi -x}}\mathrm{d}\xi ,C_G\left(x\right)={\displaystyle \frac{1}{\pi }}\mathcal{P}{\int }_{-\infty }^{\infty }{\displaystyle \frac{S_G\left(\xi \right)}{\xi -x}}\mathrm{d}\xi .$$

(8)

These relations are understood as Hilbert-transform relations for the boundary values of the analytic Gaussian exponential. More precisely, we regard

$$z\mapsto {\mathrm{e}}_G\left(iz\right)$$

as the analytic continuation of the Gaussian exponential to the upper half-plane. With the present conventions this function coincides with the Faddeeva function [13]:

$${\mathrm{e}}_G\left(iz\right)=W\left(z\right)={\mathrm{e}}^{-z^2}\mathrm{erfc}(-iz).$$

In particular, for real x,

$${\mathrm{e}}_G\left(ix\right)=W\left(x\right)=C_G\left(x\right)+i{S}_G\left(x\right).$$

The principal value integrals in the Kramers–Kronig formulae are therefore taken along the real axis and follow from the analyticity of $z\mapsto {\mathrm{e}}_G\left(iz\right)$ in the upper half-plane, together with its standard decay properties there.

The role of the Gaussian-exponential will be further discussed in the following sections, where we derive additional consequences of the formalism outlined above. In the final part of the paper, the same cyclic decompositions are reconsidered within the analytic umbral framework recently proposed in [14] by one of the present authors (R.R.): the formal rational factorisations are realised as Mellin–Barnes pairings, thereby clarifying the origin of the component expansions and their contour representations. The analytic framework also indicates the natural setting in which analytic continuation and sectorial asymptotic questions can be studied, although a systematic asymptotic analysis is not pursued here.

2. Cubic Exponentials and Gaussian Trigonometry

The present section remains within the formal umbral setting. Its purpose is to show that the Gaussian case is only the quadratic member of a finite cyclic decomposition associated with roots of unity. The cubic case is treated explicitly because it is the first genuinely higher-order example and anticipates the analytic interpretation developed in Section 4.

In the previous section, the classical trigonometric functions were interpreted as the exponentials associated with the square roots of the negative unit. In what follows, we shall refer to them as second-order trigonometric functions. Analogously, the third-order partners are obtained by exponentiating along the cubic roots of unity. They are defined by

$$C_0^{\left(3\right)}\left(x\right)={\displaystyle \frac{1}{3}}\left[{\mathrm{e}}^x+{\mathrm{e}}^{\omega x}+{\mathrm{e}}^{{\omega }^{2}x}\right],$$

(9)

$$C_1^{\left(3\right)}\left(x\right)={\displaystyle \frac{1}{3}}\left[{\mathrm{e}}^x+\omega {\mathrm{e}}^{\omega x}+{\omega }^2{\mathrm{e}}^{{\omega }^{2}x}\right],$$

(10)

$$C_2^{\left(3\right)}\left(x\right)={\displaystyle \frac{1}{3}}\left[{\mathrm{e}}^x+{\omega }^2{\mathrm{e}}^{\omega x}+\omega {\mathrm{e}}^{{\omega }^{2}x}\right].$$

(11)

where we have used $\omega ={\mathrm{e}}^{i{\displaystyle \frac{2\pi }{3}}}$, ${\omega }^2+\omega =-1$, ${\omega }^3=1$. It is immediately possible to derive the following identities [6]:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}{C}_0^{\left(3\right)}\left(x\right)=C_1^{\left(3\right)}\left(x\right),{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}}{C}_0^{\left(3\right)}\left(x\right)=C_2^{\left(3\right)}\left(x\right),{\displaystyle \frac{{\mathrm{d}}^3}{\mathrm{d}{x}^3}}{C}_0^{\left(3\right)}\left(x\right)=C_0^{\left(3\right)}\left(x\right).$$

(12)

An obvious extension of the factorisation procedure outlined in the introduction is its application to the cubic exponential, namely

$${\displaystyle \frac{1}{1+\widehat{c}{x}^3}}{\phi }_0={\mathrm{e}}^{-x^3},$$

(13)

where the umbral operator and the vacuum state have the same meaning as before.

The use of a standard procedure (see below) enables us to obtain the following decomposition of the l.h.s. side of Equation (13):

$${\displaystyle \frac{1}{1+\widehat{c}{x}^3}}{\phi }_0={\displaystyle \frac{1}{3}}{\displaystyle \sum _{r=0}^2}{\displaystyle \frac{1}{1+{\omega }^r{\widehat{c}}^{\frac{1}{3}}x}}{\phi }_0$$

(14)

where $\omega$ is the same as before.

The conclusion we may draw from Equations (13) and (14) is that

$${\mathrm{e}}^{-x^3}={\displaystyle \frac{1}{3}}\left[E_{{\displaystyle \frac{1}{3}},1}(-x)+E_{{\displaystyle \frac{1}{3}},1}(-\omega x)+E_{{\displaystyle \frac{1}{3}},1}(-{\omega }^{2}x)\right].$$

(15)

The identity follows from the root-of-unity projection formula for the cyclic Mittag–Leffler decomposition, which is stated and proved in the following proposition.

Proposition 1 (Cyclic decomposition of the Mittag–Leffler exponential). 

Let $n\ge 2$ be an integer and define

$$I_n\left(\eta \right)=E_{1/n,1}\left(\eta \right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\eta }^m}{\Gamma (1+m/n)}}.$$

Then $I_n$ admits the cyclic decomposition

$$I_n\left(\eta \right)={\displaystyle \sum _{r=0}^{n-1}}{I}_{n,r}\left(\eta \right),$$

where

$$I_{n,r}\left(\eta \right)={\eta }^r{E}_{1,1+r/n}\left({\eta }^n\right)={\eta }^r{\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{nq}}{\Gamma (1+q+r/n)}}.$$

Equivalently, if ${\omega }_n=\exp (2\pi i/n)$, then

$$I_{n,r}\left(\eta \right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{\ell =0}^{n-1}}{\omega }_n^{-r\ell }{I}_n\left({\omega }_n^{\ell }\eta \right)(r=0,\dots ,n-1).$$

Proof. 

The first identity follows by splitting the defining series of $I_n$ into congruence classes modulo n. Writing

$$m=nq+r(q\ge 0,r=0,\dots ,n-1),$$

one obtains

$$I_n\left(\eta \right)={\displaystyle \sum _{r=0}^{n-1}}{\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{nq+r}}{\Gamma (1+q+r/n)}}={\displaystyle \sum _{r=0}^{n-1}}{\eta }^r{E}_{1,1+r/n}\left({\eta }^n\right).$$

This proves the cyclic decomposition.

For the projection formula, observe that

$$I_n\left({\omega }_n^{\ell }\eta \right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\omega }_n^{\ell m}{\eta }^m}{\Gamma (1+m/n)}}.$$

Therefore

$${\displaystyle \frac{1}{n}}{\displaystyle \sum _{\ell =0}^{n-1}}{\omega }_n^{-r\ell }{I}_n\left({\omega }_n^{\ell }\eta \right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\eta }^m}{\Gamma (1+m/n)}}\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{\ell =0}^{n-1}}{\omega }_n^{\ell (m-r)}\right).$$

The expression in parentheses is equal to 1 if $m\equiv r\left(\mathrm{mod}n\right)$ and to 0 otherwise. Hence only the terms $m=nq+r$ survive, and the right-hand side is precisely $I_{n,r}\left(\eta \right)$. □

Third-order trigonometry is associated with the cubic roots of unity [6]. We therefore introduce three cubic Gaussian components, which are the analogues of the even–odd Gaussian components $C_G$ and $S_G$ in the quadratic case:

$$C_0^{G_3}\left(x\right)={\displaystyle \frac{1}{3}}\left[E_{{\displaystyle \frac{1}{3}},1}(-x)+E_{{\displaystyle \frac{1}{3}},1}(-\omega x)+E_{{\displaystyle \frac{1}{3}},1}(-{\omega }^{2}x)\right],$$

(16)

$$C_1^{G_3}\left(x\right)={\displaystyle \frac{1}{3}}\left[E_{{\displaystyle \frac{1}{3}},1}(-x)+{\omega }^2{E}_{{\displaystyle \frac{1}{3}},1}(-\omega x)+\omega E_{{\displaystyle \frac{1}{3}},1}(-{\omega }^{2}x)\right],$$

(17)

$$C_2^{G_3}\left(x\right)={\displaystyle \frac{1}{3}}\left[E_{{\displaystyle \frac{1}{3}},1}(-x)+\omega E_{{\displaystyle \frac{1}{3}},1}(-\omega x)+{\omega }^2{E}_{{\displaystyle \frac{1}{3}},1}(-{\omega }^{2}x)\right],$$

(18)

where $G_3$ denotes the relationship with the cubic exponential. Note that, in the quadratic case, the even component is $C_0^{G_2}\left(x\right)=C_G\left(x\right)$, while the odd component is $C_1^{G_2}\left(x\right)=i{S}_G\left(x\right)$; equivalently, $S_G\left(x\right)$ is obtained after division by i.

The explicit expressions of these functions are:

$$C_0^{G_3}\left(x\right)=E_{1,1}(-x^3)={\mathrm{e}}^{-x^3},$$

(19)

$$C_1^{G_3}\left(x\right)=-x{E}_{1,{\displaystyle \frac{4}{3}}}(-x^3),$$

(20)

$$C_2^{G_3}\left(x\right)=x^2{E}_{1,{\displaystyle \frac{5}{3}}}(-x^3).$$

(21)

This result is a direct extension of what we already obtained in the Gaussian case. It follows from Proposition 1 with $n=3$. In particular,

$$I_{3,0}\left(\eta \right)=E_{1,1}\left({\eta }^3\right)={\mathrm{e}}^{{\eta }^3},$$

whereas

$$I_{3,1}\left(\eta \right)=\eta E_{1,{\displaystyle \frac{4}{3}}}\left({\eta }^3\right)$$

and

$$I_{3,2}\left(\eta \right)={\eta }^2{E}_{1,{\displaystyle \frac{5}{3}}}\left({\eta }^3\right).$$

Thus, for $\eta =-x$, one obtains

$$I_{3,1}(-x)=-x{E}_{1,{\displaystyle \frac{4}{3}}}(-x^3)$$

and

$$I_{3,2}(-x)=x^2{E}_{1,{\displaystyle \frac{5}{3}}}(-x^3).$$

The same mechanism applies to higher orders. For instance, the quartic umbral image of the super-Gaussian is

$${\displaystyle \frac{1}{1+\widehat{c}{x}^4}}{\phi }_0={\mathrm{e}}^{-x^4}.$$

Its full cyclic decomposition is obtained from the fourth roots of $-1$. If

$${\lambda }_r=\exp \left({\displaystyle \frac{(2r+1)\pi i}{4}}\right)(r=0,1,2,3),$$

then

$${\displaystyle \frac{1}{1+\widehat{c}{x}^4}}{\phi }_0={\displaystyle \frac{1}{4}}{\displaystyle \sum _{r=0}^3}{\displaystyle \frac{1}{1+{\lambda }_r{\widehat{c}}^{1/4}x}}{\phi }_0,$$

and hence

$${\mathrm{e}}^{-x^4}={\displaystyle \frac{1}{4}}{\displaystyle \sum _{r=0}^3}{E}_{{\displaystyle \frac{1}{4}},1}(-{\lambda }_{r}x).$$

This example shows that, in general, the relevant root system is determined by the sign in the rational umbral representative.

So far, the umbral formalism has allowed us to factorize exponential functions and to relate these factorisations to combinations of Mittag–Leffler functions, which may be interpreted as higher-order trigonometric functions. This kernel-level point of view naturally leads to the construction of Fourier-type transforms, obtained by replacing the ordinary exponential Fourier kernel with the Gaussian or, more generally, Mittag–Leffler kernels introduced above.

3. Gaussian Trigonometry and Corresponding Fourier-Type Transform

The preceding construction suggests a deformation of the Fourier kernel. In order to avoid overloading the terminology, we shall call the resulting object a Fourier-type transform. The aim of this section is not to develop a complete transform theory, with inversion and Plancherel formulae, but only to show how the umbral kernel constructed above can be used to deform the ordinary Fourier integral.

Let f belong to a class of test functions for which the integral below is absolutely convergent; for instance, one may initially take $f\in C_c^{\infty }\left(\mathbb{R}\right)$. We define, at least formally, the Mittag–Leffler Fourier-type transform by

$${\mathcal{F}}_{{\displaystyle \frac{1}{n}}}\left[f\right]\left(k\right)={\int }_{-\infty }^{\infty }f\left(x\right)E_{{\displaystyle \frac{1}{n}},1}(-ikx)\mathrm{d}x.$$

Here $E_{{\displaystyle \frac{1}{n}},1}$ is the order-n Mittag–Leffler kernel defined above. A complete characterisation of the largest admissible function spaces is not pursued here and is left for a subsequent study.

The calculations are intentionally kept at the operational level: further aspects will be discussed later in Section 4.

According to the previous discussion, the following correspondence can be envisaged:

$${\mathrm{e}}^{i\zeta }\mapsto {\mathrm{e}}_G\left(i\zeta \right)=E_{{\displaystyle \frac{1}{2}},1}\left(i\zeta \right).$$

(22)

Since the ordinary Fourier transform is defined here by

$$\mathcal{F}\left[f\right]\left(k\right)={\int }_{-\infty }^{\infty }f\left(x\right){\mathrm{e}}^{-ikx}\mathrm{d}x={\int }_{-\infty }^{\infty }f\left(x\right){\mathrm{e}}^{i(-kx)}\mathrm{d}x,$$

(23)

the Gaussian Fourier-type kernel is obtained by replacing ${\mathrm{e}}^{i\zeta }$ with ${\mathrm{e}}_G\left(i\zeta \right)$ and then taking $\zeta =-kx$. Thus

$${\mathrm{e}}_G(-ikx)=E_{{\displaystyle \frac{1}{2}},1}(-ikx).$$

(24)

More generally, we can ask whether this correspondence can be exploited to define Mittag–Leffler-based Fourier-type transforms:

$${\mathcal{F}}_{{\displaystyle \frac{1}{n}}}\left[f\right]\left(k\right)={\int }_{-\infty }^{\infty }f\left(x\right)E_{{\displaystyle \frac{1}{n}},1}(-ikx)\mathrm{d}x.$$

(25)

We consider the case $n=2$ first and start with a classical example:

$$f\left(x\right)={\mathrm{e}}^{\left|x\right|}.$$

For $n=2$, the Mittag–Leffler kernel can be written explicitly as

$$E_{{\displaystyle \frac{1}{2}},1}\left(z\right)={\mathrm{e}}^{z^2}\mathrm{erfc}(-z).$$

Therefore, for real k and x,

$$E_{{\displaystyle \frac{1}{2}},1}(-ikx)={\mathrm{e}}^{-k^2{x}^2}\left(1-i\mathrm{erfi}\left(kx\right)\right).$$

Since ${\mathrm{e}}^{-\left|x\right|}$ is even and $\mathrm{erfi}\left(kx\right)$ is odd as a function of x, the odd contribution vanishes after integration on the symmetric interval. Hence

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)={\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\left|x\right|}{\mathrm{e}}^{-k^2{x}^2}\mathrm{d}x.$$

Equivalently,

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)=2{\int }_0^{\infty }{\mathrm{e}}^{-x-k^2{x}^2}\mathrm{d}x.$$

For $k\ne 0$, the last integral gives

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)={\displaystyle \frac{\sqrt{\pi }}{\left|k\right|}}\exp \left({\displaystyle \frac{1}{4k^2}}\right)\mathrm{erfc}\left({\displaystyle \frac{1}{2\left|k\right|}}\right).$$

At $k=0$, the value is obtained by continuity and equals

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(0\right)=2.$$

We provide in the following a formal umbral derivation of the same result. We start by recalling the ordinary Fourier transform of $f\left(x\right)={\mathrm{e}}^{-\left|x\right|}$:

$$\tilde{f}\left(k\right)={\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\left|x\right|}{\mathrm{e}}^{-ikx}\mathrm{d}x={\displaystyle \frac{2}{1+k^2}}.$$

(26)

The formal umbral representation of the deformed Fourier-type integral is

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)=\left({\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\left|x\right|}{\displaystyle \frac{1}{1+ik{\widehat{c}}^{\frac{1}{2}}x}}\mathrm{d}x\right){\phi }_0.$$

(27)

The denominator is represented by

$${\displaystyle \frac{1}{1+ik{\widehat{c}}^{\frac{1}{2}}x}}={\int }_0^{\infty }{\mathrm{e}}^{-s-isk{\widehat{c}}^{{\displaystyle \frac{1}{2}}}x}\mathrm{d}s.$$

(28)

Therefore

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)=\left[{\int }_0^{\infty }{\mathrm{e}}^{-s}\left({\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\left|x\right|-isk{\widehat{c}}^{{\displaystyle \frac{1}{2}}}x}\mathrm{d}x\right)\mathrm{d}s\right]{\phi }_0.$$

(29)

The inner integral is the ordinary Fourier transform of ${\mathrm{e}}^{-\left|x\right|}$ evaluated at $sk{\widehat{c}}^{1/2}$; hence

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\left|x\right|-isk{\widehat{c}}^{{\displaystyle \frac{1}{2}}}x}\mathrm{d}x={\displaystyle \frac{2}{1+\widehat{c}{s}^2{k}^2}}.$$

(30)

Consequently

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)=\left[2{\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-s}}{1+\widehat{c}{s}^2{k}^2}}\mathrm{d}s\right]{\phi }_0.$$

(31)

Finally, using Equation (1), namely

$${\displaystyle \frac{1}{1+\widehat{c}{s}^2{k}^2}}{\phi }_0={\mathrm{e}}^{-s^2{k}^2},$$

(32)

we recover

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-\left|x\right|}\right]\left(k\right)=2{\int }_0^{\infty }{\mathrm{e}}^{-s-s^2{k}^2}\mathrm{d}s.$$

(33)

The same result is obtained after noting that ${\mathrm{e}}^{-\left|x\right|}$ is an even function, so that only the real part of the Gaussian Fourier-type kernel contributes:

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\left|x\right|}\mathrm{Re}\left({\mathrm{e}}_G(-ikx)\right)\mathrm{d}x=2{\int }_0^{\infty }{\mathrm{e}}^{-x}{\mathrm{e}}^{-k^2{x}^2}\mathrm{d}x={\displaystyle \frac{\sqrt{\pi }}{\left|k\right|}}{\mathrm{e}}^{{\left({\displaystyle \frac{1}{2k}}\right)}^2}\mathrm{erfc}\left({\displaystyle \frac{1}{2\left|k\right|}}\right).$$

(34)

The preceding calculation is only meant as a kernel-level construction. A complete transform theory would require the systematic study of the corresponding inverse transform, admissible function spaces and mapping properties.

In the previous example, we have used a result from the ordinary Fourier transform and used the umbral correspondence to state the corresponding $G_2$ transform. A different strategy is outlined below for the case

$${\mathcal{F}}_{{\displaystyle \frac{1}{2}}}\left[{\mathrm{e}}^{-a{x}^2}\right]\left(k\right)={\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{E}_{{\displaystyle \frac{1}{2}},1}(-ikx)\mathrm{d}x,$$

(35)

where $a>0$. We introduce the new vacuum $\psi$ such that

$${\widehat{c}}^r\psi ={\displaystyle \frac{r!}{\Gamma \left(\frac{r}{2}+1\right)}}.$$

(36)

In terms of $\psi$, the umbral image of the Mittag–Leffler function is an exponential, namely:

$$E_{{\displaystyle \frac{1}{2}},1}(-ikx)={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{(-ikx)}^r}{\Gamma \left(\frac{r}{2}+1\right)}}={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{(-ikx)}^r}{r!}}{\widehat{c}}^r\psi ={\mathrm{e}}^{-ikx\widehat{c}}\psi .$$

(37)

Therefore, the integral in Equation (35) can be written as

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{E}_{{\displaystyle \frac{1}{2}},1}(-ikx)\mathrm{d}x=\left({\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{\mathrm{e}}^{-ikx\widehat{c}}\mathrm{d}x\right)\psi .$$

(38)

Using

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{\mathrm{e}}^{-iqx}\mathrm{d}x=\sqrt{{\displaystyle \frac{\pi }{a}}}{\mathrm{e}}^{-{\displaystyle \frac{q^2}{4a}}}$$

(39)

yields

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{E}_{{\displaystyle \frac{1}{2}},1}(-ikx)\mathrm{d}x=\sqrt{{\displaystyle \frac{\pi }{a}}}{\mathrm{e}}^{-k^2{\widehat{c}}^2/4a}\psi .$$

(40)

Expanding the exponential and using Equation (36), we eventually find

$${\mathrm{e}}^{-k^2{\widehat{c}}^2/4a}{\phi }_0={\displaystyle \sum _{r=0}^{\infty }}{\left(-{\displaystyle \frac{k^2}{4a}}\right)}^r{\displaystyle \frac{\left(2r\right)!}{{(r!)}^2}}.$$

(41)

Using the combinatorial identities

$${\displaystyle \frac{\left(2r\right)!}{{(r!)}^2}}=\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r}}\right),{\displaystyle \sum _{r=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r}}\right)z^r={\displaystyle \frac{1}{\sqrt{1-4z}}},$$

(42)

the final result is

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{E}_{{\displaystyle \frac{1}{2}},1}(-ikx)\mathrm{d}x=\sqrt{{\displaystyle \frac{\pi }{a+k^2}}}.$$

(43)

For $a>0$ and $k\in \mathbb{R}$, the square root is initially understood as the positive real square root. The identity extends by analytic continuation to $\mathrm{Re}a>0$, with the square root taken on the principal branch and fixed by continuity from $k=0$.

This result concludes our preliminary discussion on the Gaussian Fourier-type transform. The terminology “Fourier-type” is used deliberately. The construction above provides a family of kernels obtained by replacing the exponential kernel of the ordinary Fourier transform with the umbral Mittag–Leffler kernel. We do not claim here a full analogue of the classical Fourier theory. In particular, questions of inversion, Plancherel formulae, spectral resolution and optimal mapping properties are left for a separate study. The role of the present section is only to show that the same umbral factorisation mechanism naturally produces integral kernels which deform the ordinary Fourier kernel.

4. Analytic Framework

The purpose of this section is to reinterpret the formal decompositions of the preceding sections in the analytic umbral framework. We first recall the Mellin–Barnes pairing and the corresponding spectral transmutation mechanism. We then treat explicitly the quadratic and cubic cases, in order to show how the formal cyclic decompositions are realised at the level of contour integrals. Finally, we record the general n-fold pattern and explain how the same analytic mechanism underlies the Fourier-type kernels introduced above.

The constructions developed in the previous sections are formal in nature. They are based on rational identities involving umbral operators and on the subsequent evaluation of these identities on the prescribed vacuum. In this section we recall how the same identities can be realised analytically by means of Mellin–Barnes pairings. The purpose of the analytic formulation is not to replace the formal construction, but to fix the contour, branch and residue conventions which make the umbral identities into well-defined integral representations.

The analytic framework recalled in this section is used only in the limited form needed for the present paper. The broader construction is developed in [14], but the definitions and residue computations required below are stated explicitly here, so that the Mellin–Barnes interpretation of the preceding formal identities is self-contained.

Throughout this section, the cyclic decompositions already proved at the formal level are taken as identities of entire Mittag–Leffler functions. The analytic framework gives an alternative analytic realisation of the same functions, in terms of Mellin–Barnes pairing between a jump kernel and a ground state. This has two advantages.

First, it explains why the formal exponential–rational transmutation produces Mittag–Leffler functions.

Second, it shows that the cyclic components are not merely formal projections of a series, but analytic contour objects whose local and asymptotic expansions arise from different deformations of the same Mellin–Barnes integral.

In the analytic umbral framework, the umbral operator $\mathfrak{u}$ is reinterpreted as a functional in the Borel additive complex variable u, namely:

$$\mathfrak{u}\mapsto {\mathrm{e}}^u.$$

For a suitable f and $\eta \in \mathbb{C}$, the formal operator function $f\left(\eta \mathfrak{u}\right)$ thus corresponds to the composed umbral Borel functional $f\left(\eta {\mathrm{e}}^u\right)$. After introducing the multiplicative Borel variable $z={\mathrm{e}}^u$, one defines the spectral jump function associated with f as

$$J_f(t;\eta ) :={\mathcal{M}}_z\left[f\left(\eta z\right)\right](-t)={\int }_0^{\infty }f\left(\eta z\right)z^{-t-1}\mathrm{d}z,$$

where $\mathcal{M}$ denotes the Mellin transform operator.

In particular, if $f\left(\eta {\mathrm{e}}^u\right)=\exp \left(\eta {\mathrm{e}}^u\right)$ we have that

$$J_{\exp }(t;\eta )={(-\eta )}^t\Gamma (-t),$$

with the branch fixed by the principal determination of the logarithm.

On the other hand, the jump function corresponding to the rational umbral Borel functional $f(t;\eta )=1/(1-\eta {\mathrm{e}}^u)$ is

$$J_{\mathrm{rat}}(t;\eta )={(-\eta )}^t\Gamma (-t)\Gamma (1+t)=-{(-\eta )}^t{\displaystyle \frac{\pi }{\sin \left(\pi t\right)}}.$$

The two jump functions are related by spectral duality, namely:

$$J_{\exp }(t;\eta )={\displaystyle \frac{J_{\mathrm{rat}}(t;\eta )}{\Gamma (1+t)}}.$$

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Let $\phi \left(t\right)$ be a meromorphic ground state function with at most exponential growth on vertical lines compatible with the Barnes kernel below.

The analytic umbral pairing of an umbral Borel functional $f\left(\eta {\mathrm{e}}^u\right)$ with $\phi \left(t\right)$ is defined by the Mellin–Barnes integral

$$F\left(\eta \right)=\langle f\left(\eta {\mathrm{e}}^u\right),\phi \rangle :={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathcal{C}}{J}_f(t;\eta )\phi \left(t\right)\mathrm{d}t,$$

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where ${\mathcal{C}}_t$ is an upward oriented Barnes contour separating the poles of $\Gamma (-t)$ at

$$t=0,1,2,\dots$$

from the other singularities of the integrand.

In particular, the pairing with $\phi$ of the exponential umbral Borel functional ${\mathrm{e}}^{\eta {\mathrm{e}}^u}$ is

$$〈{\mathrm{e}}^{\eta {\mathrm{e}}^u},\phi \left(t\right)〉={\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathcal{C}}_t}{(-\eta )}^t\Gamma (-t)\phi \left(t\right)\mathrm{d}t,$$

The power ${(-\eta )}^t$ is defined by

$${(-\eta )}^t=\exp \left(t\mathrm{Log}(-\eta )\right),$$

where Log denotes the principal logarithm unless otherwise stated. Thus the initial branch is taken in the cut plane

$$\eta \in \mathbb{C}\setminus [0,\infty ),$$

and other determinations are obtained by analytic continuation.

In the present paper the relevant ground state is

$${\Phi }_n\left(t\right)={\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}.$$

Therefore the analytic realisation of the n-th Mittag–Leffler object is

$$I_n\left(\eta \right)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathcal{C}}_t}{(-\eta )}^t\Gamma (-t){\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}\mathrm{d}t.$$

Using the reflection formula, this may also be written as

$$I_n\left(\eta \right)=-{\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathcal{C}}_t}{\displaystyle \frac{\pi {(-\eta )}^t}{\sin \left(\pi t\right)}}{\displaystyle \frac{1}{\Gamma (1+t/n)}}\mathrm{d}t,$$

with the sign fixed by the upward orientation of the Barnes contour.

Closing the contour to the right gives the Mittag–Leffler expansion. Indeed, the crossed poles are those of $\Gamma (-t)$ at $t=m$, $m=0,1,2,\dots$, and

$$\underset{t=m}{\mathrm{Res}}\Gamma (-t)={\displaystyle \frac{{(-1)}^{m+1}}{\Gamma (1+m)}}$$

with the sign compensated by the clockwise closure convention. Equivalently, using the displayed Barnes orientation, the residue contribution is

$$\underset{t=m}{\mathrm{Res}}\left[{(-\eta )}^t\Gamma (-t){\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}\right]=-{\displaystyle \frac{{\eta }^m}{\Gamma (1+m/n)}}.$$

The additional minus sign is cancelled by the orientation of the closed contour, and one obtains

$$I_n\left(\eta \right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\eta }^m}{\Gamma (1+m/n)}}=E_{1/n,1}\left(\eta \right).$$

The aim of this section is deliberately modest. We explicitly analyse the cases $n=2$ and $n=3$, because these are the cases used in the previous formal discussion, and then record the general n-fold pattern. We do not develop here the full sectorial asymptotic theory of these Mellin–Barnes integrals. The analytic framework is used in the present article only to provide a precise contour realisation of the formal umbral identities and to indicate the natural setting in which analytic continuation and asymptotic questions may subsequently be studied. We refer the reader to [14] for further details.

4.1. Spectral Transmutation and Mellin Dilation

The analytic counterpart of the formal identity used above is

$$〈{\mathrm{e}}^{\eta {\mathrm{e}}^u},{\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}〉=〈{\displaystyle \frac{1}{1-\eta {\mathrm{e}}^{u/n}}},{\displaystyle \frac{1}{\Gamma (1+t)}}〉=E_{1/n,1}\left(\eta \right).$$

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The first identity should be understood as the combination of two operations: a transmutation from the exponential Borel functional to the rational Borel functional, and a Mellin dilation of the spectral variable.

As proved in [14] in the analytic framework and already evident at the formal level, given any admissible ground state function $\phi \left(t\right)$, the following identity holds:

$$〈{\mathrm{e}}^{\eta {\mathrm{e}}^u},\phi \left(t\right)〉=〈{\displaystyle \frac{1}{1-\eta {\mathrm{e}}^u}},{\displaystyle \frac{\phi \left(t\right)}{\Gamma (1+t)}}〉.$$

We say that the pairing of the exponential Borel functional with $\phi \left(t\right)$ is equivalent to the pairing of the rational Borel functional with the transmuted ground state $\phi \left(t\right)/\Gamma (1+t)$. This is an immediate consequence of the spectral duality holding between the Mellin representatives of the two Borel functionals; see Equation (44).

Equation (46) is obtained by combining this transmutation with a further dilation of the Mellin–Barnes variable t, as clarified below.

With the previously described conventions, the analytic pairing corresponding to the formal cyclic exponential is

$$I_n\left(\eta \right)=〈{\mathrm{e}}^{\eta {\mathrm{e}}^u},{\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}〉={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathcal{C}}{(-\eta )}^t\Gamma (-t){\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}\mathrm{d}t.$$

Using $\Gamma (-t)\Gamma (1+t)=-\pi /\sin \left(\pi t\right)$, this can also be written as

$$\begin{array}{cc}\hfill I_n\left(\eta \right)& =〈{\displaystyle \frac{1}{1-\eta {\mathrm{e}}^u}},{\displaystyle \frac{1}{\Gamma (1+t/n)}}〉\hfill \\ \hfill & =-{\displaystyle \frac{1}{2\pi i}}{\int }_{\mathcal{C}}{(-\eta )}^t{\displaystyle \frac{\pi }{\sin \left(\pi t\right)}}{\displaystyle \frac{1}{\Gamma (1+t/n)}}\mathrm{d}t.\hfill \end{array}$$

Closing the contour to the right gives the primary expansion

$$I_n\left(\eta \right)={\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \frac{{\eta }^m}{\Gamma (1+m/n)}}=E_{1/n,1}\left(\eta \right).$$

Indeed, the poles crossed by the contour are located at

$$t=m,m=0,1,2,\dots .$$

With the orientation convention fixed above, their residue contribution is

$$\underset{t=m}{\mathrm{Res}}\left[{(-\eta )}^t\Gamma (-t){\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}\right]={\displaystyle \frac{{\eta }^m}{\Gamma (1+m/n)}}.$$

Thus the formal umbral object of the previous sections is analytically the Mittag–Leffler function $E_{1/n,1}$.

Performing the change of variable $t=ns$ yields

$$I_n\left(\eta \right)=-{\displaystyle \frac{n}{2\pi i}}{\int }_{{\mathcal{C}}_s}{(-\eta )}^{ns}{\displaystyle \frac{\pi }{\sin \left(n\pi s\right)}}{\displaystyle \frac{1}{\Gamma (1+s)}}\mathrm{d}s.$$

This Mellin–Barnes integral corresponds to the rational pairing

$$I_n\left(\eta \right)=〈{\displaystyle \frac{1}{1-\eta {\mathrm{e}}^{u/n}}},{\displaystyle \frac{1}{\Gamma (1+t)}}〉.$$

This identity is the analytic version of the formal transformations used in the previous sections.

4.2. The Quadratic Case

For $n=2$ one obtains

$$I_2\left(\eta \right)=E_{1/2,1}\left(\eta \right),$$

with the closed form

$$E_{1/2,1}\left(\eta \right)={\mathrm{e}}^{{\eta }^2}\mathrm{erfc}(-\eta ).$$

The rational decomposition

$${\displaystyle \frac{1}{1-\eta {\mathrm{e}}^{u/2}}}={\displaystyle \frac{1}{1-{\eta }^2{\mathrm{e}}^u}}+{\displaystyle \frac{\eta {\mathrm{e}}^{u/2}}{1-{\eta }^2{\mathrm{e}}^u}}$$

induces the splitting

$$I_2\left(\eta \right)=I_{2,0}\left(\eta \right)+I_{2,1}\left(\eta \right),$$

where

$$I_{2,0}\left(\eta \right)={\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{2q}}{\Gamma (1+q)}}={\mathrm{e}}^{{\eta }^2}$$

and

$$I_{2,1}\left(\eta \right)={\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{2q+1}}{\Gamma (q+3/2)}}=\eta E_{1,3/2}\left({\eta }^2\right).$$

The identity with $\eta E_{1,3/2}\left({\eta }^2\right)$ is meant in the defining Mittag–Leffler sense. In particular,

$$E_{1,3/2}\left(z\right)={\displaystyle \frac{{\mathrm{e}}^z\mathrm{erf}\left(\sqrt{z}\right)}{\sqrt{z}}},$$

with the branch fixed by analytic continuation from the origin. Hence, for $\eta =ix$, the odd component reduces to the Gaussian sine/Faddeeva companion, while the even component gives the Gaussian factor

$$I_{2,0}\left(ix\right)={\mathrm{e}}^{-x^2}.$$

This is precisely the analytic counterpart of the Gaussian trigonometric functions introduced formally in Section 1.

The same splitting is obtained from the Mellin–Barnes kernel through the elementary identity

$${\displaystyle \frac{\pi }{\sin \left(2\pi s\right)}}={\displaystyle \frac{\pi }{2}}\left(\cot \left(\pi s\right)+\tan \left(\pi s\right)\right).$$

Equivalently, since

$$\cot \left(\pi \left(s-{\displaystyle \frac{1}{2}}\right)\right)=-\tan \left(\pi s\right),$$

one may write

$${\displaystyle \frac{\pi }{\sin \left(2\pi s\right)}}={\displaystyle \frac{1}{2}}\left[\pi \cot \left(\pi s\right)-\pi \cot \left(\pi \left(s-{\displaystyle \frac{1}{2}}\right)\right)\right].$$

The reason why this identity gives the desired Mellin–Barnes splitting is that the two cotangent kernels separate the two pole lattices. After the change of variable $t=2s$, the poles of the kernel

$${\displaystyle \frac{\pi }{\sin \left(2\pi s\right)}}$$

are located at $s={\displaystyle \frac{m}{2}}(m=0,1,2,\dots )$. They split into the integer and half-integer sublattices

$$s=q,s=q+{\displaystyle \frac{1}{2}}(q=0,1,2,\dots ).$$

The first term, $\pi \cot \left(\pi s\right)$, has simple poles on the integer lattice, whereas the second term, $\pi \cot \left(\pi \right(s-1/2\left)\right)$, has simple poles on the half-integer lattice. Moreover the residues reproduce the residues of the original kernel:

$$\underset{s=q}{\mathrm{Res}}{\displaystyle \frac{\pi }{\sin \left(2\pi s\right)}}={\displaystyle \frac{1}{2}}=\underset{s=q}{\mathrm{Res}}{\displaystyle \frac{\pi }{2}}\cot \left(\pi s\right)$$

and

$$\underset{s=q+1/2}{\mathrm{Res}}{\displaystyle \frac{\pi }{\sin \left(2\pi s\right)}}=-{\displaystyle \frac{1}{2}}=\underset{s=q+1/2}{\mathrm{Res}}\left[-{\displaystyle \frac{\pi }{2}}\cot \left(\pi \left(s-{\displaystyle \frac{1}{2}}\right)\right)\right].$$

Consequently the two terms in the cotangent decomposition collect respectively the even and odd residue subseries. This is the Mellin–Barnes counterpart of the elementary cyclic splitting of the power series into the classes $m=2q$ and $m=2q+1$.

Thus the partial fraction factorisation of the rational umbral kernel and the spectral splitting of the Mellin kernel are the same operation seen in two languages.

4.3. The Cubic Case

We now pass from the quadratic decomposition to the cubic one. This is the first case in which the cyclic splitting is no longer an even–odd decomposition, but a genuine three-sector projection associated with the cubic roots of unity. The aim of this subsection is to show that the formal cubic identities obtained earlier are reproduced analytically by splitting the corresponding Mellin–Barnes kernel into three shifted pole lattices.

For $n=3$ one has

$$I_3\left(\eta \right)=E_{1/3,1}\left(\eta \right).$$

The rational decomposition is

$${\displaystyle \frac{1}{1-\eta {\mathrm{e}}^{u/3}}}={\displaystyle \frac{1}{1-{\eta }^3{\mathrm{e}}^u}}+{\displaystyle \frac{\eta {\mathrm{e}}^{u/3}}{1-{\eta }^3{\mathrm{e}}^u}}+{\displaystyle \frac{{\eta }^2{\mathrm{e}}^{2u/3}}{1-{\eta }^3{\mathrm{e}}^u}}.$$

Accordingly,

$$I_3\left(\eta \right)=I_{3,0}\left(\eta \right)+I_{3,1}\left(\eta \right)+I_{3,2}\left(\eta \right),$$

with

$$I_{3,0}\left(\eta \right)={\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{3q}}{\Gamma (1+q)}}={\mathrm{e}}^{{\eta }^3},$$

$$I_{3,1}\left(\eta \right)={\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{3q+1}}{\Gamma (q+4/3)}}=\eta E_{1,4/3}\left({\eta }^3\right),$$

and

$$I_{3,2}\left(\eta \right)={\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{3q+2}}{\Gamma (q+5/3)}}={\eta }^2{E}_{1,5/3}\left({\eta }^3\right).$$

Taking $\eta =-x$ gives the cubic Gaussian components displayed formally in Section 2:

$$I_{3,0}(-x)={\mathrm{e}}^{-x^3},$$

$$I_{3,1}(-x)=-x{E}_{1,4/3}(-x^3),$$

and

$$I_{3,2}(-x)=x^2{E}_{1,5/3}(-x^3).$$

Here the sign follows directly from $\eta =-x$, since ${\eta }^3=-x^3$.

The Mellin–Barnes splitting is governed by

$${\displaystyle \frac{\pi }{\sin \left(3\pi s\right)}}={\displaystyle \frac{1}{3}}\left({\displaystyle \frac{\pi }{\sin \left(\pi s\right)}}-{\displaystyle \frac{\pi }{\sin \left(\pi \left(s-\frac{1}{3}\right)\right)}}+{\displaystyle \frac{\pi }{\sin \left(\pi \left(s-\frac{2}{3}\right)\right)}}\right).$$

This is the analytic origin of the three cyclic components obtained formally by the cubic roots of unity.

4.4. General Cyclic Decomposition and Fourier-Transform Synthesis

The preceding cases are the first members of the general decomposition

$${\displaystyle \frac{1}{1-\eta {\mathrm{e}}^{u/n}}}={\displaystyle \sum _{r=0}^{n-1}}{\displaystyle \frac{{\eta }^r{\mathrm{e}}^{ru/n}}{1-{\eta }^n{\mathrm{e}}^u}}.$$

Consequently,

$$I_n\left(\eta \right)={\displaystyle \sum _{r=0}^{n-1}}{I}_{n,r}\left(\eta \right),$$

where

$$I_{n,r}\left(\eta \right)={\eta }^r{\displaystyle \sum _{q=0}^{\infty }}{\displaystyle \frac{{\eta }^{nq}}{\Gamma \left(1+q+\frac{r}{n}\right)}}={\eta }^r{E}_{1,1+r/n}\left({\eta }^n\right).$$

Equivalently, if $\omega ={\mathrm{e}}^{2\pi i/n}$, then

$$I_{n,r}\left(\eta \right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=0}^{n-1}}{\omega }^{-rk}{I}_n\left({\omega }^k\eta \right).$$

This root-of-unity projector selects the powers $m\equiv r\left(\mathrm{mod}n\right)$ in the primary expansion of $E_{1/n,1}\left(\eta \right)$.

The Fourier-type construction of Section 3 can be summarised in the same analytic language. We use the convention

$$\mathcal{F}\left[f\right]\left(k\right)={\int }_{-\infty }^{\infty }f\left(x\right){\mathrm{e}}^{-ikx}\mathrm{d}x.$$

The corresponding umbral-analytic, or spectral, primary definition is

$${\mathfrak{F}}_n\left[f\right]\left(k\right)=〈\mathcal{F}\left[f\right]\left(k{\mathrm{e}}^u\right),{\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}〉.$$

Indeed,

$$\mathcal{F}\left[f\right]\left(k{\mathrm{e}}^u\right)={\int }_{-\infty }^{\infty }f\left(x\right){\mathrm{e}}^{-ikx{\mathrm{e}}^u}\mathrm{d}x.$$

Therefore, whenever the pairing may be interchanged with the spatial integral, one obtains

$${\mathfrak{F}}_n\left[f\right]\left(k\right)={\int }_{-\infty }^{\infty }f\left(x\right)〈{\mathrm{e}}^{-ikx{\mathrm{e}}^u},{\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}〉\mathrm{d}x,$$

that is,

$${\mathfrak{F}}_n\left[f\right]\left(k\right)={\int }_{-\infty }^{\infty }f\left(x\right)E_{1/n,1}(-ikx)\mathrm{d}x.$$

Thus ${\mathfrak{F}}_n$ is the analytic realisation of the formal Fourier-type transform ${\mathcal{F}}_{1/n}$ introduced in Section 3. From this point of view, ${\mathfrak{F}}_n$ should be understood as a spectral deformation of the ordinary Fourier transform at the level of the forward kernel. No inversion theorem or Plancherel-type identity is asserted here.

The Gaussian example in Section 3 is a concrete instance of this principle. For $n=2$ and $f\left(x\right)={\mathrm{e}}^{-a{x}^2}$, with $a>0$, one obtains

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2}{E}_{{\displaystyle \frac{1}{2}},1}(-ikx)\mathrm{d}x=\sqrt{{\displaystyle \frac{\pi }{a+k^2}}},$$

with the branch fixed by analytic continuation from the disk of convergence of the corresponding power series. The sign in the argument of $E_{{\displaystyle \frac{1}{2}},1}$ is dictated by the Fourier convention ${\mathrm{e}}^{-ikx}$; for even test functions such as ${\mathrm{e}}^{-a{x}^2}$, the final value is unchanged if the opposite sign is used, but the present sign is the consistent one.

The analytic framework therefore reorganises the formal construction as follows: the finite cyclic decompositions of the previous sections are root-of-unity splittings of a Mellin–Barnes kernel, while the associated Fourier-type transforms are spectral deformations of the ordinary Fourier transform by the same Gamma-ratio ground state.

4.5. Analytic Synthesis of the Cyclic Construction

The analysis carried out in this section gives a contour-level interpretation of the formal cyclic decompositions developed in the preceding sections. The fundamental analytic object is the Mellin–Barnes pairing, reproducing and generalising the formal construction. We prove, in particular, that the quadratic and cubic examples are special cases of a general spectral transmutation between exponential and rational umbral functionals.

The cyclic components are analytic objects in the same sense. They may be obtained either from the root-of-unity factorisation of the rational kernel or from the splitting of the Mellin factor $\pi /\sin \left(n\pi s\right)$. This gives the contour realisation of the higher-order trigonometric components introduced formally in Section 2 and Section 3.

The same analytic point of view also clarifies the Fourier-type transform construction, whose natural definition is spectral:

$${\mathfrak{F}}_n\left[f\right]\left(k\right)=〈\mathcal{F}\left[f\right]\left(k{\mathrm{e}}^u\right),{\displaystyle \frac{\Gamma (1+t)}{\Gamma (1+t/n)}}〉.$$

Whenever the pairing can be exchanged with the spatial integral, this becomes a Fourier-type transform with a Mittag–Leffler kernel. The distinction is important: the spatial representation is useful, but it may impose stronger convergence restrictions than the primary spectral pairing.

In summary, the analytic framework does not replace the formal umbral construction. Rather, it explains why the same cyclic functions arise from rational factorisation, root-of-unity projection, Mellin–Barnes splitting, and the spectral deformation of Fourier transforms.

Several questions remain open. A full inversion theory for the umbral Fourier-type transform has not been developed here. The admissible classes of test functions and distributions should be characterised in the analytic-functional setting. Finally, the asymptotic and Stokes behaviour of the cyclic components, already visible from the Mellin–Barnes representation, deserves a separate investigation. These directions will be addressed in future work.

5. Concluding Remarks and Perspectives

The analysis developed in the previous sections shows that the umbral formalism provides a consistent operational framework for extending Fourier analysis to kernels generated by non-standard exponential functions. The basic mechanism is the replacement of the ordinary factorial structure by the Gamma factors encoded in the umbral vacuum. In the simplest case, this is expressed by the prescription

$${\widehat{c}}^r{\phi }_0={\displaystyle \frac{1}{\Gamma (r+1)}}.$$

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This rule allows one to rewrite generalised $G_n$-exponentials in terms of ordinary exponential structures, while preserving the algebraic backbone of the Fourier-transform construction.

The main result of this work is that the forward G-Fourier-type transform can be evaluated by reducing it to a standard Fourier transform, provided that the umbral substitution is performed together with the correct vacuum prescription. As shown by the Fourier-type examples discussed above, the consistency of the procedure relies on replacing the ordinary factorial structure with the Gamma factors prescribed by the umbral vacuum.

A natural and significant extension of the present framework is obtained by considering the general family of ${\mathrm{e}}^{-x^n}$-type exponentials, which can be treated within the same umbral scheme. In this case, the decomposition with respect to the n-th roots of unity provides the organising principle. If

$${\omega }_k={\mathrm{e}}^{2\pi ik/n},k=0,1,\dots ,n-1,$$

then the associated algebraic factorisation leads to a splitting of the generalised exponential into elementary cyclic components. Correspondingly, the transform decomposes into branches indexed by the roots ${\omega }_k$. The quadratic and cubic cases treated above therefore appear as particular instances of a single cyclic construction.

This roots-of-unity structure suggests that the hierarchy of G-Fourier-type transforms can be organised within a unified algebraic scheme. The role of the umbral operator is precisely to encode the combinatorial and Gamma-normalised structure attached to this cyclic decomposition. In this respect, the formalism preserves the operational simplicity of the Gaussian case, while extending it to a wider class of generalised exponential kernels.

The inverse transform requires a separate analysis. Although the forward G-Fourier-type transform is naturally defined through the umbral correspondence, the construction of an inverse transform depends on the correct identification of the dual vacuum and of the corresponding inverse umbral action. One expects that the same roots-of-unity decomposition should play a role in the reconstruction formula, possibly through conjugate cyclic branches. However, a complete and unambiguous inversion theorem has not been established here and remains a problem for future work.

The same cyclic mechanism may also be applied to other families of special functions. In particular, the interpretation of the cyclic decomposition as a spectral transmutation followed by a Mellin dilation suggests further developments for Bessel and Bessel–Clifford functions. Promising results in this direction have already been obtained and will be presented elsewhere.

It would also be interesting to understand whether analogous umbral mechanisms can be formulated for kernels governed by genuinely periodic or doubly periodic structures, such as elliptic functions, although this lies beyond the scope of the present work.

In conclusion, the umbral decomposition developed in this paper provides an operationally transparent extension of Fourier analysis from the classical Gaussian kernel to a class of generalised exponential kernels. The emergence of roots-of-unity decompositions gives a unified algebraic explanation of the quadratic, cubic and higher-order cases. The formulation of the inverse transform, together with possible extensions to more general function classes, remains an open direction for subsequent investigation.