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Revised 5 September 2023

Accepted 5 October 2023

Available Online 29 November 2023

- DOI
- https://doi.org/10.55060/s.atmps.231115.009
- Keywords
- Umbral methods

Operator theory

Bessel function

Hermite polynomials

Trigonometric functions

Fresnel integral - Abstract
This article presents an overview of Umbral calculus and of its wide range of applicability. The aim is to provide new tools, embedding umbral, symbolic and operatorial methods to be exploited in pure and applied mathematics, in the research of analytical or numerical solutions in different fields.

- Copyright
- © 2023 The Authors. Published by Athena International Publishing B.V.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (https://creativecommons.org/licenses/by-nc/4.0/).

## 1. INTRODUCTION

The term *Umbra*^{1} has been diffused by S. Roman and G.C. Rota [1] in the second half of the 20th century to underline, in the emerging field of operational calculus, the practice of replacing the representation in series of certain function
^{2}. Earlier, other mathematicians like J. Blissard in *Theory of Generic Equations* or Edouard Lucas in *Théorie nouvelle des nombres de Bernoulli et d'Euler*, established the rules which allowed for viewing different functions through the same abstract entity and showed its usefulness for a wide range of applications. In the current research, the same starting point has been used but with methods closer to the *Heaviside* point of view [2], by proposing strategies combining different technicalities and operational methods which yield the formulation of a powerful “different mathematical language” [3]. The umbral image, the key element to establish the rules to replace higher transcendental functions in terms of elementary functions, and rewrite complex problems into simplified exercises, will be the starting point to fix the criteria to take advantage from such a replacement, based on the Laplace and Borel transform Theory and the Principle of Permanence of the Formal Properties.

To introduce the umbral operator, we have to provide some preliminary definitions. The operator
*umbral*, is a shift operator
^{3}, which is a function that will change according to the problem to solve [3]. If the initial vacuum is, for example, the function:

^{4}

By that definition, the operator

## 2. SPECIAL FUNCTIONS

By using the operator in Eq. 25 and the property of

So we have formally reduced a transcendental function to an exponential form, downgrading the “complexity” of the function.

The umbral image is not unique indeed, if we chose another vacuum, the action of the umbral operator will yield a different function. For example, if we introduce the pair umbral operator/vacuum such that:

### Example 1.

This kind of new “interpretation” of a certain function opens a wide range of possibilities. To this aim, we provide some examples including the solution of non-trivial integrals.

### Example 2.

*We consider integrals involving the Bessel function
* [3,6].

*Then*∀ a , b ∈ ℝ : a x 2 + b x > 0 :

*where*

*is a modified Bessel function of the first kind, or*

Furthermore, we want to stress the possibility to evaluate the product of special functions by using the umbral images. However, we recall that in general the operator is not commutative. So if we want to calculate, for example,

By expanding the exponential in MacLaurin series we can finally write:

### 2.1. Hermite Polynomials

We consider now the two-variable polynomials of Hermite Kampé dè Fériét [4,8,9], also called heat polynomials:

From an operational point of view, they can be expressed as the result of the action of the operator

### Proposition 1.

*with*

*umbral operator and*

*vacuum such that:*

### Proof.

By considering the definition of the

### Proposition 2.

*The relevant Hermite function integral representation, with order real and negative too, can be written as:*

### Proof.

By considering the definition of the

Furthermore, if we introduce another application of the umbral methods, called Hermite calculus [15], we can reach the target to combine the previous results and calculate, for example, non-trivial integrals including combination of special functions. To give an idea of the methods we will use, we consider the following integral:

By that position, we can recast:

In this way, we have shown the flexibility of the technique. Indeed, we can raise the complexity of the integrals and solve them by methods as, for example, in the case of the anharmonic oscillator in the example below [15,16].

### Example 3.

*Let:*

*be the integral of the anharmonic oscillator. We can solve it by the Hermite calculus position*:

*in which we used the fractional order of the umbral operator and of the Hermite function*:

The same consideration can be done for each special function through appropriate umbral images. We list in the following just some examples for the Laguerre, Jacobi, Legendre, Tricomi-Bessel and Chebyshev functions, and the Voigt transform [12,17,18].

## 3. NUMBER THEORY

The umbral method can be applied in many different fields of pure and applied mathematics. A further example is indeed Number Theory. We remind that Harmonic Numbers are defined as [19,20,21,22]:

In terms of Laplace transform, we obtain:

*n*-th harmonic number through Euler's integral:

We can then introduce the function:

*Harmonic Number Umbral Vacuum:*

^{5}, introduce the

*Harmonic Based Exponential Function*as the series:

In an analogous way, we can treat other families of numbers like the Motzkin, telegraph, Stirling numbers, etc.

## 4. CLASSIC TRIGONOMETRY

Trigonometry can also be reinterpreted in terms of umbral images.

### Definition 1.

*We introduce,*
*the umbral vacuum:*

*and the shift operator*

*such that we can get:*

### Proposition 3.

(*Cos-exponential umbral image*) *If*
*and*

*as we expect*.

### Corollary 1.

Fresnel Integral [17]:

By applying the methods discussed so far, we can put together integrals including trigonometric functions and Hermite polynomials [23] and solve them in a simple way.

### Example 4.

## 5. NEW TRIGONOMETRIES AND THE GIELIS SUPERFORMULA

We wish to close this contribution with a view on new trigonometries and the Gielis Superformula.

### Definition 2.

*We introduce the composition rule*

*called the Laguerre binomial sum*.

Eq. (42) is based on the Laguerre exponential [24,25]:

### Definition 3.

*We introduce l-trigonometric (l-t) functions through the identity*:

*where*

*l − t cosine and l − t sine functions are specified by the series*

^{6}(Fig. 2):

We can also use matrices as arguments of the *l-sin* or *l-cos* functions (like in PDE system problems): see Fig. 3.

Finally, by using the Superformula of Johan Gielis [26], a wide range of novel shapes can be generated (Fig. 4).

We can identify

## ACKNOWLEDGMENTS

This work was supported by an ENEA Research Center individual fellowship and under the auspices of INDAM's GNFM (Italy).

## Footnotes

It is assonant to the term “Ombra” in Latin which means “shadow” in English.

The promotion of the index *n* in *c _{n}* to a power exponent of the operator

*ĉ*, the umbral operator, is the essence of umbra, since it is a kind of projection of one into the other:

This term is used to stress that the action of the operators, raised to some power, is that of acting on an appropriate set of functions by “filling” the initial state

The *l-t* functions are defined by the corresponding series expansion

## REFERENCES

### Cite This Article

TY - CONF AU - Silvia Licciardi PY - 2023 DA - 2023/11/29 TI - Umbral Calculus and New Trigonometries BT - Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022) PB - Athena Publishing SP - 95 EP - 106 SN - 2949-9429 UR - https://doi.org/10.55060/s.atmps.231115.009 DO - https://doi.org/10.55060/s.atmps.231115.009 ID - Licciardi2023 ER -