Le Roy, Lerch and Legendre chi functions and generalised Borel-Le Roy transform

Imagine you are a master chef trying to understand a vast, chaotic pantry of ingredients. Some ingredients are common (like flour and sugar), but others are exotic, rare, and behave in very strange ways. In the world of mathematics, these "exotic ingredients" are called Special Functions. They are complex formulas that pop up everywhere, from describing how particles move in physics to solving problems in pure math.

This paper is like a new, revolutionary cookbook written by Giuseppe Dattoli and Roberto Ricci. Instead of treating each exotic ingredient as a separate mystery, they propose a unified kitchen tool called Indicial Umbral Theory (IUT).

Here is a simple breakdown of what they are doing, using everyday analogies:

1. The Problem: A Messy Pantry

For a long time, mathematicians studied these special functions (like the Le Roy, Lerch, and Legendre functions) one by one. It was like having a different set of rules for every single spice.

Le Roy function: Useful for fractional calculus (think of it as measuring "half a step" instead of a whole step).
Lerch transcendent: Used in physics for things like Bose-Einstein statistics (how particles crowd together).
Legendre chi function: Another complex formula with its own specific rules.

The challenge was: Is there a single way to handle all of them?

2. The Solution: The "Magic Shaker" (Umbral Theory)

The authors introduce a concept called Umbral Theory. Imagine a "Magic Shaker" (the Umbral Operator).

Normally, if you have a list of numbers (a series), you have to calculate them one by one.
With this "Magic Shaker," you can treat a whole complex list of numbers as if it were a simple, single object (like a monomial).
The Analogy: Imagine you have a recipe that requires you to add 1 cup of flour, then 2 cups, then 6 cups, then 24 cups... (a factorial pattern). Instead of measuring each one, the "Magic Shaker" lets you just say, "Add the Flour-Sequence," and the machine handles the complexity automatically.

The authors reformulated this theory to make it even more powerful, allowing them to handle not just neat, converging lists, but also messy, diverging lists (infinite sums that usually blow up and don't make sense).

3. The Three Main Ingredients

The paper takes three specific "exotic ingredients" and shows how the Magic Shaker simplifies them:

The Le Roy Function:
- What it is: A function that generalizes the famous exponential function ( $e^x$ ).
- The Paper's Trick: They show that by using their Magic Shaker, you can easily calculate derivatives (slopes) and integrals (areas under the curve) of this function. It's like turning a difficult calculus problem into simple algebra. They also show how to "resum" (fix) divergent series using a Borel-Le Roy transform, which is like a "strainer" that catches the useful parts of a broken formula and throws away the noise.
The Lerch Transcendent:
- What it is: A master key that unlocks many other famous functions, including the Riemann Zeta function (famous for the unsolved Riemann Hypothesis).
- The Paper's Trick: They show that this function is just a "Gaussian" (a bell curve) being shaken by their Magic Shaker. This allows them to easily find the derivatives of the function, which would otherwise be a nightmare to calculate. They even use this to define "polylogarithms of non-integer order"—basically, creating new types of numbers that sit between whole numbers.
The Legendre Chi Function:
- What it is: A function related to the Lerch one, often used in geometry and number theory.
- The Paper's Trick: They break this function down into two simpler parts (like splitting a complex dish into a savory and a sweet component) and show how the Magic Shaker manipulates them effortlessly.

4. The "Strainer" for Broken Formulas (Resummation)

One of the most exciting parts of the paper is how they handle divergent series.

The Analogy: Imagine a recipe that calls for adding 1 cup, then 2 cups, then 4 cups, then 8 cups... of water. Eventually, the pot overflows, and the recipe makes no sense. In math, this is a divergent series.
The Fix: The authors use a technique called the Borel-Le Roy transform. Think of this as a special strainer. Even though the pot is overflowing, the strainer can separate the "good broth" (the meaningful mathematical value) from the "overflowing water" (the nonsense). This allows mathematicians to assign a real, useful value to formulas that previously seemed broken.

5. The Polygamma Connection

They also briefly touch on the Polygamma function (related to the Gamma function, which generalizes factorials). They show that even this function, which seemed a bit disconnected from their "Magic Shaker" method, can be reinterpreted as a special case of the Lerch function. It's like realizing that a mysterious spice you thought was unique is actually just a variation of a spice you already knew how to use.

The Big Picture

In simple terms, this paper says:

"We have built a universal remote control (Indicial Umbral Theory) for the most complicated mathematical functions. Instead of learning a new button for every function, you can now use this one remote to calculate, simplify, and even fix broken formulas for the Le Roy, Lerch, and Legendre functions. This makes solving complex problems in physics and math much faster and more elegant."

The authors are essentially saying that by viewing these functions through the lens of "power series" and using their "Magic Shaker," they can uncover hidden patterns and solve problems that were previously too difficult or messy to handle.

Here is a detailed technical summary of the paper "Le Roy, Lerch and Legendre chi functions and generalised Borel-Le Roy transform" by Giuseppe Dattoli and Roberto Ricci.

1. Problem Statement

The paper addresses the need for a unified mathematical framework to analyze and generalize several important special functions: the Le Roy function, the Lerch transcendent, and the Legendre chi function. While these functions are individually well-studied and have applications ranging from fractional calculus and stochastic differential equations to statistical physics (Bose-Einstein and Fermi-Dirac statistics) and number theory (Dirichlet L-series), they often lack a cohesive theoretical structure.

Furthermore, many of these functions are defined by power series with limited radii of convergence. A significant challenge is extending the domain of these functions beyond their convergence intervals and handling divergent series that arise in asymptotic expansions. The authors aim to demonstrate how Indicial Umbral Theory (IUT) can provide a rigorous algebraic and analytic framework to unify these functions, derive their properties (such as derivatives and integrals), and extend their applicability via resummation techniques.

2. Methodology: Indicial Umbral Theory (IUT)

The core methodology relies on a reformulation of Indicial Umbral Theory (IUT), grounded in the formal theory of power series and differential algebras.

Umbral Operators: The authors introduce a linear functional operator, denoted as $u$ , acting on a space of "ground states" (specific functions of a continuous variable $t$ ). The operator is defined such that $u^r[\phi(t)] = \phi(r)$ , where $r$ is an integer within the domain of convergence.
Ground States: Specific classes of functions are defined as ground states to represent the coefficients of the target special functions. For example:
- For the Le Roy function: $\phi^\mu(t) = \frac{1}{\Gamma(1+t)^\mu}$ .
- For the Lerch transcendent: $\nu_{\alpha, s}(t) = \frac{\Gamma(1+t)}{(t+\alpha)^s}$ .
Exponential Umbral Image: The special functions are expressed as the exponential action of the umbral operator on these ground states. For instance, a function $F(\zeta)$ is written as $e^{\zeta u}[\phi(t)]$ . This transforms the problem of manipulating series coefficients into the algebraic manipulation of the operator $u$ .
Borel-Le Roy Transform: The framework incorporates the Borel-Laplace transform to handle divergent series. By interpreting formal power series as asymptotic expansions, the authors use integral transforms to "resum" divergent series, effectively extending the domain of definition of the functions.

3. Key Contributions and Results

A. Unification of Special Functions

The authors successfully express the Le Roy, Lerch, and Legendre functions in a unified "umbral form":

Le Roy Function: $L(\zeta; \mu) = e^{\zeta u}[\phi^\mu(t)]$ .
Lerch Transcendent: $\Phi(\zeta; \alpha, s) = e^{\zeta u}[\nu_{\alpha, s}(t)]$ .
Legendre Chi Function: $\chi_s(\zeta) = \zeta \cosh(\zeta u)[\nu_{1, s}(t)]$ .

This unification allows for the derivation of general properties for all these functions using a single set of operator rules.

B. Derivation of Differential and Integral Properties

Using the umbral formalism, the authors derive closed-form expressions for:

Successive Derivatives: They show that the $n$ -th derivative of these functions corresponds to shifting the parameters of the ground state. For example, $\partial_\zeta^n \Phi(\zeta; \alpha, s) = (1)_n \Phi(\zeta; \alpha+n, 1+n, s)$ .
Integral Evaluations: The method simplifies the evaluation of infinite integrals involving these functions. For instance, they evaluate Gaussian integrals of the Le Roy function and the Kolokoltsov function, yielding results in terms of $\sqrt{\pi}$ and shifted parameters.
Polylogarithms: The paper establishes a link between the Lerch transcendent and polylogarithms ( $Li_s(\zeta)$ ), allowing for the definition of "polylogarithms of non-integer order" via fractional powers of the umbral operator ( $u^{1/2}$ ).

C. Generalized Borel-Le Roy Transform

A major contribution is the demonstration of how the Borel-Le Roy transform acts as a parameter-shifting operator.

The authors prove that the Borel transform of the Le Roy function $L(\zeta; \mu)$ yields $L(\zeta; \mu-1)$ .
They generalize this to the Generalized Borel-Le Roy Transform, showing that:
$\int_0^\infty dt \, e^{-t} t^\mu L(\zeta t^\beta; \alpha, \beta, \mu) = L(\zeta; \alpha, \beta, \mu-1)$
This identity provides a powerful tool for relating functions with different parameters and for resumming divergent series.

D. Extension to Divergent Series

The paper discusses the application of IUT to divergent series, specifically the Euler series. By replacing factorials with their integral representations (Gamma function integrals), the authors show how divergent series can be associated with convergent integral transforms. This suggests a pathway to define Le Roy functions for integer orders using multidimensional Borel-Le Roy transforms, linking them to Humbert-type Bessel functions.

E. Connection to Polygamma Functions

The authors reinterpret the Polygamma function $\psi^{(m)}(\alpha)$ within the IUT framework. They demonstrate that $\psi^{(m)}(\alpha)$ is essentially a specific case of the Lerch transcendent evaluated at $\zeta=1$ , and they propose a two-variable generalization $\psi^{(m)}(\alpha, \zeta)$ that fits naturally into the umbral formalism.

4. Significance and Impact

Theoretical Unification: The paper provides a robust algebraic framework that treats diverse special functions as instances of a single operator formalism, simplifying the derivation of complex identities.
Computational Efficiency: The umbral approach reduces complex series manipulations to algebraic operations on operators, making the calculation of derivatives and integrals more systematic.
Handling Divergence: By integrating Borel resummation techniques, the framework offers a mathematically sound method to extend the domain of special functions beyond their natural convergence radii, which is crucial for applications in physics (e.g., perturbation theory) where divergent series are common.
New Research Directions: The work opens avenues for defining "fractional" orders of special functions (e.g., non-integer order polylogarithms) and exploring multidimensional generalizations of the Borel-Le Roy transform.

In conclusion, Dattoli and Ricci demonstrate that Indicial Umbral Theory is not merely a notational convenience but a powerful analytical tool capable of unifying the study of Le Roy, Lerch, and Legendre functions, while providing new methods for their generalization and resummation.