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On Apostol-Type Mersenne-Bernoulli and Mersenne-Euler Polynomials

This paper introduces Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order $\alpha$ and utilizes M-calculus to derive their explicit series representations, addition theorems, difference equations, and convolution identities.

Original authors: Artatrana Suna, Prasanta Kumar Ray

Published 2026-06-30✓ Author reviewed ⓘ

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Artatrana Suna, Prasanta Kumar Ray

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a mathematician who loves building with blocks. For a long time, you've been using a standard set of blocks called Bernoulli and Euler polynomials. These are like the "Lego bricks" of advanced math, used to solve puzzles in number theory and physics. They have very specific rules about how they stack, grow, and interact with each other.

A few years ago, someone realized that if you tweak the rules slightly, you get even more interesting shapes. This led to "Apostol-type" polynomials, which are like the standard Lego bricks but with a special dial (a parameter called $\lambda$ ) that changes their shape.

The Big Idea of This Paper
The authors of this paper, Artatrana Suna and Prasanta Kumar Ray, decided to build a completely new set of blocks. Instead of using the standard rules of arithmetic, they decided to build everything based on a specific sequence of numbers called Mersenne numbers.

Think of Mersenne numbers as a special rhythm: 1, 3, 7, 15, 31... (each number is double the previous one, minus one). The authors asked: "What if we built our entire mathematical world using only this rhythm?"

To do this, they invented a new toolkit they call M-calculus.

The New Toolkit (M-Calculus)
In normal math, if you want to know how fast something is changing (a derivative), you look at the difference between two points very close together. In this new "M-world," the rules are different:

The M-Derivative: Instead of looking at $x$ and $x + \text{tiny}$ , this tool looks at the difference between $x$ and $2x$ (the next step in the Mersenne rhythm).
The M-Integral: This is the reverse process, like filling a bucket, but the bucket has Mersenne-shaped sides.
M-Factorials: Even the way they count (1, 2, 6, 24...) is changed to fit the Mersenne rhythm.

The New Characters: Apostol-Mersenne Polynomials
Using this new toolkit, the authors created two new families of characters:

Apostol-Mersenne-Bernoulli Polynomials
Apostol-Mersenne-Euler Polynomials

These aren't just random shapes; they are the "Mersenne versions" of the classic Lego bricks. The paper shows that these new characters behave in surprisingly familiar ways, just with a twist.

What Did They Discover?
The authors spent the paper proving that these new polynomials have a "personality" full of rules, similar to the old ones but adapted for the Mersenne rhythm:

The Recipe Book (Explicit Representations): They showed exactly how to build these polynomials. It's like a recipe: "Take a specific number of Mersenne-blocks, mix them with a parameter $\lambda$ , and you get the polynomial."
The Mixing Rule (Addition Theorems): If you have a polynomial at point $x$ and you want to know what happens at a "Mersenne-added" point ( $x +_M y$ ), you don't need to start from scratch. You can just mix the existing pieces together in a specific pattern.
The Step-Down Ladder (Difference Equations): They found a rule that connects a polynomial of a certain "order" (complexity) to one that is slightly simpler. It's like a ladder where you can step down from a complex version to a simpler one by applying a specific formula.
The Shrink Ray (M-Derivative): They proved that if you apply their special "M-derivative" tool to these polynomials, they shrink down to a simpler version of themselves, just like a video game character losing a level.
The Mixing Bowl (Convolution Identities): They showed how to mix two of these polynomials together to create a new result, finding a hidden balance between them.
The Weighted Bucket (Integrals): Finally, they figured out how to calculate the "area" under these curves using their new M-integral tool, giving them a way to measure the "weight" of these mathematical shapes.

Why Does This Matter?
The paper doesn't claim these polynomials will cure diseases or build bridges today. Instead, it's a foundational study. It's like an architect designing a new type of brick and proving it holds together, has corners, and can be stacked.

The authors suggest that in the future, people might use these "Mersenne bricks" to:

Study the zeros (where the polynomials hit zero) and draw pictures of them.
Connect them to other deep mathematical functions (like Zeta functions).
Apply this same "Mersenne rhythm" to other number sequences, like Pell or Jacobsthal numbers.

In short, the paper is a blueprint for a new mathematical universe built on the rhythm of Mersenne numbers, showing that even in this strange new world, the familiar laws of calculus and polynomials still hold true, just in a different, fascinating language.

Technical Summary: Apostol-Type Mersenne-Bernoulli and Mersenne-Euler Polynomials

Problem Statement and Context
Special polynomials, particularly the classical Bernoulli and Euler polynomials, are foundational in number theory, combinatorics, and mathematical physics. While significant generalizations exist, such as the Apostol-Bernoulli and Apostol-Euler polynomials (which incorporate a parameter $\lambda$ ), and sequence-based calculi have been developed for Fibonacci (F-calculus, Golden calculus) and Padovan sequences, there is a need to extend these frameworks to Mersenne numbers ( $M_n = 2^n - 1$ ). Mersenne numbers are historically significant in the study of perfect numbers and large primes. This paper addresses the gap in the literature by introducing a calculus based on Mersenne numbers (M-calculus) and utilizing it to define and analyze a new class of special polynomials: the Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order $\alpha$ .

Methodology
The authors construct a mathematical framework termed "M-calculus," which serves as an analog to classical calculus but is intrinsically linked to Mersenne numbers. The methodology proceeds in three stages:

Foundational Definitions: The paper establishes the basic operators of M-calculus:
- M-factorial ( $M_n!$ ): Defined recursively as $M_n! = M_n M_{n-1}!$ .
- M-binomial coefficients: Defined using M-factorials, satisfying specific symmetry and recurrence relations distinct from classical binomial coefficients.
- M-exponential function ( $e_M^{xt}$ ): Defined via a power series using M-factorials. Crucially, this function is not multiplicative in the classical sense; instead, it satisfies $e_M^{xt} e_M^{yt} = e_M^{(x+_M y)t}$ , where $+_M$ denotes M-binomial addition.
- M-derivative ( $D_x$ ): Defined as $D_x(f(x)) = \frac{f(2x) - f(x)}{x}$ . This operator acts on monomials as $D_x(x^n) = M_n x^{n-1}$ .
- M-integral: Defined as the inverse of the M-derivative, with specific integration rules derived from the product rule of the M-derivative.
Definition of Polynomials: Using the generating functions of the M-calculus, the authors define the Apostol-Mersenne-Bernoulli polynomials $B_n^{(\alpha)}(x; \lambda)$ and Apostol-Mersenne-Euler polynomials $E_n^{(\alpha)}(x; \lambda)$ of order $\alpha$ via the generating functions:
$\left( \frac{t}{\lambda e_M^t - 1} \right)^\alpha e_M^{xt} = \sum_{n=0}^\infty B_n^{(\alpha)}(x; \lambda) \frac{t^n}{M_n!}$
$\left( \frac{2}{\lambda e_M^t + 1} \right)^\alpha e_M^{xt} = \sum_{n=0}^\infty E_n^{(\alpha)}(x; \lambda) \frac{t^n}{M_n!}$
Derivation of Properties: The authors systematically derive properties by manipulating these generating functions, applying the M-derivative, and utilizing the M-binomial theorem.

Key Contributions and Results
The paper establishes a comprehensive set of identities and theorems for the newly defined polynomials:

Explicit Representations: The polynomials are shown to be monic of degree $n$ and can be expressed as sums involving the corresponding Apostol-Mersenne numbers and powers of $x$ weighted by M-binomial coefficients.
Addition Theorems: Generalized addition formulas are derived for both polynomial types, relating $P_n(x +_M y)$ to sums of products of lower-order polynomials and powers of $y$ .
Difference Equations: The paper establishes recurrence relations connecting polynomials of order $\alpha$ and $\alpha - 1$ . Specifically, it derives difference equations involving shifts by $+_M 1$ (e.g., $\lambda B_n^{(\alpha)}(x +_M 1; \lambda) - B_n^{(\alpha)}(x; \lambda)$ ).
Mixed Addition Formulas: Theorems are presented that link Apostol-Mersenne-Bernoulli polynomials with Apostol-Mersenne-Euler polynomials, providing mixed addition formulas that express one type in terms of the other.
Derivative and Integral Properties:
- The M-derivative acts as a lowering operator: $D_x(B_n^{(\alpha)}(x; \lambda)) = M_n B_{n-1}^{(\alpha)}(x; \lambda)$ .
- Closed-form expressions are derived for weighted M-integrals of these polynomials over the interval $[0, 1]$ , denoted as $I_{m,n}^{(\alpha)}$ and $J_{m,n}^{(\alpha)}$ . These results involve sums of Mersenne numbers and the underlying Apostol-Mersenne numbers.
Convolution Identities: The authors derive specific convolution identities for the Apostol-Mersenne numbers (the case where $x=0$ ), relating sums of products of these numbers to linear combinations of the numbers themselves.

Significance and Claims
The paper claims to extend the theory of special polynomials by integrating them with the arithmetic properties of Mersenne numbers through the novel M-calculus framework. The significance lies in:

Generalization: It provides a unified generalization of Apostol-type polynomials based on Mersenne sequences, analogous to previous work on Fibonacci and Padovan sequences.
Structural Completeness: The paper demonstrates that these new polynomials possess a rich algebraic structure, mirroring the classical theory with explicit formulas for derivatives, integrals, addition, and recurrence.
Future Directions: The authors modestly suggest that the established framework could be applied to other recursive sequences (such as Pell, Jacobsthal, or Perrin numbers) and that future work could computationally investigate the zeros and graphical representations of these polynomials, as well as explore connections to M-analogues of zeta functions.

The work does not propose experimental applications but rather contributes to the theoretical landscape of special functions and number theory.

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