🔢 mathematics

On Mersenne-Bernoulli and Mersenne-Euler polynomials

This paper introduces Mersenne-Bernoulli and Mersenne-Euler polynomials based on the Mersenne number sequence, deriving their identities via generating functions and M-calculus, and analyzing the factorization and inverses of their associated matrices.

Original authors: Artatrana Suna, Prasanta Kumar Ray

Published 2026-07-03

📖 5 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 or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a mathematician who loves building with LEGO bricks. For a long time, you've been using a standard set of bricks called "Bernoulli" and "Euler" blocks. These are special shapes that help you build complex structures in number theory and combinatorics (the math of counting and arranging things). They are very famous and very useful.

Now, imagine you discover a new, slightly different type of brick. These are called Mersenne bricks. They are named after a famous number pattern ( $2^n - 1$ ) that is famous for helping people find the world's largest prime numbers.

This paper is about what happens when you swap your old standard bricks for these new Mersenne bricks to build two specific types of structures: Mersenne-Bernoulli polynomials and Mersenne-Euler polynomials.

Here is a breakdown of what the authors did, using simple analogies:

1. The New Rules of the Game (M-Calculus)

In the old world, when you multiplied numbers or added them up, you followed standard rules (like $1, 2, 6, 24...$ for factorials).
In this paper, the authors invent a new rulebook called M-calculus.

The New Factorial: Instead of the usual way of multiplying numbers to get a factorial, they use a "Mersenne factorial." It's like a different way of stacking your bricks.
The New Binomial Coefficients: You know how you can choose 2 items out of 5 in a specific number of ways? The authors created a "Mersenne version" of this choice. It's a new way of counting combinations that fits their new brick set.
The New Exponential: They even created a new version of the famous "e" function (used in growth and decay) that works with these new bricks.

2. Building the New Structures (The Polynomials)

Using these new rules, the authors built two new families of polynomials (mathematical expressions with variables like $x$ ):

Mersenne-Bernoulli Polynomials: These are the new versions of the classic Bernoulli polynomials.
Mersenne-Euler Polynomials: These are the new versions of the classic Euler polynomials.

Just like the old versions, these new ones have "generating functions." Think of a generating function as a magic machine. If you put a number into the machine, it spits out the entire sequence of these new polynomials. The authors wrote down the blueprints for these machines.

3. Discovering the Patterns (Identities)

Once they built these new structures, they started looking for patterns. They found that these new polynomials behave in very specific, predictable ways:

The Shift Rule: If you shift the input of the polynomial by a certain amount (using their new Mersenne addition), the result is a mix of the old polynomial and some new numbers. It's like sliding a puzzle piece and seeing how the picture changes.
The Connection: They found a direct link between the Mersenne-Bernoulli and Mersenne-Euler versions. You can turn one into the other using a specific recipe involving their new "Mersenne numbers."
The Derivative (The Mersenne-Derivative): In normal calculus, a derivative tells you the slope of a curve. The authors invented a "Mersenne-derivative." It's a special tool that measures how fast these new polynomials are changing, but it follows the Mersenne rules. They proved that if you use this tool on their new polynomials, you get a simpler version of the same polynomial back.

4. The Matrix Puzzle (The Inverse)

This is perhaps the most visual part of the paper.

The Matrix: The authors arranged these polynomials into a grid (a matrix), like a spreadsheet. Each cell in the grid contains a specific polynomial.
The Inverse: In math, every matrix has an "inverse" (a matrix that, when multiplied by the original, gives you a blank slate or an identity matrix). Finding the inverse is usually very hard.
The Solution: The authors figured out exactly how to build the "inverse" matrix for these new polynomials. They did this by breaking the big grid down into smaller, simpler pieces (factorization). They even showed a concrete example with a 3x3 grid to prove it works.

5. The "What's Next?"

The paper ends by suggesting that while they have built these new structures and found their rules, there is still a lot of mystery left. They suggest that in the future, mathematicians could try to connect these polynomials to probability (the math of chance). Imagine if these polynomials could describe the behavior of random events, like rolling dice or flipping coins, but in this new Mersenne world.

Summary

In short, the authors took a famous mathematical concept (Bernoulli and Euler polynomials), replaced the standard arithmetic rules with a new set of rules based on Mersenne numbers, and successfully built new mathematical objects. They then mapped out the rules these new objects follow, showed how they relate to each other, and figured out how to reverse-engineer them using matrices. It is a study of new mathematical tools created by changing the fundamental rules of the game.

Technical Summary: On Mersenne-Bernoulli and Mersenne-Euler Polynomials

Problem Statement and Motivation
Special polynomials, particularly Bernoulli and Euler polynomials, are fundamental tools in combinatorics, number theory, and mathematical analysis due to their structured recurrence relations and generating functions. Recent literature has seen a surge in exploring "degenerate," "probabilistic," and "hybrid" forms of these polynomials. Motivated by these developments and the significance of Mersenne numbers ( $M_n = 2^n - 1$ ) in the search for large primes, this study introduces a new class of polynomials: the Mersenne-Bernoulli and Mersenne-Euler polynomials. The work aims to establish their fundamental properties, derive associated identities, and investigate the matrix structures formed by these polynomials.

Methodology
The authors employ a framework based on M-calculus, which adapts classical calculus concepts to the sequence of Mersenne numbers. Key methodological components include:

M-Definitions: The study defines the M-factorial ( $M_n!$ ), M-binomial coefficients ( $\binom{n}{k}_M$ ), and the M-exponential function ( $e_M^{xt}$ ). These definitions rely on the recursive properties of Mersenne numbers ( $M_{n+1} = 2M_n + 1$ ).
Generating Functions: The Mersenne-Bernoulli polynomials ( $B_{n,M}(x)$ $B_{n, M} (x)$ ) and Mersenne-Euler polynomials ( $E_{n,M}(x)$ $E_{n, M} (x)$ ) are defined via their exponential generating functions, analogous to the classical definitions but utilizing the M-exponential function:
- $\frac{t e_M^{xt}}{e_M^t - 1} = \sum_{n=0}^{\infty} B_{n,M}(x) \frac{t^n}{M_n!}$
- $\frac{2 e_M^{xt}}{e_M^t + 1} = \sum_{n=0}^{\infty} E_{n,M}(x) \frac{t^n}{M_n!}$
M-Derivative and M-Integral: A specific operator, the Mersenne-derivative ( $D_t$ ), is defined as $D_t(f(t)) = \frac{f(2t) - f(t)}{t}$ for $t \neq 0$ . This operator satisfies properties similar to the conventional derivative, such as the product rule and linearity. Correspondingly, the Mersenne-integral is defined as the anti-derivative with respect to this operator.
Matrix Factorization: The authors construct square matrices of order $n$ composed of these polynomials and utilize algebraic factorization to determine their inverses.

Key Contributions and Results

Fundamental Identities: The paper derives several core identities, including:
- Binomial-type expansions: $B_{n,M}(x +_M y) = \sum_{k=0}^n \binom{n}{k}_M B_{k,M}(x) y^{n-k}$ .
- Difference equations: $B_{n,M}(x +_M 1) - B_{n,M}(x) = M_n x^{n-1}$ and $E_{n,M}(x +_M 1) + E_{n,M}(x) = 2x^n$ .
- Explicit formulas: Expressions for $B_{n,M}(x)$ and $E_{n,M}(x)$ in terms of Mersenne-Bernoulli/Euler numbers ( $B_{k,M}, E_{k,M}$ ) and powers of $x$ .
- Recurrence relations: Recursive formulas for the numbers $B_{n,M}$ and $E_{n,M}$ derived by setting $x=0$ .
Convolution and Operator Relations:
- The study establishes relationships between the two polynomial classes, such as $E_{n-1,M}(x) = -\frac{2}{M_n} \sum_{k=1}^n \binom{n}{k}_M E_{k,M} B_{n-k,M}(x)$ .
- Using the Mersenne-derivative, the authors prove that $D_x(B_{n,M}(x)) = M_n B_{n-1,M}(x)$ and $D_x(E_{n,M}(x)) = M_n E_{n-1,M}(x)$ .
- Convolution identities involving products of these numbers are derived using the $D_t$ operator on generating functions.
Mersenne-Integrals: The paper calculates definite Mersenne-integrals of the form $\int_0^1 x^n B_{m,M}(x) dM(x)$ and $\int_0^1 x^n E_{m,M}(x) dM(x)$ , providing explicit summation formulas involving Mersenne-binomial coefficients.
Matrix Theory:
- The Mersenne-Pascal matrix $M_n(x)$ is defined, and its inverse is cited from existing literature.
- The Mersenne-Bernoulli matrix $B_n(x)$ and Mersenne-Euler matrix $E_n(x)$ are introduced.
- Inversion Formulas: The paper provides explicit inverses for these matrices:
  - $B_n^{-1} = Q_n$ , where $Q_n$ is a matrix involving Mersenne-binomial coefficients and reciprocals of Mersenne numbers.
  - $E_n^{-1} = \frac{1}{2}(M_n + I_n)$ , where $I_n$ is the identity matrix.
- Product Formulas: The matrices satisfy product relations such as $B_n(x +_M y) = B_n(x) M_n(y)$ , allowing for the factorization of polynomial matrices.

Significance and Claims
The paper claims to extend the theory of special polynomials by connecting them to Mersenne numbers, thereby providing a new avenue for research in number theory and combinatorics. The authors position their work as a systematic investigation that establishes the foundational properties of these new polynomials.

Regarding future directions, the paper modestly suggests that a significant area for further study is the probabilistic interpretation of these polynomials. Specifically, the authors propose constructing associated random variables whose moments correspond to the Mersenne-Bernoulli and Mersenne-Euler polynomials, potentially leading to new recurrence relations (such as Spivey-type) similar to those found in degenerate Bell and Dowling polynomials. The paper does not claim immediate applications in physics or engineering but focuses on the structural and algebraic properties within pure mathematics.