Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers

Based on the provided paper "Hypergeometric Bernoulli Polynomials Defined on Simplicial d-Polytopic Numbers" by Ronald Orozco López, here is a detailed technical summary.

1. Problem Statement

The paper addresses the need to generalize classical Bernoulli polynomials and hypergeometric functions within the framework of simplicial $d$-polytopic numbers. While numerous generalizations of Bernoulli polynomials exist (e.g., $q$-analogs, Fibonacci-analogs, Apostol-Bernoulli), there was a lack of a unified calculus and polynomial theory specifically constructed on the combinatorial structure of $d$-dimensional simplices. The author seeks to:

• Define a new differential calculus (Simplicial $d$-polytopic calculus) based on these numbers.
• Introduce $S_d$-hypergeometric Bernoulli polynomials and $S_d$-Bernoulli numbers.
• Establish the fundamental properties (generating functions, recurrence relations, derivative rules, and translation formulas) of these new objects.
• Connect these structures to existing special functions, specifically the Kummer confluent hypergeometric function and Touchard polynomials.

2. Methodology

The author employs a constructive approach combining combinatorics, special functions, and operator theory:

• Combinatorial Foundation: The work is built upon simplicial $d$-polytopic numbers ($\langle n \rangle_d$), defined as binomial coefficients $\binom{n+d-1}{d}$. The author utilizes $d$-Hoggatt binomial coefficients ($\binom{n}{k}_d$) and $d$-simplitorial numbers ($\langle n \rangle_d!$) to generalize standard factorial and binomial structures.
• Operator Definition: Two distinct definitions for the Simplicial $d$-polytopic derivative ($D_{S_d}$) are introduced:
  1. A summation involving Stirling numbers of the first kind and standard derivatives.
  2. An operator form involving the inverse of the Euler operator ($X^{-1}$) and powers of the Euler operator ($XD$).
     The paper proves these definitions are equivalent and satisfy the property $D_{S_d} x^n = \langle n \rangle_d x^{n-1}$.
• Generating Functions: The core of the methodology relies on defining new generating functions. The $S_d$-exponential function ($\exp_d(x)$) is defined as a series using $\langle n \rangle_d!$ in the denominator, which is shown to be a generalized hypergeometric function (${}_0F_{d-1}$).
• Polynomial Construction: Using the $S_d$-exponential function and its truncated Taylor polynomials ($T_{d,m}$), the author defines the generating functions for the new Bernoulli numbers and polynomials.

3. Key Contributions

A. Simplicial $d$-Polytopic Calculus

• Derivative Operator: Defined $D_{S_d}$ which acts as a generalized derivative. For $d=1$, it is the standard derivative; for $d=2$, it is the "triangular derivative"; for $d=3$, the "tetrahedral derivative," etc.
• Binomial Theorem: Introduced the bivariate $d$-Hoggatt polynomials $(x \dot{+}_d y)^{(n)}$, which serve as the $S_d$-analogue of the binomial expansion. These satisfy a specific recurrence relation generalizing Pascal's triangle.
• Product Rule: Derived a generalized Leibniz rule (product rule) for $D_{S_d}$, which includes additional summation terms dependent on $d$.

B. Special Functions and Identities

• $S_d$-Exponential Function: Defined $\exp_d(x)$ as the solution to a specific linear differential equation of order $d$. It is proven to be an eigenfunction of the $D_{S_d}$ operator.
• Hypergeometric Connection: Established a novel identity linking the Kummer confluent hypergeometric function (${}_1F_1$) and Touchard polynomials ($T_n$):
  $${}_1F_1(1-d; 2; -x) = \frac{x^{-1}}{d!} \sum_{k=0}^d \left[ \begin{matrix} d \\ k \end{matrix} \right] T_k(x)$$
  where $\left[ \begin{matrix} d \\ k \end{matrix} \right]$ are Stirling numbers of the first kind.

C. $S_d$-Hypergeometric Bernoulli Polynomials

• Definition: Defined $B_{d,n}(m; x)$ via the generating function:
  $$\frac{t^m \exp_d(xt) / \langle m \rangle_d!}{\exp_d(t) - T_{d,m-1}(t)} = \sum_{n=0}^\infty B_{d,n}(m; x) \frac{t^n}{\langle n \rangle_d!}$$
• Properties:
  • Translation Formula: $B_{d,n}(m; x \dot{+}_d y) = \sum \binom{n}{k}_d B_{d,n}(m; x) y^{n-k}$.
  • Derivative Rule: $D_{S_d} B_{d,n}(m; x) = \langle n \rangle_d B_{d,n-1}(m; x)$.
  • Inversion Formula: Provided a formula to express $x^n$ in terms of these polynomials.
  • Explicit Representation: Derived an explicit sum involving $d$-Hoggatt coefficients and the polynomials.

D. $S_d$-Bernoulli Numbers and Polynomials

• Defined the standard $S_d$-Bernoulli numbers ($B_{d,n}$) and polynomials ($B_{d,n}(x)$) as the case where $m=1$ (or via the generating function $\frac{t \exp_d(xt)}{\exp_d(t)-1}$).
• Provided explicit values for $d=2$ (Triangular), $d=3$ (Tetrahedral), $d=4$ (Pentachoron), and $d=5$ (Hexateron).

4. Results

• Generalization: The paper successfully generalizes the classical Bernoulli polynomials ($d=1$) to arbitrary dimensions $d$.
• Explicit Formulas: Closed-form expressions are provided for the first few $S_d$-Bernoulli numbers and polynomials for various $d$.
• Differential Equations: The $S_d$-exponential function is identified as the solution to a specific $d$-th order linear differential equation.
• Generating Functions: The generating functions for bivariate Narayana polynomials (the $d=2$ case of bivariate $d$-Hoggatt polynomials) are explicitly linked to products of modified Bessel functions.
• Equivalence: The paper proves the equivalence of the two proposed definitions for the $S_d$-derivative operator.

5. Significance

• Unification: This work unifies several disparate areas of mathematics: combinatorial number theory (polytopic numbers), special functions (hypergeometric functions, Bessel functions), and polynomial sequences (Bernoulli, Touchard, Hoggatt).
• New Calculus: It establishes a rigorous "Simplicial Calculus" that extends standard differential calculus to discrete geometric structures (simplices).
• New Identities: The derived identity connecting Kummer functions and Touchard polynomials offers new tools for analyzing these special functions.
• Applications: The framework provides a foundation for future research in numerical analysis, approximation theory, and combinatorics where higher-dimensional simplicial structures are relevant. The specific cases (Triangular, Tetrahedral) offer concrete tools for problems involving 2D and 3D lattice structures.

In summary, the paper introduces a comprehensive mathematical framework extending Bernoulli polynomials to simplicial $d$-polytopic numbers, defining a new derivative operator, and establishing deep connections between hypergeometric series, combinatorial coefficients, and special functions.