[Paper Review] Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials

Yılmaz Şimşek|arXiv (Cornell University)|Oct 10, 2019

Advanced Mathematical IdentitiesMathematics82 references18 citations

TL;DR

This paper presents novel explicit formulas for p-adic integrals, particularly the Volkenborn and fermionic p-adic integrals, linking them to fundamental special numbers and polynomials such as Bernoulli, Euler, Stirling, Daehee, Changhee, and central factorial numbers. It introduces two new sequences, $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$, which express Bernoulli and Euler numbers via central factorial numbers of the second kind, providing new computational identities through p-adic integral evaluations.

ABSTRACT

The main objective of this article is to give and classify new formulas of $p$-adic integrals and blend these formulas with previously well known formulas. Therefore, this article gives briefly the formulas of $p$-adic integrals which were found previously, as well as applying the integral equations to the generating functions and other special functions, giving proofs of the new interesting and novel formulas. The $p$-adic integral formulas provided in this article contain several important well-known families of special numbers and special polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Lah numbers, the Peters numbers and polynomials, the central factorial numbers, the Daehee numbers and polynomials, the Changhee numbers and polynomials, the Harmonic numbers, the Fubini numbers, combinatorial numbers and sums. In addition, we defined two new sequences containing the Bernoulli numbers and Euler numbers. These two sequences include central factorial numbers, Bernoulli numbers and Euler numbers. Some computation formulas and identities for these sequences are given. Finally we give further remarks, observations and comments related to content of this paper.

Motivation & Objective

• To derive and unify new explicit formulas for p-adic integrals, especially Volkenborn and fermionic p-adic integrals.
• To connect these integrals with well-known special numbers and polynomials, including Bernoulli, Euler, Stirling, Lah, Peters, Daehee, Changhee, and harmonic numbers.
• To define and analyze two new sequences, $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$, that encode Bernoulli and Euler numbers using central factorial numbers.
• To establish computational identities for these sequences through p-adic integral evaluations and generating functions.
• To provide a comprehensive framework linking p-adic analysis with combinatorial and special functions in number theory and mathematical physics.

Proposed method

• Utilizes the Volkenborn integral $\int_{\mathbb{Z}_p} f(x) d\mu_1(x)$ and the fermionic p-adic integral $\int_{\mathbb{Z}_p} f(x) d\mu_{-1}(x)$ as core tools for deriving identities.
• Applies these integrals to generating functions and polynomial expansions, particularly for monomials $x^{2n}$ and products like $x^2(x^2-1)\cdots(x^2-(n-1)^2)$.
• Employs the central factorial numbers of the second kind $T(n,k)$ to express monomials as linear combinations of falling factorial-like terms.
• Introduces the sequences $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$ via integral evaluations of $x^j x_{(k)}$, linking them to Bernoulli and Euler numbers.
• Uses Stirling numbers of the first kind $S_1(n,j)$ and absolute Lah numbers $|L(n,k)|$ to decompose the integral expressions into explicit sums.
• Combines known identities involving Bernoulli and Euler numbers with p-adic integral results to derive new closed-form expressions.

Experimental results

Research questions

• RQ1 How can the Volkenborn and fermionic p-adic integrals be systematically applied to derive new identities for special numbers and polynomials?
• RQ2 What is the role of central factorial numbers $T(n,k)$ in expressing Bernoulli and Euler numbers through p-adic integrals?
• RQ3 Can new sequences $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$ be defined such that they recover Bernoulli and Euler numbers via linear combinations of $T(n,k)$ and other number sequences?
• RQ4 How do the new sequences $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$ relate to existing sequences like Daehee and Changhee numbers?
• RQ5 What explicit computational formulas can be derived for $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$ using p-adic integrals and combinatorial coefficients?

Key findings

• The Bernoulli numbers satisfy $B_{2n} = \sum_{k=0}^{n} T(n,k) \mathcal{Y}(k,B)$, where $T(n,k)$ are central factorial numbers of the second kind.
• The Euler numbers satisfy $E_{2n} = \sum_{k=0}^{n} T(n,k) \mathcal{Y}(k,E)$, establishing a symmetric structure to the Bernoulli case.
• The sequence $\mathcal{Y}(n,B)$ is explicitly given by $\sum_{j=0}^{n} \sum_{k=1}^{n} \sum_{m=0}^{k} S_1(n,j) S_1(k,m) B_{j+m} |L(n,k)|$, linking it to Stirling and Lah numbers.
• The sequence $\mathcal{Y}(n,E)$ is given by $\sum_{j=0}^{n} \sum_{k=1}^{n} \sum_{m=0}^{k} S_1(n,j) S_1(k,m) E_{j+m} |L(n,k)|$, providing a new computational path for Euler numbers.
• The sequence $\mathcal{Y}(n,B)$ corresponds to the Daehee numbers $D_n$, and $\mathcal{Y}(n,E)$ corresponds to the Changhee numbers $Ch_n$, establishing a novel connection between p-adic integrals and these special sequences.
• The paper establishes that $\mathcal{Y}(n,B)$ and $\mathcal{Y}(n,E)$ can be computed via p-adic integrals of monomials and falling factorials, offering a new method for evaluating Bernoulli and Euler numbers.