Fermionic P-adic Integral Research Articles

Some identities involving Bernoulli, Euler and degenerate Bernoulli numbers and their applications

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable.

Some Explicit Expressions for Twisted Catalan-Daehee Numbers

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by using p-adic integrals or generating functions.

New construction of type 2 degenerate central Fubini polynomials with their certain properties

Kim et al. (Proc. Jangjeon Math. Soc. 21(4):589–598, 2018) have studied the central Fubini polynomials associated with central factorial numbers of the second kind. Motivated by their work, we introduce degenerate version of the central Fubini polynomials. We show that these polynomials can be represented by the fermionic p-adic integral on mathbb{Z}_{p}. From the fermionic p-adic integral equations, we derive some new properties related to degenerate central factorial numbers of the second kind and degenerate Euler numbers of the second kind.

Some New Families of Special Polynomials and Numbers Associated with Finite Operators

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

Ordinary and degenerate Euler numbers and polynomials

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on mathbb{Z}_{p}. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials

In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on mathbb{Z}_{p} and generating functions. In addition, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.

Some identities of special numbers and polynomials arising from p-adic integrals on mathbb{Z}_{p}

In recent years, studying degenerate versions of various special polynomials and numbers has attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas, and analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials. The present study was done by using the bosonic and fermionic p-adic integrals on mathbb{Z}_{p}.

Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

The Boole polynomials associated with the p-adic gamma function

We set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on Zp. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.

Symmetry Identities of Changhee Polynomials of Type Two

In this paper, we consider Changhee polynomials of type two, which are motivated from the recent work of D. Kim and T. Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmetry for the fermionic p-adic integral on Z p .

On the <i>p</i>-adic Gamma Function and Changhee Polynomials

The p-adic gamma function is considered to obtain its derivative and to evaluate its the fermionic p-adic integral. Furthermore the relationship between the p-adic gamma function and Changhee polynomials and also between the Changhee polynomials and p-adic Euler constants is obtained. In addition, the p-adic Euler constants are expressed in term of Mahler coefficients of the p-adic gamma function.

Some p-Adic Integrals on ℤp Associated with Trigonometric Functions

In this paper, we study some p-adic invariant and fermionic p-adic integrals on ℤp associated with trigonometric functions. By using these p-adic integrals we represent several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers. In addition, we obtain some identities relating various special numbers like zigzag, some ‘trigonometric’, Bernoulli, Euler numbers, and Euler numbers of the second kind.

Symmetric Identities for Fubini Polynomials

We represent the generating function of w-torsion Fubini polynomials by means of a fermionic p-adic integral on Zp. Then we investigate a quotient of such p-adic integrals on Zp, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials.

Identities associated with Milne\u2013Thomson type polynomials and special numbers

The purpose of this paper is to give identities and relations including the Milne–Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By using fermionic and bosonic p-adic integrals, we derive some new relations and formulas related to these numbers and polynomials, and also the combinatorial sums.

Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications

By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give some identities and relations including the generalized Apostol-Daehee and Apostol-Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the generalized Euler numbers and the Frobenious-Euler polynomials. By using the bosonic and the fermionic p-adic integrals, we derive integral represantations for the generalized Apostol-type Daehee numbers and the generalized Apostol-type Changhee numbers.

Some identities of Korobov-type polynomials associated with p-adic integrals on Z p $\mathbb{Z}_{p}$

In this paper, we consider Korobov-type polynomials derived from the bosonic and fermionic p-adic integrals on $\mathbb{Z}_{p}$ , and we give some interesting and new identities of those polynomials and of their mixed-types.

Some identities of Barnes-type special polynomials

In this paper, we consider Barnes-type special polynomials and give some identities of their polynomials which are derived from the bosonic p-adic integral or the fermionic p-adic integral on $\mathbb{Z}_{p}$ .

Some identities of degenerate Euler polynomials arising from p-adic fermionic integrals on ℤp

L. Carlitz introduced higher order degenerate Euler polynomials (see [Carlitz L. Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Math. 1979;15:51–88; Carlitz L. A degenerate Staudt-Clausen theorem. Arch Math (Basel). 1956;7:28–33]). In this paper, we give some formulae and identities of degenerate Euler polynomials which are derived from the fermionic p-adic integrals on ℤp.

A note on three variable symmetric identities for q-Euler polynomials arising from fermionic p-adic integral on Zp

Recently, Kim Kim have given some interesting new three variable symmetric identities involving Carlitz-type q-Euler polynomials(see [5]). In this paper, we investigate some properties of symmetric identies for the modified q-Euler polynomials which are slightly different the symmetric identities of Kim Kim for the Carlitztype q-Euler polynomials.