P-adic Invariant Integral Research Articles

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.

On \u03bb-linear functionals arising from p-adic integrals on mathbb{Z}_{p}

The aim of this paper is to determine the λ-linear functionals sending any given polynomial p(x) with coefficients in mathbb{C}_{p} to the p-adic invariant integral of P(x) on mathbb{Z}_{p} and also to that of P(x_{1}+cdots +x_{r}) on mathbb{Z}_{p}^{r}. We show that the former is given by the generating function of degenerate Bernoulli polynomials and the latter by that of degenerate Bernoulli polynomials of order r. For this purpose, we use the λ-umbral algebra which has been recently introduced by Kim and Kim (J. Math. Anal. Appl. 493(1):124521 2021).

A Note on Type 2 w-Daehee Polynomials

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

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.

The modified degenerate q-Bernoulli polynomials arising from p-adic invariant integral on mathbb{Z}_{p}

Dolgy et al. introduced the modified degenerate Bernoulli polynomials, which are different from Carlitz’s degenerate Bernoulli polynomials (see Dolgy et al. in Adv. Stud. Contemp. Math. (Kyungshang) 26(1):1-9, 2016). In this paper, we study some explicit identities and properties for the modified degenerate q-Bernoulli polynomials arising from the p-adic invariant integral on mathbb{Z}_{p}.

A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on [formula omitted

The degenerate Bernoulli polynomials were introduced by Carlitz and rediscovered later by Ustiniv under the name of Korobov polynomials of the second kind. In this paper, we derive a Witt-type formula for the degenerate Bernoulli polynomials which can be represented by the p-adic invariant integral on Zp. In addition, we introduce the λ-Daehee polynomials and give some relations between the degenerate Bernoulli polynomials and the λ-Daehee polynomials.

Symmetry identities of q-Bernoulli polynomials of the second kind

In this paper, we give identities of symmetry for the q-Bernoulli polynomials which are derived from the symmetric properties of the p-adic invariant integrals on ℤp.

Symmetric identities for the generalized higher-order q-Bernoulli polynomials

In this paper, we give identities of symmetry for the generalized higher-order qBernoulli polynomials attached to χ which are derived from the symmetric properties of multivariate p-adic invariant integrals on Zp.

Symmetric identities for the generalized tangent polynomials associated with p-adic integral on Zp

In this paper, we study the symmetry for the generalized tangent numbers Tn,χ and polynomials Tn,χ(x). We obtain some interesting identities of the power sums and the generalized polynomials Tn,χ(x) using the symmetric properties for the p-adic invariant integral on Zp. Mathematics Subject Classification: 11B68, 11S40, 11S80

P-Adic invariant integral on Zp associated with the Changhee's q-Bernoulli polynomials

In this paper, we study some properties of Changhee's q-Bernou lli polynomials which are derived from p-adic invariant integral on Zp .B y using these properties, we give some interesting identities related to higher- order q-Bernoulli polynomials.

A NOTE ON THE WEIGHTED LEBESGUE-RADON-NIKODYM THEOREM WITH RESPECT TO $p$ -ADIC INVARIANT INTEGRAL ON $\mathbb{Z}_p$

In this paper, we give the weighted Lebesgue-Radon-Nikodym theorem with respect to p-adic invariant integral on Zp.

A NOTE ON THE q-ANALOGUES OF EULER NUMBERS AND POLYNOMIALS

In this paper, we consider the q-analogues of Euler numbers and polynomials using the fermionic p-adic invariant integral on <TEX>$\mathbb{Z}_p$</TEX>. From these numbers and polynomials, we derive some interesting identities and properties on the q-analogues of Euler numbers and polynomials.

Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤ p

The objective of the paper is to indicate a symmetry of the multivariate p-adic invariant integral on ℤ p , which leads to a relation between the power sum polynomials and higher-order Euler polynomials.

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ℤ p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ℤ p . By using this symmetry of fermionic p-adic invariant integral on ℤ p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ℤ p .

On the twisted q-Euler numbers and polynomials associated with basic q- l-functions

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariant integral on Z p in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q- l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.

On the analogs of Euler numbers and polynomials associated with p-adic q-integral on [formula omitted] at [formula omitted

The purpose of this paper is to construct λ-Euler numbers and polynomials by using fermionic expression of p-adic q-integral at q = − 1 . From these λ-Euler polynomials, we derive the harmonic sums of higher order. Finally, we investigate several interesting properties and relationships involving the classical as well as the generalized Euler numbers and polynomials. As an application, we will treat p-adic invariant integrals on Z p involving trigonometric functions.