A note on p-adic q-Euler measure

On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials

In this paper, we give a fermionic p-adic integral representions of Benstein polynomials associated with Euler numbers and polynomials. Finally, we give some interesting identities for the Euler numbers by using the properties of our integral represention.

On p-adic twisted Euler (h,q)-l-function

In this paper, we will constructed p-adic twisted q-l-functions which is a part of answer of the question in [8]. Finally, we will treat many interesting properties related to twisted q-Euler numbers and polynomials.

An Identity of the Symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Z_p

The main purpose of this paper is to prove an identity of symmetry for the Frobenius-Euler polynomials.

New q-Euler numbers and polynomials associated with p-adic q-integrals

In this paper we study q-Euler numbers and polynomials by using p-adic q-fermionic integrals on Z_p. The methods to study q-Euler numbers and polynomials in this paper are new.

Ob carlitz's type q-Euler numbers associated with the fermionic p-adic integrals on Zp

In this paper we consider the Witt's fprmula related to Carlitz's type q-Euler numbers and polynomials.

The fermionic p-adic integrals on Zp associated with extended q-Euler numbers and polynomials

The purpose of this paper is to present a systemic study of some families of q-Euler numbers and polynomials of Norlund's type by using multivariate fermionic p-adic integral on Zp. Moreover, the study of these higher-order q-Euler numbers and polynomials implies some interesting q-analogue of stirling numbers identities.

Some identities of symmetry for q-Euler polynomials under the symmetric group of degeree n arising from fermionic p-adic q-integrals on Zp

In this paper, we investigate some new symmetric identities for the q-Euler polynomials under the symmetric group of degree n which are derived from fermionic p-adic q-integrals on Zp.

A Weak Euler Formula for l-adic Galois Double Zeta values

The fact that the double zeta values at n and m can be written as a sum of products of two zeta values and of zeta value at m+n, whenever n+m is odd is due to Euler. We shall show a weak version of this result for the Galois l-adic realization.

The Digamma function, Euler-Lehmer constants and their $p$-adic counterparts

The goal of this article is twofold. We first extend a result of Murty and Saradha \cite{MS} related to the digamma function at rational arguments. Further, we extend another result of the same authors \cite{MS1} about the nature of $p$-adic Euler-Lehmer constants.

On $p$-adic Euler $L$-functions

In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. S…

A New Changhee q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals

Using non-archimedean q-integrals on Zp defined in [15, 16], we define a new Changhee q-Euler polynomials and numbers which are different from those of Kim [7] and Carlitz [2]. We define generating functions of multiple q-Euler numbers and polynomials. Furthermore we construct multivariate Hurwitz type zeta function which interpolates the multivariate q-Euler numbers or polynomials at negative integers.

A generalization of Darmon's conjecture for Euler systems for general p-adic representations

Darmon's conjecture on a relation between cyclotomic units over real quadratic fields and certain algebraic regulators was recently solved by Mazur and Rubin by using their theory of Kolyvagin systems. In this paper, we formulate a "non-explicit" version of Darmon's conjecture for Euler systems defined for general $p$-adic representations, and prove it. In the process of the proof, we introduce a notion of "algebraic Kolyvagin systems", and develop their properties.

Extended q-Dedekind-type Daehee-Changhee sums associated with Extended q-Euler polynomials

In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.

On $p$-adic multiple Barnes-Euler zeta functions and the corresponding log gamma functions

Suppose that $ω_1,\ldots,ω_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ with parameter vector $\barω=(ω_1,\ldots,ω_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ ζ_{E,N}(s,x;\barω)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+ω_1t_1+\cdots+ω_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ which we denote by $ζ_{p,E,N}(…

广义欧拉数的 $p$-adic 同余及其与偶 Zeta 值的关系

$p$-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values

本文从 $p$-adic 和分析角度研究广义欧拉数，引入同余欧拉数并建立其 $p$-adic 同余，包括解决与 Lehmer 数相关的猜想，并利用复分析方法用同余欧拉数表示偶 Zeta 值。

A unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p adic q integral on Zp at q=-1

The present paper deals with unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1. Some earlier results of Ozden's papers in terms of unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p-adic q-integral on Zp at q = 1 can be deduced. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Euler polynomials and Genocchi polynomials. To be more precise we summarize our results as follows, we obtain s…

On $p$-adic Hurwitz-type Euler zeta functions

The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$…

On the structure of the module of Euler systems for a $p$-adic representation

We investigate a question of Burns and Sano concerning the structure of the module of Euler systems for a general $p$-adic representation. Assuming the weak Leopoldt conjecture, and the vanishing of $μ$-invariants of natural Iwasawa modules, we obtain an Iwasawa-theoretic classification criterion for Euler systems which can be used to study this module. This criterion, taken together with Coleman's conjecture on circular distributions, leads us to pose a refinement of the aforementioned question for which we provide strong, and unconditional, evidence. We furthermore answer this question in th…

$p$-进族中的 Asai--Flach Euler 系统

The Asai--Flach Euler system in $p$-adic families

本文证明 Asai 表示的 Euler 系统可在 Hida 族中 $p$-进插值，为 Bloch--Kato 猜想解析秩零情形提供关键输入。

Rankin-Selberg Euler systems and p-adic interpolation

We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.

Note on q-Dedekind type sums related to q-Euler polynomials

it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.

Local epsilon-isomorphisms for rank two p-adic representations of Gal(overline{Q}_p/Q_p) and a functional equation of Kato's Euler system

In this article, we prove (many parts of) the rank two case of the Kato's local epsilon-conjecture using the Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We show that a Colmez's pairing defined in his study of locally algebraic vectors gives us the conjectural epsilon-isomorphisms for (almost) all the families of p-adic representations of Gal(overline{Q}_p/Q_p) of rank two, which satisfy the desired interpolation property for the de Rham and trianguline case. For the de Rham and non trianguline case, we also show this interpolation property for the "critical" range of Hodge-Ta…

Analysis of the p-adic q-Volkenborn Integrals: an approach to Apostol-type special numbers and polynomials

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 som…

Completed cohomology and Kato's Euler system for modular forms

In this paper, we compare two different constructions of $p$-adic $L$-functions for modular forms and their relationship to Galois cohomology: one using Kato's Euler system and the other using Emerton's $p$-adically completed cohomology of modular curves. At a more technical level, we prove the equality of two elements of a local Iwasawa cohomology group, one arising from Kato's Euler system, and the other from the theory of modular symbols and $p$-adic local Langlands correspondence for $GL_2(\mathbb{Q}_p)$. We show that this equality holds even in the cases when the construction of $p$-adic …

Identities of symmetry for expansion of q-Euler polynomials

In this paper, we investigate some symmetric properties of the multivariate p-adic fermionic integrals on Zp and derive various identities concerning the expansions of q-Euler polynomials from the symmetric properties of the multivariate p-adic fermionic integrals on Zp

A Note on p-Adic Invariant Integral in the Rings of p-Adic Integers

In [2], I constructed the p-adic q-integral on Zp. In this paper, we consider the properties of the p-adic invariant p-adic q-integral in the ring of p-adic integers at q=-1. Finally we give the some applications of p-adic q-integration at q=-1. These properties are useful and worthwhile to study Euler numbers and polynomials.

A note on the algebra of p-adic multi-zeta values

We prove that the algebra of p-adic multi-zeta values are contained in another algebra which is defined explicitly in terms of series.

A note on $p$-adic locally analytic functions with application to behavior of the $p$-adic valuations of Stirling numbers

The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number. The proof is based on elementary facts from $p$-adic analysis.

Hirzebruch--Zagier 循环的反环分 Euler 系统 I：范数关系与 $p$-进插值

An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation

本文为实二次域上 $p$-ordinary Hilbert 模形式的 Asai 表示构造反环分 Euler 系统，证明其在 Hida 族中变化，并应用于 Bloch--Kato 猜想与 Iwasawa 主猜想。

A note on the p-adic log-gamma functions

In this paper we prove that q-Euler numbers are occured in the coefficients of some stirling type seies for p-adic analytic q-log gamma function

$p$-adic model theory, $p$-adic integrals, Euler products, and zeta functions of groups

We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …

A note on Fermionic p-adic integrals on Zp and umbral calculus

In this paper we study some properties of the fermionic p-adic integrals on Zp arising from the umbral calculus

p-adic interpolating function associated with modified Dirichlet's type of twisted q-euler numbers and polynomials with weight alpha

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between mod…

A short note on p-adic families of Hilbert Modular Forms

We extend previous work of the author using an idea of Buzzard and give an elementary construction of non-ordinary $p$-adic families of Hilbert Modular Eigenforms.

关于p进高维马勒测度的一个注记

A note on $p$-adic higher Mahler measures

本文引入p进高维马勒测度，推广Kurokawa、Lalín和Ochiai的工作，并证明Akatsuka相关结果的p进版本。

Explicit Formulas involving q-Euler Numbers and Polynomials

In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring.

A note on p-adic Rankin--Selberg L-functions

We prove an interpolation formula for the values of certain $p$-adic Rankin--Selberg $L$-functions associated to non-ordinary modular forms.

A note on the Fourier transform of p-adic q-integral

The $p$-adic $q$-integral (= $I_q$-integral) was defined by author in the previous paper [1, 3]. In this paper, we consider $I_q$-Fourier transform and investigate some properties which are related to this transform.

Refined and l-adic Euler characteristics of nearly perfect complexes

We lift the Euler characteristic of a nearly perfect complex to a relative algebraic K-group by passing to its l-adic Euler characteristics.

$p$-adic Ducci Sequences: a short note

In this short note we formalized the definition for the Ducci operator $D$ in the context of the $p$-adic field $\mathbb{Q}_p$ as a natural extension of the classical one. Moreover we will describe the behavior of the operator and will provide some simple results as a counterpart to the classical one.

A note on $p$-adic denseness of quotients of values of quadratic forms

Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers. The aim of this note is to give another proof of this classification.

Dirichlet's type of twisted Eulerian polynomials in connection with Eulerian-L-function

In the present paper, we effect Dirichlet's type of twisted Eulerian polynomials by using p-adic fermionic q-integral on the p-adic integer ring. Also, we introduce some new interesting identities for them. As a result of them, by using contour integral on the generating function of Dirichlet's type of twisted Eulerian polynomials and so we define twisted Eulerian-L-function which interpolates of Dirichlet's type of Eulerian polynomials at negative integers which we state in this paper.

A note on Catalan numbers associated with p-adic integral on Zp

In this paper, we study Catalan numbers which can be represented by the p-adic integral on Zp and we investigate some properties and formulae related to Catalan numbers and special numbers.

A note on hyperelliptic curves with ordinary reduction over 2-adic fields

We study a class of semistable ordinary hyperelliptic curves over 2-adic fields and the special fibre of their minimal regular model. We show that these curves can be controlled using `cluster pictures', similarly to the case of odd residue characteristic.

A note on critical $p$-adic $L$-functions

We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable $p$-adic $L$-functions around critical points via Emerton's representation theoretic approach.

Extended Fermionic p-Adic q-Integrals On Zp In Connection With Applications Of Umbral Calculus

The purpose of this paper is to derive some applications of umbral calculus by using extended fermionic p-adic q-integral on Zp. From those applications, we derive some new interesting properties on the new family of Euler numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the weighted Euler polynomials are given in the present paper.

A note on the weighted q-Hardy-Littlewood-type maximal operator with respect to q-Volkenborn integral in the p-adic integer ring

The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator.

p进非齐次Littlewood猜想注记

A note on the $p$-adic inhomogeneous Littlewood conjecture

本文针对p进非齐次Littlewood猜想，在Bugeaud等人2011年解决p进Littlewood猜想度量版本的基础上，建立了完全非齐次推广，并在混合情形下得到Harrap等人2013年结果的部分非齐次推广。

A note on cusp forms as $p$-adic limits

Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the $p$-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.

A note on logarithmic growth of solutions of $p$-adic differential equations without solvability

For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork's conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya's decomposition theorem of $p$-adic differential equations defined over annulus.

A note on regularized Bernoulli distributions and p-adic Dirichlet expansions

We consider Bernoulli distributions and their regularizations, which are measures on the $p$-adic integers $\mathbb{Z}_p$. It is well known that their Mellin transform can be used to define $p$-adic $L$-functions. We show that for $p>2$ one of the regularized Bernoulli distributions is particularly simple and equal to a measure on $\mathbb{Z}_p$ that takes the values $\pm \frac{1}{2}$ on clopen balls. We apply this to $p$-adic $L$-functions for Dirichlet characters of $p$-power conductor and obtain Dirichlet series expansions similar to the complex case. Such expansions were studied by D. Delb…

Sum Expressions for Kubota-Leopoldt $p$-adic $L$-functions

When $p$ is an odd prime, Delbourgo observed that any Kubota-Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$, and without the Euler factor when the character is nontrivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero-Greenberg formula for the derivative $L_p'(0,χ)$. We will also discuss the convergence of sum expressions in terms of elementary $p$-adic analysis, as well as their relation to Stickelberger elements; s…

Extended fermionic $p$-adic integrals on $\mathbb{Z}_p$

In the paper, using the extended fermionic $p$-adic integral on $\mathbb{Z}_p$, the authors find some applications of the umbral calculus. From these applications, the authors derive some identities on the weighted Euler numbers and polynomials. In other words, the authors investigate systematically the class of Sheffer sequences in connection with the generating function of the weighted Euler polynomials.

Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type and evaluation of p-adic multiple L-functions at arbitrary integers

We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…