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. Delbourgo, and this contribution provides an approach via $p$-adic measures.
We construct the q-extension of the Hurwitz's type L-function which interpolates the q-extension of generalized Bernoulli polynomials attached to $chi$.
In this paper, we study some properties of Changhee's q-Bernou lli polynomials which are derived from p-adic invariant integral on Zp. By using these properties, we give some interesting identities related to higher- order q-Bernoulli polynomials.
In this paper we give some interesting equation of p-adic q-integrals on Zp. From those p-adic q-integrals, we present a systemic study of some families of extended Carlitz q-Bernoulli numbers and polynomials in p-adic number field.
Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems. The estimates and criteria all depend on our previous analysis of powers of 2 in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.
The Erdös-Moser equation is a Diophantine equation proposed more than 60 years ago which remains unresolved to this day. In this paper, we consider the problem in terms of divisibility of power sums and in terms of certain Egyptian fraction equations. As a consequence, we show that solutions must satisfy strong divisibility properties and a restrictive Egyptian fraction equation. Our studies lead us to results on the Bernoulli numbers and allow us to motivate Moser's original approach to the problem.
In recent years, studying degenerate versions of various special polynomials and numbers have 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 to those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas and analogues for the Bernoulli's interpretation of powers of the first $m$ positive integers in terms of Bernoulli …
In this paper, by the properties of p-adic invariant integral on Zp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of p-adic invariant integral on Zp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.
The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using p-adic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which interpolate generalized twisted q-Bernoulli numbers. This is the generalization of Kim's h-extension of p-adic q-L-function which was constructed in [5] and is a partial answer for the open question which was remained in [3]
In this paper, we use our previous study of the higher order Bernoulli numbers $B_n^{(l)}$ to investigate the $p$-adic properties of the Stirling numbers of the second kind $S(n,k)$. For example, we give a new, greatly simplified proof of the formula $ν_2(S(2^h,k))=d_2(k)-1$ if $1\le k \le 2^h$, and generalize this result to arbitrary primes $p$. We also consider the Stirling numbers of the first kind $s(n,k)$, with new results analogous to those for the Stirling numbers of the second kind. New mod $p$ congruences for Stirling numbers of both kinds are also given.
Recently, $λ$-Bernoulli and $λ$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $λ$-Bernoulli and the $λ$-Euler numbers by using the bosonic $p$-adic $q$-integral and the fermionic $p$-adic $q$-integral. The investigation of these $λ$-$q$-Bernoulli and $λ$-$q$-Euler numbers leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning $q$-Bernoulli and $q$-Euler numbers. By using derivative operator to the generating functions of $λ$-$q$-B…
By using $q$-Volkenborn integration and uniform differentiable on $\mathbb{Z}%_{p}$, we construct $p$-adic $q$-zeta functions. These functions interpolate the $q$-Bernoulli numbers and polynomials. The value of $p$-adic $q$-zeta functions at negative integers are given explicitly. We also define new generating functions of $q$-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of some basic (or $q$-) $L$% -series. These generating functions also interpolate Barnes' type Changhee $% q $-Bernoulli numbers with attached to Dirichlet character as well. By a…
The purpose of this paper is to construct p-adic Dedekind sums and Hardy-Berndt type sums. We also construct generating function of the twisted Bernoulli polynomials and functions. Furthermore, we give some discussions on elliptic analogue of the Apostol-Dedekind sums.
This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special…
By using q-Volkenborn integral on Z_{p}, we (simsek, simsekCanada) constructed new generating functions of the (h,q)-Bernoulli polynomials and numbers. By applying the Mellin transformation to the generating functions, we constructed integral representation of the twisted (h,q)-Hurwitz function and twisted (h,q)-two-variable L-function. By using these functions, we construct twisted new (h,q)-partial zeta function which interpolates the twisted (h,q)-Bernoulli polynomials and generalized twisted (h,q)-Bernoulli numbers at negative integers. We give relation between twisted (h,q)-partial zeta f…
Let $B_n$ ($n = 0, 1, 2, ...$) denote the usual $n$-th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not\equiv B_{l+p-1}/(l+p-1) \modp{p^2}$. We show that for every $r \geq 1$ the congruence $B_{m_r}/m_r \equiv 0 \modp{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \modp{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $χ_{(p, l)}$ which is a zer…
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.
The Euler--Riemann zeta function is a largely studied numbertheoretic object, and the birthplace of several conjectures, such as the Riemann Hypothesis. Different approaches are used to study it, including $p$-adic analysis : deriving information from $p$-adic zeta functions. A generalized version of $p$-adic zeta functions (Riemann zeta function) are $p$-adic $L$-functions (resp. Dirichlet $L$-functions). This paper describes formalization of $p$-adic $L$-functions in an interactive theorem prover Lean 3. Kubota--Leopoldt $p$-adic $L$-functions are meromorphic functions emerging from the spec…
The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $p$-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.
The aim of the present paper is to establish relations between Iwasawa and Bernoulli numbers based on some results by M. Kervaire and M. P. Murthy about the structure of the $K_0$ groups of the integer group rings of cyclic groups of prime power order $p^n .$ In particular, we will prove that $λ_{i}\leq p-1$ under assumption that the generalized Bernoulli number $B_{1,ω^{-i}}$ is not divisible by $p^2$. Here $ω$ is the Teichmüller character of $\mathbb{Z}/(p-1)\mathbb{Z}$. $λ_{i}=1$ if $B_{1,ω^{-i}}$ is divisible by $p^2$. We will prove that $S_{n,i}\cong \mathbb{Z}/(p^{n+k_i})$, where $S_n$ i…
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 polyno…
Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$.
The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these generating functions, their functional equations, and their partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials and others. Among…
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…
We study $p$-adic $L$-functions $L_p(s,χ)$ for Dirichlet characters $χ$. We show that $L_p(s,χ)$ has a Dirichlet series expansion for each regularization parameter $c$ that is prime to $p$ and the conductor of $χ$. The expansion is proved by transforming a known formula for $p$-adic $L$-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the $p$-adic Dirichlet series. We also provide an alternative proof of the expansion using $p$-adic measures and give an explicit formula for the values of the regularized Bernoulli …
We introduce $p$-adic Kummer spaces of continuous functions on $\mathbb{Z}_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and certain decompositions. As a result, these functions have always a fixed point, functions of certain subclasses have always a unique simple zero in $\mathbb{Z}_p$. The fixed points and the zeros are effectively computable by given algorithms. This theory can be transferred to values of Dirichlet $L$-functions at negative integer arguments. That leads to a conjecture about their stru…
By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.
We construct the two-variable $p$-adic $q$-$L$-function which interpolates the generalized $q$-Bernoulli polynomials associated with primitive Dirichlet character $χ$. Indeed, this function is the $q$-extension of two-variable $p$-adic $L$-function due to Fox, corresponding to the case $q=1 .$ Finally, we give some $p$-adic integral representation for this two-variable $p$-adic $q$-$L$-function and derive to $q$-extension of the generalized formula of Diamond and Ferro and Greenberg for the two-variable $p$-adic $L$-function in terms of the $p$-adic gamma and $\log$ gamma function.
Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left<f\right>_q$) are quasimodular forms. We revisit a family of such functions, denoted $Q_k$, and study the $p$-adic properties of their $q$-brackets. To do this, we define regularized versions $Q_k^{(p)}$ for primes $p.$ We also use Jacobi forms to show that the $\left<Q_k^{(p)}\right>_q$ are quasimodular and find explicit expressions for them in terms of the $\left<Q_k\right>_q$.
The article is dedicated to the memory of George Voronoi. It is concerned with ($p$-adic) $L$-functions (in partially ($p$-adic) zeta functions) and cyclotomic ($p$-adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. Ünver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple…
In this article, we give an explicit construction of the $p$-adic Fourier transform by Schneider and Teitelbaum, which allows for the investigation of the integral property. As an application, we give a certain integral basis of the space of $K$-locally analytic functions on the ring of integers $\mathcal{O}_K$ for any finite extension $K$ of $\mathbb{Q}_p$, generalizing the basis constructed by Amice for locally analytic functions on $\mathbb{Z}_p$. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes originally investigated by K…
In this paper, we consider normalized newforms $f\in S_k(Γ_0(N),\varepsilon_f)$ whose non-constant term Fourier coefficients are congruent to those of an Eisenstein series modulo some prime ideal above a rational prime $p$. In this situation, we establish a congruence between the anticyclotomic $p$-adic $L$-function of Bertolini-Darmon-Prasanna and the Katz two-variable $p$-adic $L$-function. From this, we derive congruences between images under the $p$-adic Abel-Jacobi map of certain generalized Heegner cycles attached to $f$ and special values of the Katz $p$-adic $L$-function. In particular…
Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmoni…
Particles and $p-$adic integrals of Spin$\left(\frac{1}{2}\right)$: spin Lie group, $\mathcal{R}(ρ,q)-$gamma and $\mathcal{R}(ρ,q)-$ beta functions, ghost and applications
We derive $p$-adic expansions for the generalized Harmonic numbers $H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers $B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are not new, we obtain them elementarily, without resorting to the theory of $p$-adic L-functions as was the case previously. Moreover, we show that \begin{equation*}\sum_{j=0}^{n-1}\left(\frac{(2^{j+1}-1)}{(j+1)}\frac{(2^{j+2}-1)}{(j+2)}\frac{B_{j+2}}{2^{j}}H^{(j+1)}_{\frac{p-1}{2}}+2(-1)^j\frac{q_p^{j+1}}{j+1}\right)p^j\equiv 0 \pmod {p^n} \end{equation*} holds under the condi…
We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in t…
We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(θ)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[θ]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $β(j)$ (i.e., on the values of the Goss $ζ$-function at…
We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety $A$ defined over $F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and $\mathcal{F}$. We do the same, with characteristic ideals and $p$-adic $L$-functions, in the case of class groups (using known re…
Given a number field $K$ with a Hecke character $χ$, for each place $ν$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $χ$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $χ$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $ν$, the field theory t…
Carlitz [2] initiated a study on degenerate versions of Bernoulli and Euler numbers which has been extended recently to the researches on various degenerate versions of quite a few special numbers and polynomials. They have been explored by using several different tools including generating functions, combinatorial methods, $p$-adic analysis, umbral calculus, special functions, differential equations and probability theory as well. \par The degenerate Poisson random variables are degenerate versions of the Poisson random variables. In [6], studied are the degenerate binomial and degenerate Poi…
