Euler Type Generalization of Wilson's Theorem

Note on the generalization of the higher order q-Genocchi numbers and q-Euler numbers

In this paper we present the generalization of the higher order q-Euler numbers and q-Genocchi numbers and w-Genocchi numbers and polynomials of high order using the multivariate fermionic p-adic integral on Zp. We have the interpolation functions of these numbers and polynomials. We obtain the distribution relations for q-extensions of w-Euler and w-Genocchi polynomials. We also have the interesting relation for q-extensions of these polynomials. We define (h, q)-extensions of w-Euler and w-Genocchi polynomials of high order. We have the interpolation functions for (h, q)-extensions of these …

关于多元多项式系统的欧拉φ函数的推广

Generalizations of Euler's $φ$-function with respect to systems of polynomials of several variables

该研究引入多元多项式系统相关的欧拉φ函数新推广，用直接方法证恒等式、研特殊情况，还探讨酉类似物，为相关领域研究提供新内容与思路。

A Generalization of Euler's Criterion to Composite Moduli

A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.

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.

Shifted Euler constants and a generalization of Euler-Stieltjes constants

The purpose of this article is twofold. First, we introduce the constants $ζ_k(α,r,q)$ where $α\in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $γ(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent…

Another generalization of Euler's arithmetic function and Menon's identity

We define the $k$-dimensional generalized Euler function $φ_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots +a_k$ are prime to $n$. We investigate some of properties of the function $φ_k(n)$, and obtain a corresponding Menon-type identity.

Explicit double shuffle relations and a generalization of Euler's decomposition formula

We give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms. As an application, we generalize the well-known decomposition formula of Euler that expresses the product of two Riemann zeta values as a sum of double zeta values to a formula that expresses the product of two multiple polylogarithm values as a sum of other multiple polylogarithm values.

poly-Dedekind type DC sums involving poly-Euler functions

The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind type DC sums, which are obtained from the Dedekind type DC sums by…

A Note On Multi Poly-Euler Numbers And Bernoulli Polynomials

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.

A tiling proof of Euler's pentagnal number theorem and generalizations

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating fu…

广义欧拉多项式的一个恒等式

An Identity for Generalized Euler Polynomials

本文提出广义欧拉多项式的一个新恒等式，并借此推广了欧拉数、欧拉多项式、Genocchi多项式与Genocchi数之间的多类经典关系。

Lehmer-Euler数的同余性质

Congruence properties of Lehmer-Euler numbers

本文研究Lehmer在1935年利用三次单位根定义的广义欧拉数，给出其同余性质、递推公式和显式公式，并构造新的多项式序列，得到与欧拉数及中心阶乘数相关的恒等式。

Generalization of Some Arithmetical Properties of Fermat-Euler Dynamical Systems

We study and generalize some arithmetical properties of the classes (2^k+) and (2^k-) introduced by V. I. Arnold: a number n belongs to the class (N+) if N|φ(n) and 2^{\frac{φ(n)}{N}} \equiv 1 mod n where φ(n) is the Euler function, and belongs to the class (M-) if M|φ(n) and 2^{\frac{φ(n)}{M}} \equiv -1 mod n. The classes (2+), (2-),(4+), (4-), (8+)and (8-) are studied by V. I. Arnold and here we will show general properties of the classes (2^k+) and (2^k-) and we will see that the properties which is proved by V. I. Arnold are special cases of ours.

Truncated Euler polynomials

We define a truncated Euler polynomial $E_{m,n}(x)$ as a generalization of the classical Euler polynomial $E_n(x)$. In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

A generalization to number fields of Euler's theorem on the series of reciprocals of primes

Let $X$ be a set of positive integers, and let $\mathbb Z_K$ be the ring of integers of a number field $K$ of degree $n$. Denote by $N(I)$ the absolute norm of an ideal $I$ of $\mathbb Z_K$, and by $\mathcal A$ the set of principal ideals $a\mathbb Z_K$ such that $a$ is an atom of $\mathbb Z_K$ and $a$ divides $m$ for some $m \in X$. Building upon the ideas of Clarkson from [Proc. Amer. Math. Soc. 17 (1966), 541], we show that, if the series $\sum_{m \in X} 1/m$ diverges, then so does the series $\sum_{\mathfrak a \in \mathcal A} |N(\mathfrak a)|^{-1/n}$. Most notably, this generalizes a class…

广义欧拉-克罗内克常数的超越性

Transcendence of generalized Euler-Kronecker constants

本文推广了数域的欧拉-克罗内克常数，并研究了这些广义常数的算术性质。

On Euler systems for the multiplicative group over general number fields

We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.

关于欧拉函数推广的一个问题

On a problem on a generalization of Euler's totient function

本文通过基于与GPT-5.5 Pro的广泛交互的论证，证明了Büyükaşik等人关于欧拉函数推广的猜想，即集合D_1 = {1, 3, 15}。

On Padé approximations and global relations of some Euler-type series

We shall consider some special generalizations of Euler's factorial series. First we construct Padé approximations of the second kind for these series. Then these approximations are applied to study global relations of certain p-adic values of the series.

Counting invertible sums of squares modulo $n$ and a new generalization of Euler totient function

In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…

The generalized-Euler-constant function $γ(z)$ and a generalization of Somos's quadratic recurrence constant

We define the generalized-Euler-constant function $γ(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $γ=γ(1)$ and the "alternating Euler constant" $\log\frac{4}π=γ(-1)$. We extend Euler's two zeta-function series for $γ$ to polylogarithm series for $γ(z)$. Integrals for $γ(z)$ provide its analytic continuation to $\C-[1,\infty)$. We prove several other formulas for $γ(z)$, including two functional equations; one is an inversion relation between $γ(z)$ and $γ(1/z)$. We generalize Somos's quadratic recurrence constant an…

On Euler's decomposition formula for qMZVs

In this short and elementary note we derive a q-generalization of Euler's decomposition formula for the qMZVs recently introduced by Y. Ohno, J. Okuda, and W. Zudilin. This answers a question posed by these authors in [10].

Le système d'Euler de Kato en famille (I)

This article is the first article of a serie of articles on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems and a family of Kato's explicit reciprocity laws over the weight space, which interpolate the corresponding classical objects.

Functional equations for double series of Euler-Hurwitz-Barnes type with coefficients

We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex coefficients. The first one is of general nature, while the second one is valid when the coefficients are Fourier coefficients of a cusp form.

完全数的推广与一些欧几里得-欧拉类型结果

A Perfect Number Generalization and Some Euclid-Euler Type Results

本文提出$\mathcal{S}$-完全数的概念，并研究了其在不同权重集合下的性质与存在性。

Hypergeometric Euler numbers

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers h…

New degenerated polynomials arising from non-classical Umbral Calculus

We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some familiar identities and connection between these kinds of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity for the degenerate Bernoulli numbers.

特定模商奇点双解析解的欧拉示性数

Euler characteristic of crepant resolutions of specific modular quotient singularities

本文研究正特征下特殊线性群有限子群商奇点双解析解欧拉示性数的McKay对应推广，利用有限域上wild McKay对应证明其对特定半直积群成立，并提出模情形下的猜想。

Romanoff定理的推广

Generalization of Romanoff's theorem

本文推广Romanoff定理，并给出一个与Euler函数相关的和的结果。

A $(p,q)$-Analogue of Poly-Euler Polynomials and Some Related Polynomials

In the present article, we introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We give several combinatorial identities and properties of these new polynomials. Moreover, we show some relations with the $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials. The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue.

Extensions of Wilson's lemma and the Ax-Katz theorem

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to present a useful lemma on polynomials modulo prime powers, and applied this lemma to reprove the Ax-Katz theorem on solutions of congruences modulo p and deduce various results on codewords in p-ary linear codes with weights. In light of the recent generalizations of Fleck's congruence given by D. Wan, and D. M. Davis and Z. W. Sun, we obtain new extensions of Wil…

On certain partition bijections related to Euler's partition problem

We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a previous paper by the author , enable us to state and prove new generalizations of two of these results.

A Unification of Two Refinements of Euler's Partition Theorem

We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.

Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials with applications to series involving zeros of Bessel functions

We introduce Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials as generalizations of Bernoulli and Apostol-Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to sum a bunch of series involving the zeros of Bessel functions.

An Elliptic $BC_n$ Bailey Lemma, Multiple Rogers--Ramanujan Identities and Euler's Pentagonal Number Theorems

An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system $BC_n$ are proved as applications, including a $_6ϕ_5$ summation formula, a generalized Watson transformation and an unspecialized Rogers--Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers--Selberg identities. Standard determinant evaluations are then used to compute $B_n$ and $D_n$ generalizations of t…

The $(k,l)$-Euler theorem and the combinatorics of $(k,l)$-sequences

In 1997, Bousquet-Mélou and Eriksson stated a broad generalization of Euler's distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvester's bijection. Subsequently, as an open questio…