Generalizations of Euler Numbers and Euler Numbers of Higher Order

The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.

HIGHER-ORDER MULTIVARIABLE EULER'S POLYNOMIALAND HIGHER-ORDER MULTIVARIABLEBERNOULLI'S POLYNOMIAL

In this paper, the definitons of both higher-order multivariable Euler's numbersand polynomial. higher-order multivariable Bernoulli's numbers and polynomial aregiven and some of their important properties are expounded. As a result, themathematical relationship between higher-order multivariable Euler's polynomial(numbers) and higher-order higher -order Bernoulli's polynomial (numbers) are thusobtained.

Recurrent formula of Bernoulli numbers and the relationships among the coefficients of beam,Bernoulli numbers and Euler numbers

Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.

Multiparameter Higher Order Daehee and Bernoulli Numbers and Polynomials

This paper gives a new generalization of higher order Daehee and Bernoulli numbers and polynomials. We define the multiparameter higher order Daehee numbers and polynomials of the first and second kind. Moreover, we derive some new results for these numbers and polynomials. The relations between these numbers and Stirling and Bernoulli numbers are obtained. Furthermore, some interesting special cases of the generalized higher order Daehee and Bernoulli numbers and polynomials are deduced.

Mass Constituents of a Flat Lattice Multiverse: Conclusion from Similarity between Two Universal Numbers, the Rocksalt-Type 2<i>D</i>Madelung Constant and the Golden Mean

In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean &phi;=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by &phi;, one reaches again 1 + &phi;as the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ&minus;1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number &phi;may indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant &pi;, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.

一个包含勾股数的Euler函数非线性方程的正整数解

利用欧拉函数的性质与初等数论的方法,讨论包含勾股数的Euler函数非线性方程φ(xyz)=aφ(x)+bφ(y)+cφ(z)-m(a,b,c为勾股数且gcd(a,b,c)=1),当(a,b,c)=(3,4,5)且m=16时的正整数解情况,并证明该方程共有28组正整数解。

Special Numbers on Analytic Functions

The aim of this paper is to give some analytic functions which are related to the generating functions for the central factorial numbers. By using these functions and p-adic Volkenborn integral, we derive many new identities associated with the Bernoulli and Euler numbers, the central factorial numbers and the Stirling numbers. We also give some remarks and comments on these analytic functions, which are related to the generating functions for the special numbers.

New Extension of Unified Family Apostol-Type of Polynomials and Numbers

The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of generalized polynomials.

Sums of Involving the Harmonic Numbers and the Binomial Coefficients

Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.

Formulas for Coefficients of Hundred Cyclotomics and Numbers Battle

In this paper, we will establish a formula for calculating the 3144 coefficients coe(n, i) of the first hundred cyclotomic of index?n in xi. We will only determine 1003 for an index n odd and a degree . The others will be deduced, we’ll see how. The formula is , without exception if u(n)=-1?or if 4 doesn’t divide and with its 165 exceptions of which 7 when u(n)=0?and 158 when u(n)=1?that will be shared in 154 and 4 pairs (n, i), which we will specify the conditions and values of the coefficients. According to u(n), according to the class of i modulo p, the first factor of the prime factor decomposition of n when u(n)=1?and according to gcd(n, i), the formula will or will not be valid and replaced otherwise by the good value that will be 0 for 152 pairs (n,i) or 1 in the 13 other exceptions.

The “3 Genomic Numbers” Discovery: How Our Genome Single-Stranded DNA Sequence Is “Self-Designed” as a Numerical Whole

This article proves the existence of a hyper-precise global numerical meta-architecture unifying, structuring, binding and controlling the billion triplet codons constituting the sequence of single-stranded DNA of the entire human genome. Beyond the evolution and erratic mutations like transposons within the genome, it’s as if the memory of a fossil genome with multiple symmetries persists. This recalls the “intermingling” of information characterizing the fractal universe of chaos theory. The result leads to a balanced and perfect tuning between the masses of the two strands of the huge DNA molecule that constitute our genome. We show here how codon populations forming the single-stranded DNA sequences can constitute a critical approach to the understanding of junk DNA function. Then, we suggest revisiting certain methods published in our 2009 book “Codex Biogenesis”. In fact, we demonstrate here how the universal genetic code table is a powerful analytical filter to characterize single-stranded DNA sequences constituting chromosomes and genomes. We can then show that any genomic DNA sequence is featured by three numbers, which characterize it and its 64 codon populations with correlations greater than 99%. The number “1” is common to all sequences, expressing the second law of Chargaff. The other 2 numbers are related to each specific DNA sequence case characterizing life species. For example, the entire human genome is characterized by three remarkable numbers 1, 2, and Phi = 1.618 the golden ratio. Associated with each of these three numbers, we can match three axes of symmetry, then “imagine” a kind of hyperspace formed by these codon populations. Then we revisit the value (3-Phi)/2 which is probably universal and common to both the scale of quarks and atomic levels, balancing and tuning the whole human genome codon population. Finally, we demonstrate a new kind of duality between “form and substance” overlapping the whole human genome: we will show that—simultaneously with the duality between genes and junk DNA—there is a second layer of embedded hidden structure overlapping all the DNA of the whole human genome, dividing it into a second type of duality information/redundancy involving golden ratio proportions.

The History of the Derivation of Euler’s Number

This paper summarizes the works of numerous inspiring hard-working mathematicians in chronological order who have toiled the fields of mathematics to bring forward the harvest of Eulers number, also known as Napiers number or more infamously, e.

On Polynomials Rn(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on

In this paper, one introduces the polynomials Rn(x) and numbers Rn and derives some interesting identities related to the numbers and polynomials: Rn and Rn(x). We also give relation between the Stirling numbers, the Bell numbers, the Rn and Rn(x).

Fast Object Extraction and Euler Number on Block Represented Images

The identification of objects in binary images is a fundamental task in image analysis and pattern recognition tasks. The Euler number of a binary image is an important topological measure which is used as a feature in image analysis. In this paper, a very fast algorithm for the detection and localization of the objects and the computation of the Euler number of a binary image is proposed. The proposed algorithm operates in one scan of the image and is based on the Image Block Representation (IBR) scheme. The proposed algorithm is more efficient than conventional pixel based algorithms in terms of execution speed and representation of the extracted information.

A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given.

Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus

Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all πm as series on k-m and for expanding functions into series of Euler polynomials.

The Theoretical Significance and Reality of Imaginary Number

It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.

Euler示性数的相交数表示

Euler示性数作为拓扑学中的一个重要不变量,其涉及了组合中的Euler定理、代数拓扑中的Euler-Poincaré公式和微分拓扑中的Poincaré-Hopf指标定理。本文主要研究Euler示性数的一种几何拓扑表示,即相交数表示。一方面,设M是一个可定向的n维光滑紧流形,记Δ:M→M×M为对角嵌入,其像为N_1;另一方面,设s:M→TM为一个零截口,s (M)可嵌入M×M中,其像为N_2。本文证明了M的Euler示性数等于N_1与N_2的相交数,即χ(M)=N_1·N_2。

Bernoulli数和Euler数的关系

本文给出了Bernoulli数和Euler数之间的关系 ,从而深化和补充了文献 [1～

高阶Bernoulli多项式和高阶Euler多项式的关系

利用发生函数的方法,讨论了高阶Bernoulli数和高阶Euler数,高阶Bernoulli多项式和高阶Euler多项式之间的关系,得到了经典Bernoulli数和Euler数,经典Bernoulli多项式和Euler多项式之间的新型关系.