Kim, An explicit formula on the generalized Bernoulli number
with order n, Indian J.
j] is the j-th Bernoulli number
, [alpha] > 0 and [beta] > 0 satisfy [alpha][beta] = [[pi].
They included a way to calculate Bernoulli numbers
(a mathematical sequence of numbers) using the machine.
Mohamed Altoumaimi from Iraq solved the Bernoulli numbers
problem at the age of 16 in 2009 - a formula had puzzled the greatest mathematicians for 300 years.
In just four months, Mohamed Altoumaimi found a formula to explain and simplify the so-called Bernoulli numbers
, a sequence of calculations connected to numbe theory named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said.
13] Oiu-Ming Luo, Bai-Ni, Feng Qi, and Lokenath Debnath, Generalizations of Bernoulli numbers
Stein appeals to graduates, advanced undergraduates and non-specialists in number theory as be describes the modular forms of weights and levels, Dirichlet characters, Eisenstein series and Bernoulli numbers
, dimensions formulas, linear algebra, general modular symbols, computing with newforms, and computing periods.
He'd written a paper about Bernoulli numbers
and I realised that I had noticed some patterns which even he hadn't spotted.
Among these are the Pythagorean theorem, how Archimedes discovered integration, the Bernoulli numbers
and some wonderful discoveries of Euler, and Kepler's laws and Newton's law of gravitation.
The purpose of this paper is to construct complex analytic multiple L-function and to define the generalized multiple Bernoulli numbers
with [chi], which can be viewed in the interpolating of the multiple L-function at negative integers.
They cover the elementary methods, Bernoulli numbers
, including the Riemann Zeta function and the Euler-MacLaurin sum formula, modular forms and Hecke's theory of modular forms, representations of numbers as sums of squares, including the singular series and Liouville's methods and elliptical modular forms, arithmetic progression, and applications such as computing sums of two to four squares, resonant cavities and diamond cutting.
were first introduced by Jacques Bernoulli (1654-1705), in the second part of his treatise published in 1713, Ars conjectandi , at the time, Bernoulli numbers
were used for writing the infinite series expansions of hyperbolic and trigonometric functions.