#bernoulli numbers

[jackseatscarbs]

“This brain of mine is more than merely mortal, as time will show.”

[bookpiofficial]

Sums of Powers of Integers and Bernoulli Numbers Clarified | Chapter 02 | Theory and Applications of Physical Science Vol. 2

We exposes a very simple method for calculating at the same time the sums of powers of the first integersand the Bernoulli numbers.  This is possible thank to integrations of the equationwhich lead to a formula saying that the vector is the transform of the vector by a matrix built from the Pascal triangle and obtainable by a simple algorithm. Very useful relations between the sums, the Bernoulli numbers are deduced, leading straightforwardly to known and new properties of them.  The proof of the Faulhaber formulae on powers sums are outlined briefly at the end.

We exposes a very simple method for calculating at the same time the sums of powers of the first integersand the Bernoulli numbers.  This is possible thank to integrations of the equationwhich lead to a formula saying that the vector is the transform of the vector by a matrix built from the Pascal triangle and obtainable by a simple algorithm. Very useful relations between the sums, the Bernoulli numbers are deduced, leading straightforwardly to known and new properties of them.  The proof of the Faulhaber formulae on powers sums are outlined briefly at the end.

Author(s) Details

Do Tan Si HoChiMinh-city Physical Association, Vietnam and Université libre de Bruxelles and UEM, Belgium.

View Book: http://bp.bookpi.org/index.php/bpi/catalog/book/115

[wolframalpha]

Wolfram Language Sandbox provides free, instant access to the Wolfram Language in a notebook view.

Happy Ada Lovelace Day! Read up on Bernoulli Numbers & Ada Lovelace with this interactive notebook & Stephen Wolfram’s blog here.

[eka-mark]

[I]n the book called Katsuyo Sanpo (“essentials of the art of calculation”) by the outstanding Japanese mathematician Takakazu Seki, published also posthumously, in 1712 (and thus 1 year before Bernoulli [posthumously published the same result]!), the formula for the sums of powers and the inductive definition of the Bernoulli numbers are given. His formula and definition are completely the same as Bernoulli’s.

Bernoulli Numbers and Zeta Functions -  Arakawa & Ibukiyama & Kaneko

[adaplay]

What is the exact algorithm Ada Lovelace wrote?

From this answer to the question: “What is the exact algorithm Ada Lovelace wrote?”

It appears in "Note G" of the document linked here.  The first third or so of the document in the link, way before the "Note G" part, is Ada Lovelace's translation of a published transcript of a lecture that Charles Babbage gave in Italy. Babbage was English, of course, but the published transcript of the lecture was in French, so Babbage asked Lovelace to translate it into English so it could be published in English, too. The rest of the document is Lovelace's "Notes" on the translation, which in fact are much more detailed than the lecture itself. "Note G" (the part I linked to) contains the bit that people usually refer to as the world's first computer program. It begins with a general discussion of what the analytical engine can, and cannot, do. The discussion of the specific problem solved by her algorithm begins at the sentence "We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers." [In slightly more modern language: there's this sequence of numbers called the "Bernoulli numbers". It's rather difficult to calculate the elements of the sequence of numbers by hand--- it's exactly the kind of computation you would go to a computer or calculator for today. Because it's a difficult thing to do by hand, she uses it as an illustration of a calculation that the computer could handle.] She then mentions various ways of computing the sequence of Bernoulli numbers, all of which would have been well known to a mathematical audience of the time. She points out that one of these methods---- the one implicitly described by the equation she labels (8.)--- is particularly well suited to implementation in Babbage's computer. Then she shows how to do implement it in Babbage's computer. The specific discussion of how she implements it in Babbage's computer begins roughly around the sentence "The diagram represents the columns of the engine when just prepared for computing B2n-1 (in the case of n=4); while the table beneath them presents a complete simultaneous view of all the successive changes which these columns then severally pass through in order to perform the computation." and continues until the end of the document. The diagram she refers to is given in this link (of course, it is also linked in first document, in a sidebar in the middle of "Note G"). The diagram illustrates how the computer would, in principle, step through the computation that she had set up. 

I should say that even if you know what the algorithm is doing, it is rather difficult to read the original description, because the language of the 1840s was rather different from the language of today. But it is very impressive: Ada Lovelace was not only the first person to write a program for a specific general-purpose computer. She was the first person to write *about* how to write a program for a specific general-purpose computer. This is a difficult even today (most programmers are terrible writers), and by modern standards, her writing holds up very well. Much better, in fact, than the design of the computer itself. (As you probably know, the "Analytical engine" was not built in Babbage's or Lovelace's lifetime, because the state of technology at the time made it too expensive to construct.)

[scirp]

Read the full paper at: http://www.scirp.org/journal/PaperInformation.aspx?PaperID=49482 DOI: 10.4236/am.2014.516246 Author(s) Haifeng Xu, Jiuru Zhou Affiliation(s) School of Mathematical Sciences, Yangzhou University, Jiangsu, China. School of Mathematical Sciences, Yangzhou University, Jiangsu, China. ABSTRACT By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, we find that this special integral is two times of another special integral. By using this fact we give an easy way to calculate the value of the alternating sum of without using the Fourier expansion. Also, we discuss the relationship between Genocchi numbers and Bernoulli numbers and get some results about Bernoulli polynomials.eww140917gjr KEYWORDS Basel Problem, Zeta Function, Bernoulli Numbers, Bernoulli Polynomials

[acehotel]

A graph of the Bernoulli Numbers by French mathematician Simon Plouffe, posted today in honor of Ada Lovelace, born this day nearly two hundred years ago to poet Lord Byron and mathematician Anne Isabella Milbanke, the "princess of parallelograms," who school Ada intensively in mathematics in order to disuade her father's moody and rebellious nature from taking hold of her. Ada wrote a scientific paper in 1843 that anticipated the development of computer software, artificial intelligence and computer music, and she devised a method of using punchcards to calculate Bernoulli numbers, thus becoming the first computer programmer. She is known today as the Patron Saint of Female Hackers.

[eka-mark]

Happy birthday to Ada Lovelace, author of the first algorithm intended to be run on a machine (and thus considered the first computer programmer).

In particular, she wrote an algorithm for computing the Bernoulli numbers on Charles Babbage's Analytical Engine.