Conflicting Conventions for Bernoulli Numbers?

In summary, the Wikipedia article on Bernoulli's numbers discusses two conventions for the numbers, which only differ at m=1. The article also provides explicit definitions for m>1, and explains that the expressions are equal except for m=1. It also mentions that the extra terms in the second expression can cancel out due to Pascal's triangle and binomial coefficients. The speaker also expresses uncertainty and asks for clarification from DrClaude.
  • #1
nomadreid
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In the Wiki article on Bernoulli numbers, it gives two expressions that, if I understand correctly, are supposed to be equal except at one point. But I am not sure I understand it correctly
In the Wikipedia article https://en.wikipedia.org/wiki/Bernoulli_number on Bernoulli’s numbers, it explains that there are two conventions which differ only at m=1. Then it says…

Bernoulli1.PNG


Under “explicit definitions”, it gives, for m>1

Bernoulli2.PNG


So, it seems pretty straightforward that they are saying that (except for m=1) these two expressions are equal, but that all the extra terms in the second expression (+) not included in the first one (-) would cancel out seems so incredible that I think I might be misinterpreting something. Am I?

Thanks
 
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  • #2
The expansion of ##(v+1)^m## follows Pascal's triangle, hence binomial coefficients, so yes, the terms can cancel out.
 
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Thanks, DrClaude
 

1. What are Bernoulli numbers?

Bernoulli numbers are a sequence of rational numbers that arise in many areas of mathematics, including number theory, algebra, and calculus. They are named after the Swiss mathematician Jacob Bernoulli, who first studied them in the early 18th century.

2. What is the confusion surrounding Bernoulli numbers?

The confusion surrounding Bernoulli numbers is due to the fact that there are two different sequences of numbers that are both referred to as Bernoulli numbers. One sequence, denoted by Bn, is used in number theory and has applications in the study of prime numbers. The other sequence, denoted by Bn*, is used in calculus and has applications in the study of power series and integrals.

3. How are the two sequences of Bernoulli numbers related?

The two sequences of Bernoulli numbers are related by a simple formula: Bn* = (-1)nBn. This means that the even-indexed terms of the number theory sequence are equal to the corresponding terms in the calculus sequence, while the odd-indexed terms have opposite signs.

4. Why is it important to distinguish between the two sequences of Bernoulli numbers?

Distinguishing between the two sequences of Bernoulli numbers is important because they have different properties and applications. For example, the number theory sequence is used to study prime numbers and has connections to the Riemann zeta function, while the calculus sequence is used to study power series and has applications in differential equations and numerical analysis.

5. How can I avoid confusion when working with Bernoulli numbers?

To avoid confusion when working with Bernoulli numbers, it is important to be clear about which sequence you are using and to use the appropriate notation. It may also be helpful to understand the relationship between the two sequences and how they are used in different areas of mathematics. Consulting reliable sources and seeking guidance from experts can also help clarify any confusion.

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