Timeline for What does the generating function $x/(1 - e^{-x})$ count?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 9 at 2:10 | comment | added | Tom Copeland | The more general relation that can be used to define the Bernoulli numbers is the derivational operational definition of the Bernoulli polynomials $B_n(x) = (b.+x)^n$ where $(b.)^k =b_k = B_k(0)$ are the Bernoulli numbers: for an analytic function / Taylor series (even generalized functions) $f(x)$, $\frac{\partial}{\partial x}f(x) = f(B.(x+1))-f(B.(x))$. WIth $f(x)= e^{xt}$, this defines the e.g.f. for the Bernoulli numbers. With $f(x) = x^n$, this gives your relation. | |
| Dec 18, 2009 at 8:59 | comment | added | David Corfield | John Baez and I had a discussion about giving a species interpretation of X/(1 - e^X) here groups.google.com/group/sci.math.research/browse_thread/thread/… | |
| Dec 18, 2009 at 8:43 | comment | added | Qiaochu Yuan | You might already know this, but that 1/12 is part of the "reason" for the appearances of 12 and 24 in mathematics, as described by John Baez here: math.ucr.edu/home/baez/week126.html | |
| Dec 18, 2009 at 5:48 | comment | added | Theo Johnson-Freyd | Ah, yes, I have seen that definition. What I've never done is calculated out more than the first two or so terms, and 1, 1/2, 1/12 is meaningless, and when today I got 1, 1/2, 1/12, 0, -1/720, I still didn't have anything with which to recognize it. | |
| Dec 18, 2009 at 3:34 | history | answered | Tom Leinster | CC BY-SA 2.5 |