Bernoulli number?

[Leroy Quet]

>I recommend the book "Concrete Mathematics" by Graham, Knuth and
>Patashnik. It is excellent for such things and written on a rather
>elementary level. Amazon.com at 
>
>http://www.amazon.com/exec/obidos/tg/detail/-/0201558025/qid=1076086909/sr
>=1-1/ref=sr_1_1/104-2346331-5851113?v=glance&s=books
>
>has some reviews that may help you decide whether or not to get the book.
>
>--Edwin
>
>
>On Fri, 6 Feb 2004, y.kohmoto wrote:
>
>>     Hi, Seqfans
>> 
>>     Reading the mails from Seqfan mailing list, I have understood something
>> interesting related with Bernoulli Number was  discovered.
>>     But I don't understand well.
>>     What is Bernoulli number?
>>     Why do they think A001067 and A046968 should be same?
>>     Someone please explain them to me , and tell me a good book of Bernoulli
>> Number.
>> 
>>     Yasutoshi
>> 
>>     PS
>>     I must say I am an amateur  mathematician.
>>     What I know about Mathematics is what I learned in high school.
>> 

I might as well give you SOME of the Bernoulli basics...


B_0 = 1, B_1  = -1/2, B_2 = 1/6, B_3 = 0,.....
 
1) Bernoulli numbers {B_k} are rationals (positive, negative, and 0) 
which satisfy the EGF:

sum{k=0 to oo} B_k x^k /k! = 

x/(e^x -1).

(where e is base of natural logarithms, e =  2.7...)

2) Bernoulli numbers occur in the well-known sum identity:

For m = positive  integer,

sum{k=1 to n} k^m =

m! *sum{k=1 to m+1} B_{m+1-k} (-1)^(m+1-k) n^k /(k!(m+1-k)!).

(where m!/(k!(m+1-k)!) can be replaced with "C(m+1,k)/(m+1)",  where C is 
a binomial coefficient.)

3) sum{k=1 to oo} 1/k^(2m) =

zeta(2m) =

B_{2m} (-1)^(m+1) 2^(m2-1) pi^(m2) /(m2)!,

IF m is a positive integer.

(B_{2m}*(-1)^(m+1) can be replaced with |B_{2m}| here, if you wish.)

4) From this last identity, we see that

|B{2m}| ~ (2m)! pi^(-2m) 2^(1-2m),

since  zeta(r) -> 1   as  r  -> oo.

So, this gives the radius of convergence of the EGF in (1).


5) One more thing, an older alternative definition of Bernoulli numbers 
defines them with a different indexing so as to avoid those that are 
equal to 0.

(Since, by the definition I use above, B_m = 0 if m =  any odd integer >= 
3.
So, B_{2m} by the definition I use above is B_m by the older definition, 
perhaps....I am a little unsure how the older definition handles the fact 
that B_1 =  -1/2 {not 0}, as by the current definition. And perhaps the 
older definition's Bernoulli numbers are all positive...I am unsure here 
too. In any case, most  mathematicians now define Bernoulli as I have 
above, with some terms equal to 0 and some negative and some positive.)

Anyone feel like adding to this knowledge, or like correcting any of my 
errors, please feel free.

thanks,
Leroy Quet