Now, let consider n=3,

When n=3, the sums of power for p=3 reduces into:

Let

Then

Or

Now consider this equation

Where

and

Assuming w is an even, thus

and

Therefore

Solving the equation yields

Let s=1, then

w=12

Solving the equation yields,

, andand z=2a=6

Therefore, there is a solution for this equation, which is given as follows

Now consider when n=2 and using the same procedure.

When n=2, the sums of power for p=3 reduces into:

Let

Then

Assuming w is an even, thus

and

Solving the equation yields:

since w=2a=2(s/2)=s,

This is a trivial solution,

or

Consider

and

Solving the equations yields:

Imaginary Solution.]]>Let the expansion of (x+y) as follows:

Now let the T-th term of arithmetic progression as (a+bi).

Thus,

Summing the terms above yields:

Example:

=>

=>

As the p is getting larger, the calculation would be becoming tedious.

]]>You had an extra \left in there.

Btw, I would imagine it to be O_{m,k}, because the value of j in the expression isn't really constant...

]]>For odd power:

For even power:

Basically, there are many Bernoulli's formulations and the finding of new Bernoulli's formulation not that significant. In my paper, the development of new bernoulli's formulation is only a small portion and without this formulation I can get others formulation to get the numbers. Since, Sums of power got bernoulli's numbers in it, I managed to manipulate it to get new forms of bernoulli's formulation but the main purpose is to develop sums of power for arithmetic progression. I do believe finding new sums of power for arithmetic progression is a big thing before people get to know it. It can be used for numerical analysis, Riemman's zeta function, Fermat's Last Theorem, generating function for finding prime numbers etc. I had demonstrated few examples of the use of this formulation.

Fulhaber is known as one of the greatest mathematicians because he developed sums of power for integers. This encourage me to work on bringing this formulation to the world as it is the umbrella for all sums of power because it can do integers, non-integers, integer power, complex power and many more.

Here the bernoulli's formulation:

]]>You go to vixra dot org/numth/ and type sums of power and all five papers are mine. Select "A Treaty of Symmetric Function Part 1 Sums of Power" and read version 2 (v2).

I am trying to understand what you are trying to accomplish (I remember reading something about symmetry), and it's very interesting! I have recently just begun to study the sums of power generalized formula (well, I actually made one from scratch before I saw any existing ones), and one thing from "Part I" caught my attention (among other things, but...)

"The Derivation of Bernoulli Number for the Generalized Equation for Sums of Power", which begins on page 38 and concludes on page 40, inspired me to create my own Bernoulli Number generating function (I think that's what it is) which only relies on the preceding non-zero values from my own general sum of power formula (but it can generate the values for all *m*).

This was the result.

So the first Bernoulli number we can make with this is B[2] = 1/6 (which I believe is what yours starts off with too!)

I was also just adjusted this to make a polynomial (I guess it's considered a Bernoulli Polynomial, although my polynomial is not the same as the Bernoulli Polynomials described on Wikipedia):

, if all a "Bernoulli Polynomial" means is that one will get a Bernoulli number when 1 is substituted in for the variable of the polynomial.

Anyway,

By showing a Bernoulli number function from your generalized sum of power polynomial in your paper, are you saying that your Bernoulli function may be more useful in actual applications to more advanced concepts in number theory than the rest of the Bernoulli generating functions which existed already (and mine which at least exists now, if someone already didn't find it before)? If so, what is that exactly? (I'm not being critical at all. I am just curious because it appears that your work has been created for a "higher purpose" than just another sums of power equation).

Now, I have just mainly skimmed through Part I, so if you already explained this in any of those 5 or so papers, then I would appreciate direction to the paper and page number which answers my question (just in case you don't want to say the same thing twice).

]]>Where:

and

]]>Yes, I know who he is.

I will help you with the latex if you need it. Just post the pieces you are having problems with. Good luck with publication!

]]>Thanks for your comment. For your information, I have emailed the link to John Coates, Andrew Wiles advisor when he was in Cambridge. He said it could be new since it could do sums of power for arithmetic progression but he hasn't got time to look into the 17-19th century papers and p-adic articles. I need to change the format of the paper to latex before I could submit the paper to a mathematical journal. Thanks again with the latex info, I thought it was difficult to use it.

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