Integer Power Sum for p = 0, 1, 2,

[sennyk]

The other day I was thinking about the integer power sum and the general solution for each value of p. I came up with a method that will allow me to calculate the general solution. I thought that I may have stumbled upon something novel, because I couldn't find any reference to this method anywhere. I was hoping that someone could direct me to some published material that describes this method and sheds some light about the history of this method.

NOTE: I'm a wannabe mathematician that is currently employed as an engineer. Please excuse me if this is too elementary.

 

[sennyk]

I'd appreciate any feedback that I can get. I can't find this method documented anywhere.

 

[lurflurf]

Trivial
involves Bernoulli numbers
Consult for instance any book on calculus or discrete math.
This can be done many ways
-undetermined coefficients
prove the sum of a polynomial is a polynomial
do specific cases by solving linear equations
-repeated summation
write
x^(n+1)=x^n+...+x^n (x times)
intercge sums
-use generating functions
-use the homogeneous operator
[xD](x^k)=k*x^k
{[xD]^l}(x^k)=(k^l)*x^k
-use the equivelence of summation and antidifferencing

 

[lurflurf]

(1631) Academiae Algebrae, J. Faulhaber
http://mathworld.wolfram.com/FaulhabersFormula.html

 

[sennyk]

I've seen Faulhaber's formula. I've read the mathworld methods at least 3 times. I'm looking for more info my specific method. I've pulled out my calculus and discrete math books, but I've seen nothing that uses my method to find the solution to the power series.

Have you seen the method that I posted? If so, please direct me to some reading about who first used it and maybe how that person arrived upon this method.

Thanks,

Ken

 

[lurflurf]

Why would there be references or a know inventor? Your method is just a backwards way of doing the method of undetermined coefficients. It is obvious and easily proven (and well known) that the sum of a polynomial is another polynomial of degree one higher. This is the finite calculus equivelant of the infinitesimal calculus result that the integral of a polynomia is another of degree one higher. Solving a linear system is one obvious way to find a particular sum, though neither efficient nor elegant.

some bedtime reading for you
http://arxiv.org/abs/math.CA/9207222

 

[sennyk]

Thank you.

Thank you for the link; it is very interesting. More importantly thanks for the site that hosts the link.

I apologize if it was too trivial. It just appeared to me that everyone dismissed the rectangle idea after it doesn't just work for p = 2. I found the system of equations method shown by Shultz on mathworld, but it wasn't the same idea.

What I really thought was that this method was the one that I discovered without a reference, so it must be published somewhere.