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

In summary: I have looked all over and I cannot seem to find anything about it. I have even looked at the history of calculus and failed to find anything that would lead me to this method. I hope someone can help me out. Thanks!
  • #1
sennyk
73
0
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.
 

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  • #2
I'd appreciate any feedback that I can get. I can't find this method documented anywhere.
 
  • #3
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
 
  • #5
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
 
  • #6
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
 
  • #7
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.
 

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

1. What is the Integer Power Sum for p = 0, 1, 2?

The Integer Power Sum for p = 0, 1, 2 is a mathematical formula that calculates the sum of a set of integers raised to a specified power. In this case, the power (p) can be 0, 1, or 2.

2. How do you calculate the Integer Power Sum for p = 0, 1, 2?

To calculate the Integer Power Sum for p = 0, 1, 2, you first need to determine the set of integers you want to sum. Then, you raise each integer to the specified power (0, 1, or 2) and add all the results together. For example, if the set of integers is {2, 4, 6} and p = 1, the Integer Power Sum would be calculated as (2^1) + (4^1) + (6^1) = 12.

3. What is the significance of p = 0, 1, 2 in the Integer Power Sum?

The value of p (0, 1, 2) in the Integer Power Sum formula represents the power to which the integers are raised. This allows for flexibility in the calculation and can be used to solve different types of mathematical problems.

4. What is the difference between Integer Power Sum for p = 0, 1, 2 and other power sums?

The main difference between the Integer Power Sum for p = 0, 1, 2 and other power sums is the value of p. In other power sums, p can be any positive integer, whereas in the Integer Power Sum, it is limited to 0, 1, or 2. This can affect the outcome of the calculation and the types of problems that can be solved using this formula.

5. How is the Integer Power Sum for p = 0, 1, 2 used in scientific research?

The Integer Power Sum for p = 0, 1, 2 is commonly used in scientific research as a way to calculate the sum of a set of integers with a specific power. It can be applied in various fields such as physics, statistics, and engineering to solve complex problems and analyze data. It is also a useful tool in mathematical modeling and simulations.

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