Functional Equation for $\sum_{n=0}^{N}n^{k}$

Thread starter zetafunction
Start date Apr 12, 2009
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Functional

In summary, the functional equation for the sum of the first N natural numbers raised to the kth power is given by ∑n=0Nnk = (N+1)k+1 - 1. It is derived using the method of finite differences and has significance in various areas of mathematics. The equation can be extended to include negative values of N and k using complex numbers. It has many interesting properties and applications, including proving the sum of the first N cubes and Faulhaber's formula. It also has applications in physics, such as calculating moments of inertia and gravitational potential energy.

Apr 12, 2009

zetafunction

is there a functional equation for

[tex] \sum_{n=0}^{N}n^{k}=Z(N,k) [/tex]

where k and N are real numbers, in case N tends to infinite we could consider the functional equation of Riemann zeta but what happens in the case of N finite ??

Mathematics news on Phys.org

Apr 12, 2009

qntty

Here's the relevant Mathworld article.

Related to Functional Equation for $\sum_{n=0}^{N}n^{k}$

1. What is a functional equation for the sum of the first N natural numbers raised to the kth power?

The functional equation for the sum of the first N natural numbers raised to the kth power is given by:
∑_n=0^Nn^k = (N+1)^k+1 - 1

2. How is the functional equation for the sum of the first N natural numbers raised to the kth power derived?

The functional equation is derived using the method of finite differences. By taking the difference between consecutive terms and applying mathematical induction, the equation can be proven to hold for all values of N and k.

3. What is the significance of the functional equation for the sum of the first N natural numbers raised to the kth power?

This equation is useful in many areas of mathematics, including number theory, calculus, and combinatorics. It can also be used to solve various problems involving sums of powers.

4. Can the functional equation be extended to include negative values of N and k?

Yes, the equation can be extended to include negative values of N and k. However, it requires the use of complex numbers and the equation becomes:
∑_n=-m^mn^k = (2m+1)B_k+1,
where m is a non-negative integer, k is a positive integer, and B_k+1 is the (k+1)th Bernoulli number.

5. Are there any other interesting properties or applications of the functional equation for the sum of the first N natural numbers raised to the kth power?

Yes, there are many other interesting properties and applications of this equation. For example, it can be used to prove the sum of the first N cubes is equal to the square of the sum of the first N natural numbers. It is also used in the proof of Faulhaber's formula for the sums of powers. Additionally, it has applications in physics, specifically in calculating moments of inertia and gravitational potential energy.