Faulhaber's formula

Faulhaber's formula is an important idea in mathematics. It helps us find the sum of numbers raised to a certain power. It was named after Johann Faulhaber, a mathematician from the early 1600s.

The formula shows how to add up numbers like 1^p + 2^p + 3^p all the way up to n^p. Here, p is any whole number. It gives the total as a special kind of equation involving n.

This formula uses binomial coefficients and Bernoulli numbers. It works for any power p. This makes it a useful tool for solving many types of problems in number theory and other areas of math. Today, Faulhaber's formula is still used and studied by mathematicians and students around the world.

The formula connects simple addition of powers to more complex mathematical ideas. It shows beautiful patterns in numbers. It helps us understand how sums grow and relate to polynomials. This makes it a key part of learning about sequences and series in mathematics.

The result: Faulhaber's formula

Faulhaber's formula helps us find the sum of numbers raised to a power. For example, if we want to add up all the numbers from 1 to n and raise each to the power p, there is a special way to write this sum as a polynomial (a kind of math expression) in n.

Some simple examples are well known. When p is 0, we are just adding 1 to itself n times, which gives n. When p is 1, we get the triangular numbers, like adding 1+2+3... up to n. When p is 2, we get the square pyramidal numbers, which are sums of squares like 1²+2²+3²... up to n.

triangular numbers

square pyramidal numbers

polynomial

second Bernoulli numbers

binomial coefficient

Examples

Faulhaber’s formula helps us add up powers of numbers in a simple way. For example, if we want to add the fourth powers of the first few whole numbers (like (1^4 + 2^4 + 3^4 + \ldots + n^4)), the formula tells us this sum equals (\frac{1}{5} \left(n^5 + \frac{5}{2}n^4 + \frac{5}{3}n^3 - \frac{1}{6}n\right)).

Here are some simple examples of Faulhaber’s formula:

• Adding the zeroth powers: (\sum_{k=1}^{n} k^{0} = n)
• Adding the first powers: (\sum_{k=1}^{n} k^{1} = \frac{1}{2} \left(n^2 + n\right))
• Adding the second powers: (\sum_{k=1}^{n} k^{2} = \frac{1}{3} \left(n^3 + \frac{3}{2}n^2 + \frac{1}{2}n\right))
• Adding the third powers: (\sum_{k=1}^{n} k^{3} = \frac{1}{4} \left(n^4 + 2n^3 + \frac{3}{2}n^2\right))
• Adding the fourth powers: (\sum_{k=1}^{n} k^{4} = \frac{1}{5} \left(n^5 + \frac{5}{2}n^4 + \frac{5}{3}n^3 - \frac{1}{6}n\right))
• Adding the fifth powers: (\sum_{k=1}^{n} k^{5} = \frac{1}{6} \left(n^6 + 3n^5 + \frac{5}{2}n^4 - \frac{1}{2}n^2\right))
• Adding the sixth powers: (\sum_{k=1}^{n} k^{6} = \frac{1}{7} \left(n^7 + \frac{7}{2}n^6 + \frac{7}{2}n^5 - \frac{7}{6}n^3 + \frac{1}{6}n\right))

History

Ancient period

The history of adding up numbers raised to a power goes back a long time. One of the earliest examples is finding the sum of the first n numbers. Ancient mathematicians discovered that:

• Adding the first n numbers gives you ½n² + ½n. This connects to triangular numbers.
• Adding the first n odd numbers gives you n², showing these sums create perfect squares. This was linked to figurate numbers and shapes called gnomons.
• Adding the squares of the first n numbers was explored by Archimedes in his work Spirals.
• Adding the cubes of the first n numbers was connected to a theorem by Nicomachus of Gerasa.

Middle period

Many mathematicians later studied this problem, including Aryabhata, Al-Karaji, Ibn al-Haytham, Thomas Harriot, Johann Faulhaber, Pierre de Fermat, and Blaise Pascal. They found ways to express these sums as polynomials.

In 1713, Jacob Bernoulli published a method to calculate the sum of powers using special numbers now called Bernoulli numbers. This work built on earlier discoveries and provided a general way to find these sums.

Modern period

In 1982, A.W.F. Edwards showed a new way to find these sums using patterns in triangles of numbers, similar to Pascal's triangle. This method avoids directly using Bernoulli numbers and opens up new paths for research.

Polynomials calculating sums of powers of arithmetic progressions 

Faulhaber's formula helps us find the sum of powers of numbers in an arithmetic progression. An arithmetic progression is a list of numbers where each number is a fixed amount bigger than the one before it. For example, the numbers 1, 3, 5, 7... form an arithmetic progression because each number is 2 more than the one before it.

The formula can be used to calculate sums like:

• The sum of the first n numbers raised to a power p: 1^p + 2^p + ... + n^p
• The sum of odd numbers raised to a power p: 1^p + 3^p + ... + (2n-1)^p

There are different ways to find these formulas, such as using matrices and special numbers called Bernoulli numbers. These methods help mathematicians discover patterns and work out these sums quickly.

Faulhaber polynomials

Faulhaber polynomials are special math patterns. They help us add up numbers that have been raised to a power.

For example, if we want to add up all the numbers from 1 to n and each number is raised to the power of 3, there is a neat formula. This formula gives us the answer without having to add them one by one.

Faulhaber found that for odd powers, like 3 or 5, the sum can be written using a special pattern. This pattern uses another simple sum called a. This makes calculating these sums much easier and faster!

Expressing products of power sums as linear combinations of power sums

Sometimes, we can take two math sums and multiply them to get a new sum. For example, if we multiply the sum of squares by the sum of fourth powers, we get a mix of sums of cubes, fifth powers, and seventh powers. This shows us how these sums are connected.

There are some rules that tell us how these products work. For example, if we square the sum of the first powers, we get a mix of sums of third powers. These rules help us find new ways to calculate these sums, making math simpler.

Variations

Faulhaber’s formula can be changed in a few interesting ways. One way is by switching the position of the numbers, which gives a new way to write the sum. Another way is by subtracting a value and changing another part of the formula.

We can also rewrite the formula using special number patterns called Stirling numbers and falling factorials. These help count arrangements of numbers. Using ideas like "telescoping" and the binomial theorem, we can find simpler versions or related identities.

Relationship to Riemann zeta function

Faulhaber's formula connects to the Riemann zeta function. When we add up all the powers of integers forever, it relates to special values of the zeta function. For negative whole numbers, the zeta function gives results that match patterns in Faulhaber's work.

The formula shows how sums of powers can be written using the zeta function. This link helps mathematicians learn more about these sums and the zeta function.

Main article: Riemann zeta function Main article: Hurwitz zeta function

Umbral form

In the umbral calculus, Bernoulli numbers are treated like powers of a special object. This makes Faulhaber's formula easier to understand.

Using this idea, the formula shows how the sum of powers of numbers can be written using Bernoulli numbers and special notation. This form is important in modern mathematics.