Faulhaber's formula

Faulhaber's formula is an important idea in mathematics that 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, where p is any whole number, and express that total as a special kind of equation involving n.

This formula uses something called binomial coefficients and Bernoulli numbers to create a polynomial that gives the exact sum. It works for any power p, making it a powerful 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, showing beautiful patterns in numbers. It helps us understand how sums grow and relate to polynomials, making 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 gives us simple ways to add up powers of numbers. For example, if we want to add up the fourth powers of the first few whole numbers (like (1^4 + 2^4 + 3^4 + \ldots + n^4)), the formula tells us that this sum equals (\frac{1}{5} \left(n^5 + \frac{5}{2}n^4 + \frac{5}{3}n^3 - \frac{1}{6}n\right)).

The first few examples of Faulhaber’s formula are:

• 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 sequence of numbers where each term increases by a constant difference. For example, the sequence 1, 3, 5, 7... is an arithmetic progression with a difference of 2.

The formula can 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 methods to derive these formulas, including using matrices and special numbers called Bernoulli numbers. These methods allow mathematicians to find patterns and compute these sums efficiently.

Faulhaber polynomials

The term Faulhaber polynomials refers to a special kind of math pattern. These patterns help us find the sum of numbers raised to a certain power. For example, if we want to add up all the numbers from 1 to n each raised to the power of 3, there is a neat formula that tells us the answer without having to add them one by one.

Faulhaber noticed that for odd powers, like 3 or 5, the sum can be written as a special kind of pattern using 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

Products of two power sums can be written as combinations of other power sums. For example, multiplying the sum of squares by the sum of fourth powers gives a mix of sums of cubes, fifth powers, and seventh powers. This helps us understand how these sums relate to each other.

Some general rules show how these products break down. For instance, squaring the sum of first powers leads to a combination of sums of third powers. These relationships can be used to find new ways to calculate these sums, making complex math easier to handle.

Variations

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

There are also ways to rewrite the formula using special number patterns called Stirling numbers and falling factorials. These help count certain arrangements of numbers, giving a deeper understanding of the formula. Additionally, using ideas like "telescoping" and the binomial theorem, we can find simpler versions or related identities that work in similar ways.

Relationship to Riemann zeta function

Faulhaber's formula can also be connected to the Riemann zeta function. When we look at the sum of all powers of integers going to infinity, this relates to special values of the zeta function. For negative whole numbers, the zeta function gives specific results that match patterns found in Faulhaber's work.

The formula shows how sums of powers can be expressed using the zeta function, linking discrete sums to deeper areas of mathematical analysis. This connection helps mathematicians understand both the behavior of these sums and the properties of the zeta function itself.

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

Umbral form

In the umbral calculus, Bernoulli numbers are treated as if they were powers of an object. This helps rewrite Faulhaber's formula in a simpler way.

Using this idea, the formula becomes easier to understand. It shows how the sum of powers of numbers can be expressed using Bernoulli numbers and a special notation. This form has been studied and given more meaning in modern mathematics.