In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Faulhaber's formula, named after the early 17th century mathematician
Johann Faulhaber Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician.
Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. Bes ...
, expresses the sum of the ''p''-th powers of the first ''n'' positive integers
:
as a (''p'' + 1)th-degree
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
function of ''n'', the coefficients involving
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s ''B
j'', in the form submitted by
Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
and published in 1713:
:
where
is a
falling factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
.
History
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients later discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below.
[ The arxiv.org paper has a misprint in the formula for the sum of 11th powers, which was corrected in the printed version]
Correct version.
/ref>
A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until , two centuries later.
Faulhaber polynomials
The term ''Faulhaber polynomials'' is used by some authors to refer to something other than the polynomial sequence given above. Faulhaber observed that if ''p'' is odd, then
:
is a polynomial function of
:
In particular:
:
:
:
:
:
The first of these identities (the case ''p'' = 3) is known as Nicomachus's theorem
In number theory, the sum of the first cubes is the square of the th triangular number. That is,
:1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2.
The same equation may be written more compactly using the mathematical notation for summa ...
.
More generally,
:
Some authors call the polynomials in ''a'' on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by because the Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
is 0 for odd .
Faulhaber also knew that if a sum for an odd power is given by
:
then the sum for the even power just below is given by
:
Note that the polynomial in parentheses is the derivative of the polynomial above with respect to ''a''.
Since ''a'' = ''n''(''n'' + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in ''n'' having factors ''n''2 and (''n'' + 1)2, while for an even power the polynomial has factors ''n'', ''n'' + ½ and ''n'' + 1.
''Summae Potestatum''
In 1713, Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
published under the title ''Summae Potestatum'' an expression of the sum of the powers of the first integers as a ()th-degree polynomial function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
of , with coefficients involving numbers , now called Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s:
:
Introducing also the first two Bernoulli numbers (which Bernoulli did not), the previous formula becomes
:
using the Bernoulli number of the second kind for which , or
:
using the Bernoulli number of the first kind for which
For example, as
:
one has for ,
:
Faulhaber himself did not know the formula in this form, but only computed the first seventeen polynomials; the general form was established with the discovery of the Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
(see History section). The derivation of Faulhaber's formula is available in ''The Book of Numbers'' by John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
and Richard K. Guy.
There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
, one gets ''Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Fren ...
's identity'':
:
This in particular yields the examples below – e.g., take to get the first example. In a similar fashion we also find
:
Examples
: (the triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s)
: (the square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broa ...
s)
: (the triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s squared)
:
:
:
From examples to matrix theorem
From the previous examples we get:
Writing these polynomials as a product between matrices gives
Surprisingly, inverting the matrix of polynomial coefficients yields something more familiar:
In the inverted matrix, Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
can be recognized, without the last element of each row, and with alternating signs.
Let be the matrix obtained from by changing the signs of the entries in odd diagonals, that is by replacing by , let be the matrix obtained from with a similar transformation, then
and also
This is because it is evident that
and that therefore polynomials of degree of the form subtracted the monomial difference they become .
This is true for every order, that is, for each positive integer , one has and
Thus, it is possible to obtain the coefficients of the polynomials of the sums of powers of successive integers without resorting to the numbers of Bernoulli but by inverting the matrix easily obtained from the triangle of Pascal.
Proof with exponential generating function
Let
:
denote the sum under consideration for integer
Define the following exponential generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
with (initially) indeterminate
:
We find
:
This is an entire function in so that can be taken to be any complex number.
We next recall the exponential generating function for the Bernoulli polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.
These polynomials occur in ...
:
where denotes the Bernoulli number with the convention . This may be converted to a generating function with the convention by the addition of to the coefficient of in each ( does not need to be changed):
:
It follows immediately that
:
for all .
Alternate expressions
* By relabelling we find the alternative expression
* We may also expand in terms of the Bernoulli polynomials to find which implies Since whenever is odd, the factor may be removed when .
* It can also be expressed in terms of Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
and falling factorials asConcrete Mathematics
''Concrete Mathematics: A Foundation for Computer Science'', by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatmen ...
, 1st ed. (1989), p. 275. This is due to the definition of the Stirling numbers of the second kind as mononomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
s in terms of falling factorials, and the behaviour of falling factorials under the indefinite sum In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x or \Delta^ , is the linear operator, inverse of the forward difference operator \Delta . It relates to the forward difference operator ...
.
* Faulhaber's formula was generalized by Guo and Zeng to a -analog.
Relationship to Riemann zeta function
Using , one can write
:
If we consider the generating function in the large limit for , then we find
:
Heuristically, this suggests that
:
This result agrees with the value of the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
for negative integers on appropriately analytically continuing .
Umbral form
In the classical umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
one formally treats the indices ''j'' in a sequence ''B''''j'' as if they were exponents, so that, always considering the variant ,[ in this case we can apply the ]binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
and say
In the ''modern'' umbral calculus, one considers the linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
''T'' on the vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of polynomials in a variable ''b'' given by
Then one can say
See also
*
Notes
External links
*
*
* A very rare book, but Knuth has placed a photocopy in the Stanford library, call number QA154.8 F3 1631a f MATH. ()
* (Winner of
Lester R. Ford Award
*
*{{Cite news
, last=Orosi
, first=Greg
, year=2018
, title=A Simple Derivation Of Faulhaber's Formula
, url=http://www.math.nthu.edu.tw/~amen/2018/AMEN-170803.pdf
, periodical = Applied Mathematics E-Notes
, volume =18
, pages =124–126
A visual proof for the sum of squares and cubes
Finite differences