In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers
The coefficients of Faulhaber's formula in its general form involve the Bernoulli numbersBj. The Bernoulli numbers begin
where here we use the convention that . The Bernoulli numbers have various definitions (see Bernoulli number#Definitions), such as that they are the coefficients of the exponential generating function
The first seven examples of Faulhaber's formula are
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.[1]
In 1713, Jacob Bernoulli published under the title Summae Potestatum an expression of the sum of the p powers of the n first integers as a (p + 1)th-degree polynomial function ofn, with coefficients involving numbers Bj, now called Bernoulli numbers:
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
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.
A rigorous proof of these formulas and Faulhaber's assertion that such formulas would exist for all odd powers took until CarlJacobi(1834), two centuries later.
Proof with exponential generating function
Let
denote the sum under consideration for integer
Define the following exponential generating function with (initially) indeterminate
We find
This is an entire function in so that can be taken to be any complex number.
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 .
Faulhaber polynomials
The term Faulhaber polynomials is used by some authors to refer to another polynomial sequence related to that given above.
Write
Faulhaber observed that if p is odd then is a polynomial function of a.
Some authors call the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by a2 because the Bernoulli numberBj is 0 for odd j > 1.
Inversely, writing for simplicity , we have
and generally
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 than1), the sum is a polynomial in n having factors n2 and (n+1)2, while for an even power the polynomial has factors n, n+½ and n+1.
Expressing products of power sums as linear combinations of power sums
Products of two (and thus by iteration, several) power sums can be written as linear combinations of power sums with either all degrees even or all degrees odd, depending on the total degree of the product as a polynomial in , e.g. . Note that the sums of coefficients must be equal on both sides, as can be seen by putting , which makes all the equal to 1. Some general formulae include:
Note that in the second formula, for even the term corresponding to is different from the other terms in the sum, while for odd , this additional term vanishes because of .
Writing these polynomials as a product between matrices gives
where
Surprisingly, inverting the matrix of polynomial coefficients yields something more familiar:
In the inverted matrix, Pascal's triangle 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 m, 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.[3][4]
Variations
Replacing with , we find the alternative expression:
Subtracting from both sides of the original formula and incrementing by , we get
where can be interpreted as "negative" Bernoulli numbers with .
We may also expand in terms of the Bernoulli polynomials to find
which implies
Since whenever is odd, the factor may be removed when .
This is due to the definition of the Stirling numbers of the second kind as mononomials in terms of falling factorials, and the behaviour of falling factorials under the indefinite sum.
Interpreting the Stirling numbers of the second kind, , as the number of set partitions of into parts, the identity has a direct combinatorial proof since both sides count the number of functions with maximal. The index of summation on the left hand side represents , while the index on the right hand side is represents the number of elements in the image of f.
There is also a similar (but somehow simpler) expression: using the idea of telescoping and the binomial theorem, one gets Pascal's identity:[6]
This in particular yields the examples below – e.g., take k = 1 to get the first example. In a similar fashion we also find
Faulhaber's formula was generalized by Guo and Zeng to a q-analog.[7]
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 for negative integers on appropriately analytically continuing .
Umbral form
In the umbral calculus, one treats the Bernoulli numbers , , , … as if the index j in were actually an exponent, and so as if the Bernoulli numbers were powers of some object B.
Using this notation, Faulhaber's formula can be written as
Here, the expression on the right must be understood by expanding out to get terms that can then be interpreted as the Bernoulli numbers. Specifically, using the binomial theorem, we get
Classically, this umbral form was considered as a notational convenience. In the modern umbral calculus, on the other hand, this is given a formal mathematical underpinning. One considers the linear functionalT on the vector space of polynomials in a variable b given by Then one can say
A General Formula
The series as a function of m is often abbreviated as . Beardon (see External Links) have published formulas for powers of . For example, Beardon 1996 stated this general formula for powers of , which shows that raised to a power N can be written as a linear sum of terms ... For example, by taking N to be 2, then 3, then 4 in Beardon's formula we get the identities . Other formulae, such as and are known but no general formula for , where m, N are positive integers, has been published to date. In an unpublished paper by Derby (2019) [9] the following formula was stated and proved:
.
This can be calculated in matrix form, as described above. In the case when m = 1 it replicates Beardon’s formula for . When m = 2 and N = 2 or 3 it generates the given formulas for and . Examples of calculations for higher indices are and .
Johann Faulhaber (1631). Academia Algebrae - Darinnen die miraculosische Inventiones zu den höchsten Cossen weiters continuirt und profitiert werden. A very rare book, but Knuth has placed a photocopy in the Stanford library, call number QA154.8 F3 1631a f MATH. (online copy at Google Books)
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula
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 (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example, for n = 4,
In mathematics, the Bernoulli numbersBn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.
In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation.
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions:
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
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 Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas, who used the technique extensively.
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.
In mathematics, the falling factorial is defined as the polynomial
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity as n. That is,
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.
In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891).
The Bernoulli polynomials of the second kindψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function:
The polynomials calculating sums of powers of arithmetic progressions are polynomials in a variable that depend both on the particular arithmetic progression constituting the basis of the summed powers and on the constant exponent, non-negative integer, chosen. Their degree always exceeds the constant exponent by one unit and have the property that when the polynomial variable coincides with the number of summed addends, the result of the polynomial function also coincides with that of the sum.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.