This page is currently being merged. After a discussion, consensus to merge this pageinto Faulhaber's formula was found. You can help implement the merge by following the instructions at Help:Merging and the resolution on the discussion. Process started in October 2022. |
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.
The problem therefore consists in finding i.e. polynomials as a function of calculating sums of addends:
with [1] and integers positive, first term of an arithmetic progression and the common difference. The two parameters can be not only integers but also rational, real and even complex.
The history of the problem begins in antiquity and coincides with that of some of its special cases. The case coincides with that of the calculation of the arithmetic series, the sum of the first values of an arithmetic progression. This problem is quite simple but the case already known by the Pythagorean school for its connection with triangular numbers is historically interesting:
For the first cases encountered in the history of mathematics are:
L'insieme of the cases, to which the two preceding polynomials belong, constitutes the classical problem of powers of successive integers.
Over time, many other mathematicians became interested in the problem and made various contributions to its solution. These include Aryabhata, Al-Karaji, Ibn al-Haytham, Thomas Harriot, Johann Faulhaber, Pierre de Fermat and Blaise Pascal who recursively solved the problem of the sum of powers of successive integers by considering an identity that allowed to obtain a polynomial of degree already knowing the previous ones. [2]
In 1713 the family of Jacob Bernoulli posthumously publishes his Artis Conjectandi [3] where the first 10 polynomials of this infinite series appear together with a general formula dependent on particular numbers that were soon named after him. The formula was instead attributed to Johann Faulhaber [4] for his worthy contributions recognized by Bernoulli himself. It was also immediately clear that the polynomials calcolating the sum of powers of successive integers starting from zero were very similar to those starting from one. This is because it is evident that and that therefore polynomials of degree of the form [3] subtracted the monomial difference they become .
However, a proof of Faulhaber's formula was missing, which was given more than a century later by Carl G. Jacobi [5] who benefited from the progress of mathematical analysis using the development in infinite series of an exponential function generating Bernoulli numbers.
In 1982 A.W.F. Edwards publishes an article [6] in which he shows that Pascal's identity can be expressed by means of triangular matrices containing the Pascal's triangle deprived of 'last element of each line:
The example is limited by the choice of a fifth order matrix but is easily extendable to higher orders. The equation can be written as: and multiplying the two sides of the equation to the left by , inverse of the matrix A, we obtain which allows to arrive directly at the polynomial coefficients without directly using the Bernoulli numbers. Other authors after Edwards dealing with various aspects of the power sum problem take the matrix path [9] and studying aspects of the problem in their articles useful tools such as the Vandermonde vector. [10] Other researchers continue to explore through the traditional analytic route [11] and generalize the problem of the sum of successive integers to any geometric progression [12] [13]
The coefficients of the polynomials are found through recursive formulas and in other ways that are interesting for number theory as the expression of the result of the sum as a function of Bernoulli polynomials or the formulas involving the Stirling numbers and the r-Whitney numbers of the first and second kind [14] Finally, Edwards' matrix approach was also generalized to any arithmetic progressions [15]
The general problem has recently been solved [15] through the use of binomial matrices easily constructible knowing the binomial coefficients and the Pascal's triangle. It is shown that, having chosen the parameters and which determine the arithmetic progression and a positive integer we find polynomials corresponding to the following sums of powers:
with the polynomial coefficients elements of the row of the triangular matrix of order .
Here is the solving formula in the particular case which gives the polynomials of a given arithmetic progression with exponents from 0 to 3:
The equation that can be easily extended to different values of m (non-negative integers) is summarized and generalized as follows:
or also by placing with
Here is the rigorous definition of the matrices and the Vandermonde vector:
for it results therefore
Matrix A is that of Edwards [8] already seen, a lower triangular matrix that reproduces, in the non-null elements, the triangle of Pascal deprived of the last element of each row. The elements of on the other hand are the monomials of the power development for .
is the neutral element of the row by column product so that the general equation in this case becomes:
that is the one discovered by Edwards [8]
To arrive from this particular case to prove the general one, it is sufficient to multiply on the left the two members of the equation by the matrix after having ascertained the following identity [15]
We use the previous formula to solve the problem of adding powers of successive odds:. [16] The odds correspond to the arithmetic progression with the first element and as reason We set m = 4 to find the first five polynomials calculating sums of powers of odd. Calculated we obtain:
We have therefore
At this point the general equation for and the damage done product:
using the last line () we get then
and using the other rows:
Chosen and calculated and T(1,1) which corresponds to Pascal's triangle:
Chosen and calculated and T(0,1) unit matrix:
Chosen again , calculated , exploited the result of the previous paragraph and the associative property:
The matrix can be expressed as a function of the Bernoulli polynomials in the following way [17] [18]
which for becomes
from which the generalized Faulhaber formula is derived:
and also the well-known special cases:
where the Bernoulli polynomials calculated in 0 are the Bernoulli numbers and those calculated in 1 are its variant with changed of sign. [19]
Being for the property of translation of Bernoulli's polynomials, the generalized Faulhaber formula can become:
very widespread, unlike the other, in the literature. [14] Hence also the two special cases:
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 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 Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements . It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes from 1 and 2. Starting from 0 and 1, the sequence begins
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion
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.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. A list of the spherical harmonics is available in Table of spherical harmonics.
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, a modular form is a (complex) analytic function on the upper half-plane, , that satisfies:
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 square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A.
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, , defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, , which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers to arbitrary complex-valued inputs. We consider the "convolution polynomial" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.
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
Gregory coefficientsGn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm
In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function or weighted sums over the higher-order derivatives of these functions.
The Bernoulli polynomials of the second kindψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: