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 ...
, a
Newtonian series, named after
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
, is a sum over a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
written in the form
:
where
:
is the
binomial coefficient
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 and is written \tbinom. It is the coefficient of the t ...
and
is the
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 ...
. Newtonian series often appear in relations of the form seen in
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 ...
.
List
The generalized
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 ...
gives
:
A proof for this identity can be obtained by showing that it satisfies the differential equation
:
The
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
:
:
The
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 ...
are given by the finite sum
:
This formula is a special case of the ''k''th
forward difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
of the
monomial
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 ...
''x''
''n'' evaluated at ''x'' = 0:
:
A related identity forms the basis of the
Nörlund–Rice integral:
:
where
is the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and
is the
Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^(1 ...
.
The
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s have
umbral identities:
:
and
:
The umbral nature of these identities is a bit more clear by writing them in terms of the
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 ...
. The first few terms of the sin series are
:
which can be recognized as resembling the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
for sin ''x'', with (''s'')
''n'' standing in the place of ''x''
''n''.
In
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
it is of interest to sum
:
where ''B'' are 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, ...
. Employing the generating function its Borel sum can be evaluated as
:
The general relation gives the Newton series
:
where
is the
Hurwitz zeta function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by
:\zeta(s,a) = \sum_^\infty \frac.
This series is absolutely convergent for the given values of and and can ...
and
the
Bernoulli polynomial
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 ...
. The series does not converge, the identity holds formally.
Another identity is
which converges for
. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)
:
See also
*
Binomial transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to th ...
*
List of factorial and binomial topics {{Short description, none
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation).
* Abel's binomial theorem
* Alternating factorial
*Antichain
*Beta function
*Bhargava factorial
*Binomial coefficient
**P ...
*
Nörlund–Rice integral
*
Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not co ...
References
* Philippe Flajolet and Robert Sedgewick,
Mellin transforms and asymptotics: Finite differences and Rice's integrals, ''Theoretical Computer Science'' ''144'' (1995) pp 101–124.
{{Isaac Newton
Finite differences
Factorial and binomial topics
Newton series
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...