# Table of Newtonian series

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In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence ${\displaystyle a_{n}}$ written in the form

Mathematics includes the study of such topics as quantity, structure, space, and change.

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index; it is the natural number from which the element is the image. It depends on the context or a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the nth element of the sequence is denoted by this symbol with n as subscript; for example, the nth element of the Fibonacci sequence is generally denoted Fn.

## Contents

${\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}}$

where

${\displaystyle {s \choose n}}$

is the binomial coefficient and ${\displaystyle (s)_{n}}$ is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.

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 nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula

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 (1861) and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas, who used the technique extensively.

## List

The generalized binomial theorem gives

${\displaystyle (1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .}$

A proof for this identity can be obtained by showing that it satisfies the differential equation

${\displaystyle (1+z){\frac {d(1+z)^{s}}{dz}}=s(1+z)^{s}.}$

The digamma function:

${\displaystyle \psi (s+1)=-\gamma -\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}{s \choose n}.}$

The Stirling numbers of the second kind are given by the finite sum

In mathematics, particularly in combinatorics, a Stirling number of the second kind is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or . Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.

${\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}={\frac {1}{k!}}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}j^{n}.}$

This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

${\displaystyle \Delta ^{k}x^{n}=\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}(x+j)^{n}.}$

A related identity forms the basis of the Nörlund–Rice integral:

${\displaystyle \sum _{k=0}^{n}{n \choose k}{\frac {(-1)^{n-k}}{s-k}}={\frac {n!}{s(s-1)(s-2)\cdots (s-n)}}={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)}}=B(n+1,s-n),s\notin \{0,\ldots ,n\}}$

where ${\displaystyle \Gamma (x)}$ is the Gamma function and ${\displaystyle B(x,y)}$ is the Beta function.

In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if n is a positive integer,

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

The trigonometric functions have umbral identities:

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n}=2^{s/2}\cos {\frac {\pi s}{4}}}$

and

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n+1}=2^{s/2}\sin {\frac {\pi s}{4}}}$

The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial ${\displaystyle (s)_{n}}$. The first few terms of the sin series are

${\displaystyle s-{\frac {(s)_{3}}{3!}}+{\frac {(s)_{5}}{5!}}-{\frac {(s)_{7}}{7!}}+\cdots }$

which can be recognized as resembling the Taylor series for sin x, with (s)n standing in the place of xn.

In analytic number theory it is of interest to sum

${\displaystyle \!\sum _{k=0}B_{k}z^{k},}$

where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as

${\displaystyle \sum _{k=0}B_{k}z^{k}=\int _{0}^{\infty }e^{-t}{\frac {tz}{e^{tz}-1}}dt=\sum _{k=1}{\frac {z}{(kz+1)^{2}}}.}$

The general relation gives the Newton series

${\displaystyle \sum _{k=0}{\frac {B_{k}(x)}{z^{k}}}{\frac {1-s \choose k}{s-1}}=z^{s-1}\zeta (s,x+z),}$[ citation needed ]

where ${\displaystyle \zeta }$ is the Hurwitz zeta function and ${\displaystyle B_{k}(x)}$ the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is ${\displaystyle {\frac {1}{\Gamma (x)}}=\sum _{k=0}^{\infty }{x-a \choose k}\sum _{j=0}^{k}{\frac {(-1)^{k-j}}{\Gamma (a+j)}}{k \choose j},}$ which converges for ${\displaystyle x>a}$. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)

${\displaystyle f(x)=\sum _{k=0}{{\frac {x-a}{h}} \choose k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).}$

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