# Table of Newtonian series

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In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence $a_{n}$ 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.

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

${s \choose n}$ is the binomial coefficient and $(s)_{n}$ 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.

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The generalized binomial theorem gives

$(1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .$ A proof for this identity can be obtained by showing that it satisfies the differential equation

$(1+z){\frac {d(1+z)^{s}}{dz}}=s(1+z)^{s}.$ The digamma function:

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

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

$\Delta ^{k}x^{n}=\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}(x+j)^{n}.$ A related identity forms the basis of the Nörlund–Rice integral:

$\sum _{k=0}^{n}{n \choose k}{\frac {(-1)^{n-k}}{s-k}}={\frac {n!}{s(s-1)(s-2)\cdots (s-n)}}={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)}}=B(n+1,s-n),s\notin \{0,\ldots ,n\}$ where $\Gamma (x)$ is the Gamma function and $B(x,y)$ 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:

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

$\sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n+1}=2^{s/2}\sin {\frac {\pi s}{4}}$ The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial $(s)_{n}$ . The first few terms of the sin series are

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

$\!\sum _{k=0}B_{k}z^{k},$ where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as

$\sum _{k=0}B_{k}z^{k}=\int _{0}^{\infty }e^{-t}{\frac {tz}{e^{tz}-1}}dt=\sum _{k=1}{\frac {z}{(kz+1)^{2}}}.$ The general relation gives the Newton series

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

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

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

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

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

In mathematics, the falling factorial is defined as the polynomial In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function: In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments s with Re(s) > 1 and q with Re(q) > 0 by

In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta-function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after the Czech mathematician Mathias Lerch.

In mathematics, the Barnes G-functionG(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function.

In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by

In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. As such, it commonly appears in the theory of finite differences, and also has been applied in computer science and graph theory to estimate binary tree lengths. It is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation.

In combinatorics, the binomial transform is a sequence transformation that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.

In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises.

In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a q-analog of the Pochhammer symbol. It is defined as In mathematics, Ramanujan's master theorem is a technique that provides an analytic expression for the Mellin transform of an analytic function.

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

The Bernoulli polynomials of the second kindψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: