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In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence 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 *n*th element of the sequence is denoted by this symbol with *n* as subscript; for example, the *n*th element of the Fibonacci sequence is generally denoted *F*_{n}.

where

is the binomial coefficient and 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 *n* ≥ *k* ≥ 0 and is written It is the coefficient of the *x*^{k} 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

The generalized binomial theorem gives

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

The digamma function:

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.

This formula is a special case of the *k*th forward difference of the monomial *x*^{n} 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:

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

where is the Gamma function and 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:

and

The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial . The first few terms of the sin series are

which can be recognized as resembling the Taylor series for sin *x*, with (*s*)_{n} standing in the place of *x*^{n}.

In analytic number theory it is of interest to sum

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

The general relation gives the Newton series

^{[ citation needed ]}

where is the Hurwitz zeta function and the Bernoulli polynomial. 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)

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 **ℂ** 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-function***G*(*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 *n*th 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

In mathematics, **Ramanujan's master theorem** is a technique that provides an analytic expression for the Mellin transform of an analytic function.

**Gregory coefficients***G*_{n}, 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 kind***ψ _{n}*(

- Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals
^{[ permanent dead link ]}",*Theoretical Computer Science**144*(1995) pp 101–124.

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