Table of Newtonian series

Table of Newtonian series

In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form

:f(s) = sum_{n=0}^infty (-1)^n {schoose n} a_n = sum_{n=0}^infty frac{(-s)_n}{n!} a_n


:{s choose k}

is the binomial coefficient and (s)_n is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.


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

:left{egin{matrix} n \ k end{matrix} ight}=frac{1}{k!}sum_{j=1}^{k}(-1)^{k-j}{k choose j} j^n.

This formula is a special case of the "k" 'th forward difference of the monomial x^n evaluated at "x"=0:

: Delta^k x^n = sum_{j=1}^{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)^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)

where Gamma(x) is the Gamma function and B(x,y) is the Beta function.

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 x^n.

ee also

* Binomial transform
* List of factorial and binomial topics


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

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Binomial series — In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f(x) = (1 + x) α, where α ∈ C is an arbitrary complex number. Explicitly, and the binomial series is the power series… …   Wikipedia

  • List of QI episodes (C series) — infobox tvseason season name = QI Series C caption = The front cover of the QI series C DVD, featuring Stephen Fry (left) and Alan Davies (right). dvd release date = 1 September 2008 country = UK network = BBC first aired = 30 September 2005 last …   Wikipedia

  • Trigonometric functions — Cosine redirects here. For the similarity measure, see Cosine similarity. Trigonometry History Usage Functions Generalized Inverse functions …   Wikipedia

  • List of mathematics articles (T) — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie …   Wikipedia

  • Finite difference — A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences… …   Wikipedia

  • Binomial coefficient — The binomial coefficients can be arranged to form Pascal s triangle. In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the… …   Wikipedia

  • Difference operator — In mathematics, a difference operator maps a function, f ( x ), to another function, f ( x + a ) − f ( x + b ).The forward difference operator :Delta f(x)=f(x+1) f(x),occurs frequently in the calculus of finite differences, where it plays a role… …   Wikipedia

  • Nörlund-Rice integral — In mathematics, the Nörlund 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… …   Wikipedia

  • Nörlund–Rice integral — In mathematics, the Nörlund–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… …   Wikipedia

  • List of factorial and binomial topics — This is a list of factorial and binomial topics in mathematics, by Wikipedia page. See also binomial (disambiguation).*Alternating factorial *Antichain *Beta function *Binomial coefficient *Binomial distribution *Binomial proportion confidence… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”