Table of Newtonian series

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

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 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.

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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

\left\{\begin{matrix} n \\ k \end{matrix}\right\} =\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 xn 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 Γ(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 xn.

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