Table of Newtonian series

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In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence [math]\displaystyle{ a_n }[/math] written in the form

[math]\displaystyle{ f(s) = \sum_{n=0}^\infty (-1)^n {s\choose n} a_n = \sum_{n=0}^\infty \frac{(-s)_n}{n!} a_n }[/math]


[math]\displaystyle{ {s \choose n} }[/math]

is the binomial coefficient and [math]\displaystyle{ (s)_n }[/math] is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus.


The generalized binomial theorem gives

[math]\displaystyle{ (1+z)^s = \sum_{n = 0}^{\infty}{s \choose n}z^n = 1+{s \choose 1}z+{s \choose 2}z^2+\cdots. }[/math]

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

[math]\displaystyle{ (1+z) \frac{d(1+z)^s}{dz} = s (1+z)^s. }[/math]

The digamma function:

[math]\displaystyle{ \psi(s+1)=-\gamma-\sum_{n=1}^\infty \frac{(-1)^n}{n} {s \choose n}. }[/math]

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

[math]\displaystyle{ \left\{\begin{matrix} n \\ k \end{matrix}\right\} =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k \choose j} j^n. }[/math]

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

[math]\displaystyle{ \Delta^k x^n = \sum_{j=0}^{k}(-1)^{k-j}{k \choose j} (x+j)^n. }[/math]

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

[math]\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\} }[/math]

where [math]\displaystyle{ \Gamma(x) }[/math] is the Gamma function and [math]\displaystyle{ B(x,y) }[/math] is the Beta function.

The trigonometric functions have umbral identities:

[math]\displaystyle{ \sum_{n=0}^\infty (-1)^n {s \choose 2n} = 2^{s/2} \cos \frac{\pi s}{4} }[/math]


[math]\displaystyle{ \sum_{n=0}^\infty (-1)^n {s \choose 2n+1} = 2^{s/2} \sin \frac{\pi s}{4} }[/math]

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

[math]\displaystyle{ s - \frac{(s)_3}{3!} + \frac{(s)_5}{5!} - \frac{(s)_7}{7!} + \cdots }[/math]

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

[math]\displaystyle{ \!\sum_{k=0}B_k z^k, }[/math]

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

[math]\displaystyle{ \sum_{k=0}B_k z^k= \int_0^\infty e^{-t} \frac{t z}{e^{t z}-1}d t= \sum_{k=1}\frac z{(k z+1)^2}. }[/math]

The general relation gives the Newton series

[math]\displaystyle{ \sum_{k=0}\frac{B_k(x)}{z^k}\frac{{1-s\choose k}}{s-1}= z^{s-1}\zeta(s,x+z), }[/math]

where [math]\displaystyle{ \zeta }[/math] is the Hurwitz zeta function and [math]\displaystyle{ B_k(x) }[/math] the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is [math]\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}, }[/math] which converges for [math]\displaystyle{ x\gt a }[/math]. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)

[math]\displaystyle{ f(x)=\sum_{k=0}{\frac{x-a}h \choose k} \sum_{j=0}^k (-1)^{k-j}{k\choose j}f(a+j h). }[/math]

See also