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

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

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.
List

The generalized binomial theorem gives

\( {\displaystyle (1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .} \)

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

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

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

\( \Delta^k x^n = \sum_{j=0}^{k}(-1)^{k-j}{k \choose j} (x+j)^n. \)

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

\( {\displaystyle \sum _{k=0}^{n}{n \choose k}{\frac {(-1)^{n-k}}{s-k}}={\frac {n!}{s(s-1)(s-2)\cdots (s-n)}}={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)}}=B(n+1,s-n),s\notin \{0,\ldots ,n\}} \)

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

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

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

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

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)

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