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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)

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

Binomial transform
List of factorial and binomial topics
Nörlund–Rice integral
Carlson's theorem

References

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.