Table of Newtonian series

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

$f(s)=\sum _^{\infty }(-1)^a_=\sum _^{\infty }{\frac {(-s)_}}a_$ where is the binomial coefficient and $(s)_$ is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.

List

The generalized binomial theorem gives

$(1+z)^=\sum _^{\infty }z^=1+z+z^+\cdots .$ A proof for this identity can be obtained by showing that it satisfies the differential equation

$(1+z){\frac }}=s(1+z)^.$ The digamma function:

$\psi (s+1)=-\gamma -\sum _^{\infty }{\frac {(-1)^}}.$ The Stirling numbers of the second kind are given by the finite sum

$\left\{{\beginn\\k\end}\right\}={\frac }\sum _^(-1)^j^.$ This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:

$\Delta ^x^=\sum _^(-1)^(x+j)^.$ A related identity forms the basis of the Nörlund–Rice integral:

$\sum _^{\frac {(-1)^}}={\frac }={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)}}=B(n+1,s-n),s\notin \$ where $\Gamma (x)$ is the Gamma function and $B(x,y)$ is the Beta function.

The trigonometric functions have umbral identities:

$\sum _^{\infty }(-1)^=2^\cos {\frac {\pi s}}$ and

$\sum _^{\infty }(-1)^=2^\sin {\frac {\pi s}}$ The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial $(s)_$ . The first few terms of the sin series are

$s-{\frac {(s)_}}+{\frac {(s)_}}-{\frac {(s)_}}+\cdots$ 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

$\!\sum _B_z^,$ where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as

$\sum _B_z^=\int _^{\infty }e^{-t}{\frac -1}}dt=\sum _{\frac {(kz+1)^}}.$ The general relation gives the Newton series

$\sum _{\frac (x)}}}{\frac }=z^\zeta (s,x+z),$ [citation needed]

where $\zeta$ is the Hurwitz zeta function and $B_(x)$ the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is ${\frac {\Gamma (x)}}=\sum _^{\infty }\sum _^{\frac {(-1)^}{\Gamma (a+j)}},$ 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)

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