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

(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:

\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

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).

References

Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101–124.

Table of Newtonian series

List

See also

References