In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence
written in the form
![{\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/195cd2748cffbc1ac3bc4b8fccea32eae7056ad1)
where
![{\displaystyle {s \choose n}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3bad54d485792dd0225f6c2a7f286ad98f72d304)
is the binomial coefficient and
is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus.
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 .}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/482daf744ec8259342875949e90d2acb8692820f)
A proof for this identity can be obtained by showing that it satisfies the differential equation
![{\displaystyle (1+z){\frac {d(1+z)^{s}}{dz}}=s(1+z)^{s}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/629f909fb0b712b55abed2f667048bf28f82b802)
The digamma function:
![{\displaystyle \psi (s+1)=-\gamma -\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}{s \choose n}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/8dc746e368eab10d85b38f33deb50f56bb611ba4)
The Stirling numbers of the second kind are given by the finite sum
![{\displaystyle \left\{{\begin{matrix}n\\k\end{matrix}}\right\}={\frac {1}{k!}}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}j^{n}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/50151d0991f6d70500a3e505ccedf1ecc5e5daa8)
This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:
![{\displaystyle \Delta ^{k}x^{n}=\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}(x+j)^{n}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/1ae8fea85bae422f2ba5c524b93c16730f08663c)
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\}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/99517a7978fb21b7c7e1a534016037960a74f3df)
where
is the Gamma function and
is the Beta function.
The trigonometric functions have umbral identities:
![{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n}=2^{s/2}\cos {\frac {\pi s}{4}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c4030f60cf520fb97f24b7923ac3d00913af44b1)
and
![{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n+1}=2^{s/2}\sin {\frac {\pi s}{4}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/9649493922c68862bd1b14cee86b6c8f308a2b71)
The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial
. The first few terms of the sin series are
![{\displaystyle s-{\frac {(s)_{3}}{3!}}+{\frac {(s)_{5}}{5!}}-{\frac {(s)_{7}}{7!}}+\cdots }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/4105decc5685514d7900be089df518c4a9e9dd4e)
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
![{\displaystyle \!\sum _{k=0}B_{k}z^{k},}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/56b8a6bc360c618c945e1eec19ac37f0baa207d9)
where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as
![{\displaystyle \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}}}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a95518883e902af820bb8570ab2db7fdc54c738f)
The general relation gives the Newton series
[citation needed]
where
is the Hurwitz zeta function and
the Bernoulli polynomial. The series does not converge, the identity holds formally.
Another identity is
which converges for
. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)
![{\displaystyle f(x)=\sum _{k=0}{{\frac {x-a}{h}} \choose k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/1ed6b7aea1071e103a151ca5de9828900388a9e8)
See also[edit]
References[edit]