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Table of Newtonian series

From Wikipedia, the free encyclopedia

Inmathematics, aNewtonian series, named afterIsaac Newton, is a sum over asequencean{\displaystyle a_{n}} written in the form

f(s)=∑n=0∞(−1)n(sn)an=∑n=0∞(−s)nn!an{\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}}

where

(sn){\displaystyle {s \choose n}}

is thebinomial coefficient and(s)n{\displaystyle (s)_{n}} is thefalling factorial. Newtonian series often appear in relations of the form seen inumbral calculus.

List

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The generalizedbinomial theorem gives

(1+z)s=∑n=0∞(sn)zn=1+(s1)z+(s2)z2+⋯.{\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)d(1+z)sdz=s(1+z)s.{\displaystyle (1+z){\frac {d(1+z)^{s}}{dz}}=s(1+z)^{s}.}

Thelog{\displaystyle \log } of thegamma function, and its derivative thedigamma function, can both have Newtonian series found by taking theirbinomial transform as sequences over the integers:

log⁡(Γ(s+1))=∑n=1∞(sn)∑k=1n(−1)n−k(n−1k−1)log⁡(k)ψ(s+1)+γ=Hs=∑n=1∞(sn)(−1)n−1n{\displaystyle {\begin{aligned}\log(\Gamma (s+1))&=\sum _{n=1}^{\infty }{s \choose n}\sum _{k=1}^{n}(-1)^{n-k}{n-1 \choose k-1}\log(k)\\\psi (s+1)+\gamma =H_{s}&=\sum _{n=1}^{\infty }{s \choose n}{\frac {(-1)^{n-1}}{n}}\end{aligned}}}

These are both valid in the right half-planeℜ(s)>0{\displaystyle \Re (s)>0}, as proven byCharles Hermite in 1900[1] andMoritz Abraham Stern in 1847 (seeDigamma function#Newton series) respectively.

TheStirling numbers of the second kind are given by the finite sum

{nk}=1k!∑j=0k(−1)k−j(kj)jn.{\displaystyle \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 thekthforward difference of themonomialxn evaluated at x = 0:

Δkxn=∑j=0k(−1)k−j(kj)(x+j)n.{\displaystyle \Delta ^{k}x^{n}=\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}(x+j)^{n}.}

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

∑k=0n(nk)(−1)n−ks−k=n!s(s−1)(s−2)⋯(s−n)=Γ(n+1)Γ(s−n)Γ(s+1)=B(n+1,s−n),s∉{0,…,n}{\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\}}

whereΓ(x){\displaystyle \Gamma (x)} is theGamma function andB(x,y){\displaystyle B(x,y)} is theBeta function.

Thetrigonometric functions haveumbral identities:

∑n=0∞(−1)n(s2n)=2s/2cos⁡πs4{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n}=2^{s/2}\cos {\frac {\pi s}{4}}}

and

∑n=0∞(−1)n(s2n+1)=2s/2sin⁡πs4{\displaystyle \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 thefalling factorial(s)n{\displaystyle (s)_{n}}. The first few terms of the sin series are

s−(s)33!+(s)55!−(s)77!+⋯{\displaystyle s-{\frac {(s)_{3}}{3!}}+{\frac {(s)_{5}}{5!}}-{\frac {(s)_{7}}{7!}}+\cdots }

which can be recognized as resembling theTaylor series for sin x, with (s)n standing in the place of xn.

Inanalytic number theory it is of interest to sum

∑k=0Bkzk,{\displaystyle \!\sum _{k=0}B_{k}z^{k},}

whereB are theBernoulli numbers. Employing the generating function its Borel sum can be evaluated as

∑k=0Bkzk=∫0∞e−ttzetz−1dt=∑k=1z(kz+1)2.{\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}}}.}

The general relation gives the Newton series

∑k=0Bk(x)zk(1−sk)s−1=zs−1ζ(s,x+z),{\displaystyle \sum _{k=0}{\frac {B_{k}(x)}{z^{k}}}{\frac {1-s \choose k}{s-1}}=z^{s-1}\zeta (s,x+z),}[citation needed]

whereζ{\displaystyle \zeta } is theHurwitz zeta function andBk(x){\displaystyle B_{k}(x)} theBernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is1Γ(x)=∑k=0∞(x−ak)∑j=0k(−1)k−jΓ(a+j)(kj),{\displaystyle {\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 forx>a{\displaystyle x>a}. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)

f(x)=∑k=0(x−ahk)∑j=0k(−1)k−j(kj)f(a+jh).{\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).}

See also

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References

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  1. ^Davis, Philip J. (1959)."Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz".The American Mathematical Monthly.66 (10):849–869.doi:10.2307/2309786.
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