In mathematics , a Newtonian series , named after Isaac Newton , is a sum over a sequence a n {\displaystyle a_{n}} written in the form
f ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( s n ) a n = ∑ n = 0 ∞ ( − s ) n n ! a n {\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
( s n ) {\displaystyle {s \choose n}} is the binomial coefficient and ( s ) n {\displaystyle (s)_{n}} is the falling factorial . Newtonian series often appear in relations of the form seen in umbral calculus .
List The generalized binomial theorem gives
( 1 + z ) s = ∑ n = 0 ∞ ( s n ) z n = 1 + ( s 1 ) z + ( s 2 ) z 2 + ⋯ . {\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 ) s d z = s ( 1 + z ) s . {\displaystyle (1+z){\frac {d(1+z)^{s}}{dz}}=s(1+z)^{s}.} The digamma function :
ψ ( s + 1 ) = − γ − ∑ n = 1 ∞ ( − 1 ) n n ( s n ) . {\displaystyle \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
{ n k } = 1 k ! ∑ j = 0 k ( − 1 ) k − j ( k j ) j n . {\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 the k th forward difference of the monomial x n evaluated at x = 0:
Δ k x n = ∑ j = 0 k ( − 1 ) k − j ( k j ) ( 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 the Nörlund–Rice integral :
∑ k = 0 n ( n k ) ( − 1 ) n − k s − 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\}} where Γ ( x ) {\displaystyle \Gamma (x)} is the Gamma function and B ( x , y ) {\displaystyle B(x,y)} is the Beta function .
The trigonometric functions have umbral identities:
∑ n = 0 ∞ ( − 1 ) n ( s 2 n ) = 2 s / 2 cos π s 4 {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n}=2^{s/2}\cos {\frac {\pi s}{4}}} and
∑ n = 0 ∞ ( − 1 ) n ( s 2 n + 1 ) = 2 s / 2 sin π s 4 {\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 the falling factorial ( s ) n {\displaystyle (s)_{n}} . The first few terms of the sin series are
s − ( s ) 3 3 ! + ( s ) 5 5 ! − ( s ) 7 7 ! + ⋯ {\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 x n .
In analytic number theory it is of interest to sum
∑ k = 0 B k z k , {\displaystyle \!\sum _{k=0}B_{k}z^{k},} where B are the Bernoulli numbers . Employing the generating function its Borel sum can be evaluated as
∑ k = 0 B k z k = ∫ 0 ∞ e − t t z e t z − 1 d t = ∑ k = 1 z ( k z + 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 = 0 B k ( x ) z k ( 1 − s k ) s − 1 = z s − 1 ζ ( 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 ] where ζ {\displaystyle \zeta } is the Hurwitz zeta function and B k ( x ) {\displaystyle B_{k}(x)} the Bernoulli polynomial . The series does not converge, the identity holds formally.
Another identity is 1 Γ ( x ) = ∑ k = 0 ∞ ( x − a k ) ∑ j = 0 k ( − 1 ) k − j Γ ( a + j ) ( k j ) , {\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 for x > 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 − a h k ) ∑ j = 0 k ( − 1 ) k − j ( k j ) f ( a + j h ) . {\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).}
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