In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a n written in the form
f ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( s n ) a n = ∑ n = 0 ∞ ( − s ) n n ! a n where
( s 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.
The generalized binomial theorem gives
( 1 + z ) s = ∑ n = 0 ∞ ( s n ) z n = 1 + ( s 1 ) z + ( s 2 ) z 2 + ⋯ . 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 . The digamma function:
ψ ( s + 1 ) = − γ − ∑ n = 1 ∞ ( − 1 ) n n ( s 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 . This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:
Δ k x n = ∑ j = 0 k ( − 1 ) k − j ( k j ) ( x + j ) n . A related identity forms the basis of the Nörlund–Rice integral:
∑ k = 0 n ( n k ) ( − 1 ) 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 ) where Γ ( x ) is the Gamma function and 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 and
∑ n = 0 ∞ ( − 1 ) n ( s 2 n + 1 ) = 2 s / 2 sin π 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 − ( s ) 3 3 ! + ( s ) 5 5 ! − ( s ) 7 7 ! + ⋯ 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
∑ 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 . 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 ) , where ζ is the Hurwitz zeta function and 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 ) , 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 ) = ∑ k = 0 ( x − a h k ) ∑ j = 0 k ( − 1 ) k − j ( k j ) f ( a + j h ) . 