In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by ∑ x or Δ − 1 , is the linear operator, inverse of the forward difference operator Δ . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus
Δ ∑ x f ( x ) = f ( x ) . More explicitly, if ∑ x f ( x ) = F ( x ) , then
F ( x + 1 ) − F ( x ) = f ( x ) . If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function C(x) with period 1. Therefore each indefinite sum actually represents a family of functions. However the solution equal to its Newton series expansion is unique up to an additive constant C.
Indefinite sums can be used to calculate definite sums with the formula:
∑ k = a b f ( k ) = Δ − 1 f ( b + 1 ) − Δ − 1 f ( a ) ∑ x f ( x ) = ∫ 0 x f ( t ) d t + ∑ k = 1 ∞ c k Δ k − 1 f ( x ) k ! + C where
c k = ∫ 0 1 Γ ( x + 1 ) Γ ( x − k + 1 ) d x are the Cauchy numbers of the first kind.
∑ x f ( x ) = − ∑ k = 1 ∞ Δ k − 1 f ( x ) k ! ( − x ) k + C where
( x ) k = Γ ( x + 1 ) Γ ( x − k + 1 ) is the falling factorial.
∑ x f ( x ) = ∑ n = 1 ∞ f ( n − 1 ) ( 0 ) n ! B n ( x ) + C , provided that the right-hand side of the equation converges.
If lim x → + ∞ f ( x ) = 0 , then
∑ x f ( x ) = ∑ n = 0 ∞ ( f ( n ) − f ( n + x ) ) + C . ∑ x f ( x ) = ∫ 0 x f ( t ) d t − 1 2 f ( x ) + ∑ k = 1 ∞ B 2 k ( 2 k ) ! f ( 2 k − 1 ) ( x ) + C Often the constant C in indefinite sum is fixed from the following condition.
Let
F ( x ) = ∑ x f ( x ) + C Then the constant C is fixed from the condition
∫ 0 1 F ( x ) d x = 0 or
∫ 1 2 F ( x ) d x = 0 Alternatively, Ramanujan's sum can be used:
∑ x ≥ 1 ℜ f ( x ) = − f ( 0 ) − F ( 0 ) or at 1
∑ x ≥ 1 ℜ f ( x ) = − F ( 1 ) respectively
Indefinite summation by parts:
∑ x f ( x ) Δ g ( x ) = f ( x ) g ( x ) − ∑ x ( g ( x ) + Δ g ( x ) ) Δ f ( x ) ∑ x f ( x ) Δ g ( x ) + ∑ x g ( x ) Δ f ( x ) = f ( x ) g ( x ) − ∑ x Δ f ( x ) Δ g ( x ) Definite summation by parts:
∑ i = a b f ( i ) Δ g ( i ) = f ( b + 1 ) g ( b + 1 ) − f ( a ) g ( a ) − ∑ i = a b g ( i + 1 ) Δ f ( i ) If T is a period of function f ( x ) then
∑ x f ( T x ) = x f ( T x ) + C If T is an antiperiod of function f ( x ) , that is f ( x + T ) = − f ( x ) then
∑ x f ( T x ) = − 1 2 f ( T x ) + C Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given. e.g.
∑ k = 1 n f ( k ) In this case a closed form expression F(k) for the sum is a solution of
F ( x + 1 ) − F ( x ) = f ( x + 1 ) which is called the telescoping equation. It is inverse to backward difference
∇ operator.
It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.
This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.
∑ x a = a x + C ∑ x x = x 2 2 − x 2 + C ∑ x x a = B a + 1 ( x ) a + 1 + C , a ∉ Z − where
B a ( x ) = − a ζ ( − a + 1 , x ) , the generalized to real order Bernoulli polynomials.
∑ x x a = ( − 1 ) a − 1 ψ ( − a − 1 ) ( x ) Γ ( − a ) + C , a ∈ Z − where
ψ ( n ) ( x ) is the
polygamma function.
∑ x 1 x = ψ ( x ) + C where
ψ ( x ) is the digamma function.
∑ x a x = a x a − 1 + C Particularly,
∑ x 2 x = 2 x + C ∑ x log b x = log b Γ ( x ) + C ∑ x log b a x = log b ( a x − 1 Γ ( x ) ) + C ∑ x sinh a x = 1 2 csch ( a 2 ) cosh ( a 2 − a x ) + C ∑ x cosh a x = 1 2 coth ( a 2 ) sinh a x − 1 2 cosh a x + C ∑ x tanh a x = 1 a ψ e a ( x − i π 2 a ) + 1 a ψ e a ( x + i π 2 a ) − x + C where
ψ q ( x ) is the
q-digamma function.
∑ x sin a x = − 1 2 csc ( a 2 ) cos ( a 2 − a x ) + C , a ≠ n π ∑ x cos a x = 1 2 cot ( a 2 ) sin a x − 1 2 cos a x + C , a ≠ n π ∑ x sin 2 a x = x 2 + 1 4 csc ( a ) sin ( a − 2 a x ) + C , a ≠ n π 2 ∑ x cos 2 a x = x 2 − 1 4 csc ( a ) sin ( a − 2 a x ) + C , a ≠ n π 2 ∑ x tan a x = i x − 1 a ψ e 2 i a ( x − π 2 a ) + C , a ≠ n π 2 where
ψ q ( x ) is the
q-digamma function.
∑ x tan x = i x − ψ e 2 i ( x + π 2 ) + C = − ∑ k = 1 ∞ ( ψ ( k π − π 2 + 1 − z ) + ψ ( k π − π 2 + z ) − ψ ( k π − π 2 + 1 ) − ψ ( k π − π 2 ) ) + C ∑ x cot a x = − i x − i ψ e 2 i a ( x ) a + C , a ≠ n π 2 ∑ x artanh a x = 1 2 ln ( ( − 1 ) x Γ ( − 1 a ) Γ ( x + 1 a ) Γ ( 1 a ) Γ ( x − 1 a ) ) + C ∑ x arctan a x = i 2 ln ( ( − 1 ) x Γ ( − i a ) Γ ( x + i a ) Γ ( i a ) Γ ( x − i a ) ) + C ∑ x ψ ( x ) = ( x − 1 ) ψ ( x ) − x + C ∑ x Γ ( x ) = ( − 1 ) x + 1 Γ ( x ) Γ ( 1 − x , − 1 ) e + C where
Γ ( s , x ) is the
incomplete gamma function.
∑ x ( x ) a = ( x ) a + 1 a + 1 + C where
( x ) a is the falling factorial.
∑ x sexp a ( x ) = ln a ( sexp a ( x ) ) ′ ( ln a ) x + C (see super-exponential function)
