List of indefinite sums

This is a list of indefinite sums (also known as antidifferences) of various functions. An indefinite sum ${\textstyle \sum _{x}f(x)}$ is the inverse of the forward difference operator ${\displaystyle \Delta }$, defined as ${\displaystyle \Delta f(x)=f(x+1)-f(x)}$. It satisfies the relation

${\displaystyle \Delta \sum _{x}f(x)=f(x).}$

The operator is defined only up to an additive periodic function with period 1.

For positive integer exponents, Faulhaber's formula can be used. Note that ${\displaystyle x}$ in the result of Faulhaber's formula must be replaced with ${\displaystyle x-1}$ due to the offset, as Faulhaber's formula finds ${\displaystyle \nabla ^{-1}}$ rather than ${\displaystyle \Delta ^{-1}}$.

For negative integer exponents, the indefinite sum is closely related to the polygamma function:^[1]

${\displaystyle \sum _{x}{\frac {1}{x^{a}}}={\frac {(-1)^{a-1}\psi ^{(a-1)}(x)}{(a-1)!}}+C,\,a\in \mathbb {N} }$

For fractions not listed in this section, one may use the polygamma function with partial fraction decomposition. More generally,^[1]

${\displaystyle \sum _{x}x^{a}={\begin{cases}{\frac {B_{a+1}(x)}{a+1}}+C,&{\text{if }}a\neq -1\\\psi (x)+C,&{\text{if }}a=-1\end{cases}}={\begin{cases}-\zeta (-a,x)+C,&{\text{if }}a\neq -1\\\psi (x)+C,&{\text{if }}a=-1\end{cases}}}$

where ${\displaystyle B_{a}(x)}$ are the Bernoulli polynomials, ${\displaystyle \zeta (s,a)}$ is the Hurwitz zeta function, and ${\displaystyle \psi (z)}$ is the digamma function. This is related to the generalized harmonic numbers.

As the generalized harmonic numbers use reciprocal powers, ${\displaystyle a}$ must be substituted for ${\displaystyle -a}$, and the most common form uses the inverse of the backward difference offset:^[1]

${\displaystyle \nabla ^{-1}x^{a}={H_{x}^{(-a)}}=\zeta (-a)-\zeta (-a,x+1).}$

Here, ${\displaystyle \zeta (-a)}$ is the constant ${\displaystyle C}$.

The Bernoulli polynomials are also related via a partial derivative with respect to ${\displaystyle x}$:

${\displaystyle {\frac {\partial }{\partial x}}\left(\sum _{x}x^{a}\right)=B_{a}(x)=-a\zeta (1-a,x).}$

This relationship can be expressed via the inverse backward difference operator as:

${\displaystyle {\frac {\partial }{\partial x}}\left(\nabla ^{-1}x^{a}\right){\bigg |}_{x=0}=-a\zeta (1-a,x+1){\bigg |}_{x=0}=-a\zeta (1-a)=B_{a}.}$

Further generalization comes from use of the Lerch transcendent:

${\displaystyle \sum _{x}{\frac {z^{x}}{(x+a)^{s}}}=-z^{x}\,\Phi (z,s,x+a)+C,}$

which generalizes the generalized harmonic numbers as ${\displaystyle z\Phi \left(z,s,a+1\right)-z^{x+1}\Phi \left(z,s,x+1+a\right)}$ when taking ${\displaystyle \nabla ^{-1}}$.

${\displaystyle \sum _{x}B_{a}(x)=(x-1)B_{a}(x)-{\frac {a}{a+1}}B_{a+1}(x)+C}$

${\displaystyle \sum _{x}a^{x}={\frac {a^{x}}{a-1}}+C}$^[2]

${\displaystyle \sum _{x}\log _{b}x=\log _{b}\Gamma (x)+C}$^[2]

${\displaystyle \sum _{x}\log _{b}ax=\log _{b}(a^{x-1}\Gamma (x))+C}$^[2]

${\displaystyle \sum _{x}\sinh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\cosh \left({\frac {a}{2}}-ax\right)+C}$

${\displaystyle \sum _{x}\cosh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\sinh \left(ax-{\frac {a}{2}}\right)+C}$

${\displaystyle \sum _{x}\tanh ax={\frac {1}{a}}\psi _{e^{a}}\left(x-{\frac {i\pi }{2a}}\right)+{\frac {1}{a}}\psi _{e^{a}}\left(x+{\frac {i\pi }{2a}}\right)-x+C}$

where ${\displaystyle \psi _{q}(x)}$ is the q-digamma function.

${\displaystyle \sum _{x}\sin ax=-{\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\cos \left({\frac {a}{2}}-ax\right)+C\,,\,\,a\neq 2n\pi }$^[2]

${\displaystyle \sum _{x}\cos ax={\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\sin \left(ax-{\frac {a}{2}}\right)+C\,,\,\,a\neq 2n\pi }$^[2]

${\displaystyle \sum _{x}\sin ^{2}ax={\frac {x}{2}}+{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }$

${\displaystyle \sum _{x}\cos ^{2}ax={\frac {x}{2}}-{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }$

${\displaystyle \sum _{x}\tan ax=ix-{\frac {1}{a}}\psi _{e^{2ia}}\left(x-{\frac {\pi }{2a}}\right)+C\,,\,\,a\neq {\frac {n\pi }{2}}}$

where ${\displaystyle \psi _{q}(x)}$ is the q-digamma function.

$${\displaystyle {\begin{aligned}\sum _{x}\tan x&=ix-\psi _{e^{2i}}\left(x+{\frac {\pi }{2}}\right)+C\\&=-\sum _{k=1}^{\infty }\left(\psi \left(k\pi -{\frac {\pi }{2}}+1-x\right)+\psi \left(k\pi -{\frac {\pi }{2}}+x\right)\right.\\&\quad \left.-\psi \left(k\pi -{\frac {\pi }{2}}+1\right)-\psi \left(k\pi -{\frac {\pi }{2}}\right)\right)+C\end{aligned}}}$$

${\displaystyle \sum _{x}\cot ax=-ix-{\frac {i\psi _{e^{2ia}}(x)}{a}}+C\,,\,\,a\neq {\frac {n\pi }{2}}}$

The antidifference of the normalized sinc function can be obtained by applying the Abel–Plana formula presented in Candelpergher^[1] with the shift ${\displaystyle x\mapsto x-1}$, the condition ${\displaystyle F(0)=0}$, and recurrence of ${\displaystyle F(x+1)-F(x)=f(x)}$. Using the reflection formula for the digamma function, this simplifies to: $${\displaystyle {\begin{aligned}\sum _{x}\operatorname {sinc} x&=\operatorname {sinc} (x-1)\left({\frac {1}{2}}+(x-1)\right.\\&\quad \left.\times \left(\ln(2)+{\frac {\psi ({\frac {x-1}{2}})+\psi ({\frac {1-x}{2}})}{2}}\right.\right.\\&\quad \quad \left.\left.-{\frac {\psi (x-1)+\psi (1-x)}{2}}\right)\right)+{\frac {1}{2}}+C\end{aligned}}}$$

If ${\displaystyle T}$ is a period of function ${\displaystyle f(x)}$ then

${\displaystyle \sum _{x}f(Tx)=xf(Tx)+C.}$

If ${\displaystyle T}$ is an antiperiod of function ${\displaystyle f(x)}$, that is ${\displaystyle f(x+T)=-f(x)}$ then

${\displaystyle \sum _{x}f(Tx)=-{\frac {1}{2}}f(Tx)+C.}$

${\displaystyle \sum _{x}\psi (x)=(x-1)\psi (x)-x+C}$

${\displaystyle \sum _{x}\Gamma (x)=(-1)^{x+1}\Gamma (x){\frac {\Gamma (1-x,-1)}{e}}+C}$

where ${\displaystyle \Gamma (s,x)}$ is the incomplete gamma function.

${\displaystyle \sum _{x}(x)_{a}={\frac {(x)_{a+1}}{a+1}}+C}$^[2]

where ${\displaystyle (x)_{a}}$ is the falling factorial.

${\displaystyle \sum _{x}\operatorname {sexp} _{a}(x)=\ln _{a}{\frac {(\operatorname {sexp} _{a}(x))'}{(\ln a)^{x}}}+C}$

(see super-exponential function)