In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.

Let :${\displaystyle f:\mathbb {Z} _{p}\to \mathbb {C} _{p))$ be a function from the p-adic integers taking values in the p-adic numbers. The Volkenborn integral is defined by the limit, if it exists:

${\displaystyle \int _{\mathbb {Z} _{p))f(x)\,{\rm {d))x=\lim _{n\to \infty }{\frac {1}{p^{n))}\sum _{x=0}^{p^{n}-1}f(x).}$

More generally, if

${\displaystyle R_{n}=\left\{\left.x=\sum _{i=r}^{n-1}b_{i}x^{i}\right|b_{i}=0,\ldots ,p-1{\text{ for ))r<n\right\))$

then

${\displaystyle \int _{K}f(x)\,{\rm {d))x=\lim _{n\to \infty }{\frac {1}{p^{n))}\sum _{x\in R_{n}\cap K}f(x).}$

This integral was defined by Arnt Volkenborn.

${\displaystyle \int _{\mathbb {Z} _{p))1\,{\rm {d))x=1}$

${\displaystyle \int _{\mathbb {Z} _{p))x\,{\rm {d))x=-{\frac {1}{2))}$

${\displaystyle \int _{\mathbb {Z} _{p))x^{2}\,{\rm {d))x={\frac {1}{6))}$

${\displaystyle \int _{\mathbb {Z} _{p))x^{k}\,{\rm {d))x=B_{k))$

where ${\displaystyle B_{k))$ is the k-th Bernoulli number.

The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.

${\displaystyle \int _{\mathbb {Z} _{p)){x \choose k}\,{\rm {d))x={\frac {(-1)^{k)){k+1))}$

${\displaystyle \int _{\mathbb {Z} _{p))(1+a)^{x}\,{\rm {d))x={\frac {\log(1+a)}{a))}$

${\displaystyle \int _{\mathbb {Z} _{p))e^{ax}\,{\rm {d))x={\frac {a}{e^{a}-1))}$

The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.

${\displaystyle \int _{\mathbb {Z} _{p))\log _{p}(x+u)\,{\rm {d))u=\psi _{p}(x)}$

with ${\displaystyle \log _{p))$ the p-adic logarithmic function and ${\displaystyle \psi _{p))$ the p-adic digamma function.

${\displaystyle \int _{\mathbb {Z} _{p))f(x+m)\,{\rm {d))x=\int _{\mathbb {Z} _{p))f(x)\,{\rm {d))x+\sum _{x=0}^{m-1}f'(x)}$

From this it follows that the Volkenborn-integral is not translation invariant.

If ${\displaystyle P^{t}=p^{t}\mathbb {Z} _{p))$ then

${\displaystyle \int _{P^{t))f(x)\,{\rm {d))x={\frac {1}{p^{t))}\int _{\mathbb {Z} _{p))f(p^{t}x)\,{\rm {d))x}$

• P-adic distribution

• Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen I. In: Manuscripta Mathematica. Bd. 7, Nr. 4, 1972, [1]
• Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen II. In: Manuscripta Mathematica. Bd. 12, Nr. 1, 1974, [2]
• Henri Cohen, "Number Theory", Volume II, page 276