Volkenborn integral

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

Definition

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.
Examples

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

where \( B_{k}\) 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
Properties

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

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

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

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