Volkenborn integral

Last updated May 17, 2024

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

Definition

Let : $f:\mathbb {Z} _{p}\to \mathbb {C} _{p}$ 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:

\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

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

\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

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

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

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

\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.

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

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

\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.

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

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

Properties

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

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

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References

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

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