Solid partition

The Volkenborn-integral is an integral for p-adic functions.

Definition

Suppose

f:\mathbb {Z} _{p}\rightarrow {\mathbb {Q}}_{p}

is a function from the p-adic integers to the p-adic rationals, then, under certain conditions, the Volkenborn-Integral is defined by

\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\{x=\sum _{i=r}^{n-1}b_{i}x^{i}|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.

\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}

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

\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.

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

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