Solid partition

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

Definition

Suppose

f : ℤ_{p} \to 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_{ℤ_{p}} f (x) d x = \lim_{n \to \infty} \frac{1}{p^{n}} \sum_{x = 0}^{p^{n} - 1} f (x) .

More generally, if

R_{n} = {x = \sum_{i = r}^{n - 1} b_{i} x^{i} | b_{i} = 0, \dots, p - 1 for r < n}

then

\int_{K} f (x) 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_{ℤ_{p}} 1 d x = 1

\int_{ℤ_{p}} x d x = - \frac{1}{2}

\int_{ℤ_{p}} x^{2} d x = \frac{1}{6}

\int_{ℤ_{p}} x^{k} d x = B_{k}

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

\int_{ℤ_{p}} (\binom{x}{k}) d x = \frac{(- 1)^{k}}{k + 1}

\int_{ℤ_{p}} (1 + a)^{x} d x = \frac{\log (1 + a)}{a}

\int_{ℤ_{p}} e^{a x} 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_{ℤ_{p}} \log_{p} (x + u) d u = ψ_{p} (x)

with $\log_{p}$ the p-adic logarithmic function and $ψ_{p}$ the p-adic digamma function

\int_{ℤ_{p}} f (x + m) d x = \int_{ℤ_{p}} f (x) 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) d x = \frac{1}{p^{t}} \int_{ℤ_{p}} f (p^{t} x) 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