Volkenborn integral explained

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

Definition

Let :

f:\Z _{p\to} \Complex _{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 _{\Z _{p}} f(x) {\rm d}x = \lim _{n} (1)/(p ^{n}) \sum ^{p ^{n-1}} _{x =0} f(x).

More generally, if

R _{n} = \left\{\left. x = \sum ^{n -1} _{i = r} b _{i} x ^{i} \right | b _{i=0,} \ldots, p -1 for r<n \right\}

then

\int _{K} f(x) {\rm d}x = \lim _{n} (1)/(p ^{n}) \sum _{x \in R _{n} \cap K} f(x).

This integral was defined by Arnt Volkenborn.

Examples

\int _{\Z _{p}} 1 {\rm d}x = 1 \int _{\Z _{p}} x {\rm d}x =- (1)/(2) \int _{\Z _{p}} x ^{2} {\rm d}x = (1)/(6) \int _{\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 _{\Z _{p}} {x \choose k} {\rm d}x = (( -1 ) ^{k})/(k +1) \int _{\Z _{p}} ( 1 + a) ^{x} {\rm d}x = (log ( 1+ a))/(a) \int _{\Z _{p}} e ^{a} {\rm d}x = (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)} {\rm d}u = \psi _{p(x)}

with

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

Properties

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

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

If

P ^{t} = p ^{t} \Z _{p} then \int _{P ^{t}} f(x) {\rm d}x = (1)/(p ^{t}) \int _{\Z _{p}} f(p ^{t} x) {\rm d}x

See also

• P-adic distribution

References

• Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen I. In: Manuscripta Mathematica. Bd. 7, Nr. 4, 1972, http://eudml.org/doc/154126
• Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen II. In: Manuscripta Mathematica. Bd. 12, Nr. 1, 1974, http://eudml.org/doc/154225
• Henri Cohen, "Number Theory", Volume II, page 276