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 :$$f:\Z_p\to \Complex_p$$ 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:


 * $$ \int_{\Z_p} f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x=0}^{p^n-1} f(x). $$

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

 * $$ \int_{\Z_p} 1 \, {\rm d}x = 1 $$
 * $$ \int_{\Z_p} x \, {\rm d}x = -\frac{1}{2} $$
 * $$ \int_{\Z_p} x^2 \, {\rm d}x = \frac{1}{6} $$
 * $$ \int_{\Z_p} x^k \, {\rm d}x = B_k$$

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.


 * $$ \int_{\Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1} $$
 * $$ \int_{\Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a} $$
 * $$ \int_{\Z_p} e^{a x} \, {\rm d}x = \frac{a}{e^a-1} $$

The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.


 * $$ \int_{\Z_p} \log_p(x+u) \, {\rm d}u = \psi_p(x)$$

with $$\log_p$$ the p-adic logarithmic function and $$\psi_p $$ the p-adic digamma function.

Properties

 * $$ \int_{\Z_p} f(x+m) \, {\rm d}x = \int_{\Z_p} f(x) \, {\rm d}x+ \sum_{x=0}^{m-1} f'(x)$$

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

If $$P^t = p^t \Z_p$$ then


 * $$ \int_{P^t} f(x) \, {\rm d}x = \frac{1}{p^t} \int_{\Z_p} f(p^t x) \, {\rm d}x$$