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 → C p {\displaystyle 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:

∫ Z p f ( x ) d x = lim n → ∞ 1 p n ∑ x = 0 p n − 1 f ( x ) . {\displaystyle \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 = { x = ∑ i = r n − 1 b i x i | b i = 0 , … , p − 1  for  r < n } {\displaystyle 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

∫ K f ( x ) d x = lim n → ∞ 1 p n ∑ x ∈ R n ∩ K f ( x ) . {\displaystyle \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

∫ Z p 1 d x = 1 {\displaystyle \int _{\mathbb {Z} _{p}}1\,{\rm {d}}x=1}
∫ Z p x d x = − 1 2 {\displaystyle \int _{\mathbb {Z} _{p}}x\,{\rm {d}}x=-{\frac {1}{2}}}
∫ Z p x 2 d x = 1 6 {\displaystyle \int _{\mathbb {Z} _{p}}x^{2}\,{\rm {d}}x={\frac {1}{6}}}
∫ Z p x k d x = B k {\displaystyle \int _{\mathbb {Z} _{p}}x^{k}\,{\rm {d}}x=B_{k}}

where B k {\displaystyle 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.

∫ Z p ( x k ) d x = ( − 1 ) k k + 1 {\displaystyle \int _{\mathbb {Z} _{p}}{x \choose k}\,{\rm {d}}x={\frac {(-1)^{k}}{k+1}}}
∫ Z p ( 1 + a ) x d x = log ⁡ ( 1 + a ) a {\displaystyle \int _{\mathbb {Z} _{p}}(1+a)^{x}\,{\rm {d}}x={\frac {\log(1+a)}{a}}}
∫ Z p e a x d x = a e a − 1 {\displaystyle \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.

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

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

Properties

∫ Z p f ( x + m ) d x = ∫ Z p f ( x ) d x + ∑ x = 0 m − 1 f ′ ( x ) {\displaystyle \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 Z p {\displaystyle P^{t}=p^{t}\mathbb {Z} _{p}} then

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