Let : 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:
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More generally, if
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then
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This integral was defined by Arnt Volkenborn.
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From this it follows that the Volkenborn-integral is not translation invariant.
If then
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- 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