Volkenborn integral

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



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:

More generally, if


This integral was defined by Arnt Volkenborn.


, the k-th Bernoulli number

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

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

with the p-adic logarithmic function and the p-adic digamma function


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

If then

See also[edit]


  • 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