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Generalized Volkenborn Integrals Associated with \(p\)-Adic Distributions and the Bernoulli Numbers

Abstract

Our goal is to give a formula representing the Bernoulli numbers by \(p\)-adic distributions. We consider \(p\)-adic distributions on the ring of \(p\)-adic integers which are invariant by rotations around the origin, and define a generalization of the Vokenborn integrals with respect to such distributions. It is shown the generalized Volkenborn integrals of power functions, and of negative powers of the \(p\)-adic norm converge under some conditions on the distributions, and their universal relation to the Bernoulli numbers is presented.

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Correspondence to Kumi Yasuda.

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Yasuda, K. Generalized Volkenborn Integrals Associated with \(p\)-Adic Distributions and the Bernoulli Numbers. P-Adic Num Ultrametr Anal Appl 14, 164–171 (2022). https://doi.org/10.1134/S2070046622020078

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  • DOI: https://doi.org/10.1134/S2070046622020078

Keywords

  • \(p\)-adic distributions
  • Volkenborn integrals
  • Bernoulli numbers