Volkenborn integral

The Volkenborn integral is an integral term for functions on the p-adic numbers .

definition

Be

{\ displaystyle f \ colon \ mathbb {Z} _ {p} \ rightarrow \ mathbb {C} _ {p}}

a locally-analytic function of , the ring of p-adic integers, in , the completion of the algebraic closure of , the field of -adic numbers (a function is called locally-analytic if there is a circular disk around every point within which the function can be developed into a power series). The Volkenborn integral of is then defined by ${\ displaystyle \ mathbb {Z} _ {p}}$ ${\ displaystyle \ mathbb {C} _ {p}}$ ${\ displaystyle \ mathbb {Q} _ {p}}$ ${\ displaystyle p}$ ${\ displaystyle f}$

{\ 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).}

Emergence

F. Thomas and F. Bruhat initially had the idea of integrating p-adic functions . The definition of its translation-invariant p-adic integral turned out to be too restrictive for analytical and number theoretic purposes.

In his dissertation at the University of Cologne in 1971, Arnt Volkenborn developed the generalized -adic integral that was later named after him . With the Volkenborn integral , all local analytical functions, such as the Laurent series , can be integrated. The Volkenborn integral is used in the calculation of the so-called generalized - Bernoulli numbers and other -adic functions. ${\ displaystyle p}$ ${\ displaystyle p}$ ${\ displaystyle p}$

literature

Arnt Volkenborn: A p-adic integral and its applications I. In: Manuscripta Mathematica. Vol. 7, No. 4, 1972, ISSN 0025-2611 , pp. 341-373. doi : 10.1007 / BF01644073
Arnt Volkenborn: A p-adic integral and its applications II. In: Manuscripta Mathematica. Vol. 12, No. 1, 1974, ISSN 0025-2611 , pp. 17-46. doi : 10.1007 / BF01166232
Alain M. Robert : A Course on p-adic Analysis (= Graduate Texts in Mathematics. Vol. 198). Springer, New York et al. 2000, ISBN 0-387-98669-3 , pp. 263-279.
Min-Soo Kim, Jin-Woo Son: Analytic Properties of the q-Volkenborn Integral on the Ring of p-Adic Integers. In: Bulletin of the Korean Mathematical Society. Vol. 44, No. 1, 2007, ISSN 1015-8634 , pp. 1-12, online .