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A product expansion in $ p$-adic and other non-Archimedean fields

Authors: Arnold Knopfmacher and John Knopfmacher
Journal: Proc. Amer. Math. Soc. 104 (1988), 1031-1035
MSC: Primary 11J61; Secondary 11S80
MathSciNet review: 929409
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Abstract: An algorithm is introduced and shown to lead to a unique infinite product representation for a given $ p$-adic integer $ A$ with leading coefficient 1 as a product

$\displaystyle A = \prod\limits_{n = 1}^\infty {(1 + {b_n}{p^{{r_n}}})} $

where $ 1 \leq {b_n} \leq p - 1,{r_n} \in {\mathbf{N}}$ and $ {r_{n + 1}} > {r_n}$. The degree of approximation by the natural number $ (1 + {b_1}{p^{{r_1}}}) \cdots (1 + {b_n}{p^{{r_n}}})$ is also considered. In addition we derive similar representations for elements of arbitrary complete non-Archimedean fields with discrete valuations.

References [Enhancements On Off] (What's this?)

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  • [2] A. Knopfmacher and J. Knopfmacher, Infinite products for power series, J. Approx. Theory (to appear).
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Article copyright: © Copyright 1988 American Mathematical Society

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