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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
DOI: https://doi.org/10.1090/S0002-9939-1988-0929409-9
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?)

  • [1] P. Bundschuh, 𝑝-adische Kettenbrüche und Irrationalität 𝑝-adischer Zahlen, Elem. Math. 32 (1977), no. 2, 36–40 (German). MR 0453620
  • [2] A. Knopfmacher and J. Knopfmacher, Infinite products for power series, J. Approx. Theory (to appear).
  • [3] K. Mahler, Zur Approximation $ p$-adischer Irrationalzahlen, Nieuw Arch. Wisk. 18 (1934), 22-34.
  • [4] W. H. Schikhof, Ultrametric calculus, Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, Cambridge, 1984. An introduction to 𝑝-adic analysis. MR 791759
  • [5] Th. Schneider, Über 𝑝-adische Kettenbrüche, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 181–189 (German). MR 0272720

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0929409-9
Article copyright: © Copyright 1988 American Mathematical Society