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Diophantine approximation with partial sums of power series

Volume 161 / 2013

Bruce C. Berndt, Sun Kim, M. Tip Phaovibul, Alexandru Zaharescu Acta Arithmetica 161 (2013), 249-266 MSC: Primary 11J68; Secondary 11J17, 11J70. DOI: 10.4064/aa161-3-4

Abstract

We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.

Authors

  • Bruce C. BerndtDepartment of Mathematics
    University of Illinois
    1409 West Green St.
    Urbana, IL 61801, U.S.A.
    e-mail
  • Sun KimDepartment of Mathematics
    Ohio State University
    231 West 18th Avenue
    Columbus, OH 43210, U.S.A.
    e-mail
  • M. Tip PhaovibulDepartment of Mathematics
    University of Illinois
    1409 West Green St.
    Urbana, IL 61801, U.S.A.
    e-mail
  • Alexandru ZaharescuDepartment of Mathematics
    University of Illinois
    1409 West Green St.
    Urbana, IL 61801, U.S.A.
    and
    Institute of Mathematics of the
    Romanian Academy
    P.O. Box 1-764
    Bucureşti RO-70700, Romania
    e-mail

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