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Algebraic Independence of the Power Series Defined by Blocks of Digits

โœ Scribed by Yoshihisa Uchida

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
128 KB
Volume
78
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

Let q 2 be an integer and let w be a block of 0, ..., q&1 of finite length. For a nonnegative integer n, let e(w; n) denote the number of occurrences of w in the q-adic expansion of n. Define f (w; z)= n 0 e(w; n) z n . We give necessary and sufficient conditions for the algebraic independence of functions f (w 1 ; z), ..., f (w m ; z) and their values for given blocks w 1 , ..., w m .

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Some time ago Mills and Robbins (1986, J. Number Theory 23, No. 3, 388-404) conjectured a simple closed form for the continued fraction expansion of the power series solution \(f=a_{1} x^{-1}+a_{2} x^{-2}+\cdots\) to the equation \(f^{4}+f^{2}-x f+1=0\) when the base field is GF(3). In this paper we