Download Analytic Theory of Continued Fractions II: Proceedings of a by Christopher Baltus, William B. Jones (auth.), Wolfgang J. PDF

By Christopher Baltus, William B. Jones (auth.), Wolfgang J. Thron (eds.)

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Additional resources for Analytic Theory of Continued Fractions II: Proceedings of a Seminar-Workshop held in Pitlochry and Aviemore, Scotland June 13–29, 1985

Sample text

Observations required basis of exact". for the is r e p l a c ed in three ways: to be on the real axis. strategy is established. by a simpler and cheaper one. 39 Instead fractions of c o n s i d e r i n g with elements the from class the of all boundary in those P 2-periodic c) it s u f f i c e s continued are the diagonal. same to use fractions elements observed continued opposite whose points This was in [I]. and p r o v e d of first ~E Re 2. 2) P will q be to use the continued fractions P 1 + 1 + 1 + p = a + r'e I + ...

Pad~ ~ O. (in [2]) is to e x h i b i t familiar This to those structure is M-table. functio~ R appears more thaD once in the t h e n the set E R = ((n, k) IMn,k(Z) m R} forms a s a u a r e b l o c k in t h e table. The s q u a r e b l o c k m a y be i n f i n i t e . 28 Proof: This Mn,k(Z) is a direct consequence a n d MJM~(z)n,K (see T h e o r e m of 4) a n d T h e o r e m We s a y an e n t r v M n , k ( Z ) in the M - t a b l e square block normal will if e v e r y e n t r y be u s e f u l Lemma then containing 7.

1 + w' is easy to show that S n is written If /~n/~n_l independent fraction { g 2 n + l ( Z ) }n= I, In accordance It 12 a n d ~ diverges, fraction Some l + 1 + /~2n-i ~ if odd approximants 2. p. ( ~ 2 n + ~ 2 n + l )z ~ 0, the further 2n3 + i] tf~2n_-iz i ~2n-2 continued if 1 + [7, bn(Z) , n = 3,4,5,,,. Sn(O)" With this we introduce n ~ ~. 2] can be 50 Clearly, converges tail in C. of Hence, K(an/1). 2] is tails called tails of K ( a n / l ) or N=1,2,3 ..... 3] K(an/1), complex a sequence K(an/l) of if of riaht or or not.

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