By Christopher Baltus, William B. Jones (auth.), Wolfgang J. Thron (eds.)
Read or Download Analytic Theory of Continued Fractions II: Proceedings of a Seminar-Workshop held in Pitlochry and Aviemore, Scotland June 13–29, 1985 PDF
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Additional info for Analytic Theory of Continued Fractions II: Proceedings of a Seminar-Workshop held in Pitlochry and Aviemore, Scotland June 13–29, 1985
That $0 = are n the Bn(Z) ) . 1] , we we see have (C) Therefore, by Lemma 7, given Hin2n+l)(c) (2) " (-(2n+l)) n2n+l (C) (3) N ( - 2 n ) (C) --2n+1 > 0 (4) H~n2n) (C) > 0 i, (z)) H(-n+I)(c) n (1) n = 0, and n Sn of H(-n+2)(C) ~ 0 f o r n = I, 2 ..... n the first row must be normal. were deg(B 2, ... ~ O, ~ O, Using Jacobi's identity we obtain ( ( - -(C)H2n (-2n) Hzn2n) ( 2 n - 2 ) ) (C) _ ~-2n+1 ( C ~" H (2n-1 -(2n-2))(C) each 35 and H(-2n) 2n+1 (C) > 0 by (5) All we is [H ~ (2n-l))(C) ]2 since and (3) H(-(2n-2))(C)2n can conclude nonzero since > 0 (i) , by ~(-(2n-1))(C) -2n-1 > 0 by > O for n = I, 2 .....
I C k " + ... + ~ i C n + k _ 2 has a n o n - t r i v i a l rule = O(z n+k) + ~n_iC_n+k+l SnC_n+k+l solution. 11] where ^ C-n+ k C-n+k+ I ... c_j+ k ... ck C-n+k+l C-n+k+2 "'" c_j+k+ 1 ... Ck+ 1 ck ... c_j+k+n_ 1 -.. 1]. C H(k)(c) - - M-table Proof relationship dimension H(-n+k)(c) ~ 0 and H(-n+k+l)(c) ~ 0 n n M n , k ( Z ) is e i t h e r a n o r m a l e n t r y or square which is is n o r m a l . in the n e x t the 29 (the get indicated column is o m i t t e d ) . it we m u s t choose $0 - is a l s o Bn,k(Z) unique = @0 + @1 z + a square to block but the result that for a is not on the know that appearing left c ~ O.
K(an/1) the negative real convergence criteria , if 2, 3 . . . that K(an/l) n = i , 2 , 3 ..... 2] diverges if a n = -1/4 where c prove that > 1/16, this - (C+an)/(n+p)(n+p+l), p ~ ~\z is indeed convergence/divergence , ~ n so. ~ criterion R But n = I, 2, 3 ..... and ~ ~ 0 fast e n o u g h . n f i r s t w e s h a l l p r o v e the for real continued fractions. 3] We shall following 51 G i v e n the c o n t i n u e d an = g ( n - l ) ( l + g(n)) f r a c t i o n K ( a n / l ), w h e r e • R , n = i, 2 , 3, ...
Analytic Theory of Continued Fractions II: Proceedings of a Seminar-Workshop held in Pitlochry and Aviemore, Scotland June 13–29, 1985 by Christopher Baltus, William B. Jones (auth.), Wolfgang J. Thron (eds.)