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William B. Jones, W. J. Thron, Haakon Waadeland (eds.)'s Analytic Theory of Continued Fractions: Proceedings of a PDF

By William B. Jones, W. J. Thron, Haakon Waadeland (eds.)

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Additional info for Analytic Theory of Continued Fractions: Proceedings of a Seminar-Workshop held at Loen, Norway, 1981

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GQ(n) = GQ)n) Approximation of by quadrature formula (n)_. (n). L w~ Fit~ ) SD(n) denotes the number of significant digits in j=l o o given by SD(n) = -loglOl(l-GQ(n))/l I , ! 00000 818 178 996 176 738 269 104 042 Agknowledgement. 20(-7) not defined The authors wish to express thanks to Allan Stelnhardt and Robert Jones for able assistance with the computations discussed above. 8. Correspondence and asymptotic expansions. 3) . We first show that the T-fraction (with L0 at 0 (and Let L~ n) = -c_iz .

This can be taken care of by for all but a finite number n if we say that lim An/B n for all n ~ 1 . exists, of n , or by assuming or, finally, that by assuming as we A minor change was made a few years ago n/ and that is if convergent. [14] involving of convergence. for an infinite an ~ 0 future applications in this context. A new look at the definition fraction this of Jacobsen on possible lim A B n ~ then the continued fraction is considered to be n~ Previously it was called unessentially (unwesentlich) divergent in that case.

14a) . 9 We conclude this section with the following result which shows that the denominators of a positive T-fraction give rise to a sequence of orthogonal L-polynomials. 6. 16) and ~(t) E ~(0, ~) Proof. nth denominator of a positive F > 0 , G > 0 for all n > I . Let {Qn(Z)}o be n n --Q0(z) E i . 14a) . The difference equations satisfied by the V (z) insure that the n Qn(Z) satisfy Theorem 7. 15) . 5 . 9 Gaussian Quadrature. We present here a new case in the development of Gaussian quadaratures which also is suggested by continued fractions.

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