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

ISBN-10: 3540115676

ISBN-13: 9783540115670

**Read or Download Analytic Theory of Continued Fractions: Proceedings of a Seminar-Workshop held at Loen, Norway, 1981 PDF**

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

**Sample text**

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.

System of three-term G n > 0 r for all E ~(0,~) with respect Proof. la) . If there theorem is also a partial recurrence to r be any sequence An(Z) {Qn(Z)}0 denominator, respectively [ii, p. la) formula to each other problem , corresponds ~ , respectively. 3(B) a distribution F and decomposition for classical of the form there exists normalized The numbers of Theorem of a theorem relations Then fraction [3] . {Qn(Z)}; n > i . 3 . 1 ... 1) to formal by . 16) , by comparing satisfied by the Vn(Z) .

Subset of z E K, This D . 1]) theorem the sequence The coefficients Fn,G n and T-fraction be an arbitrary 6(K) for for all such that, j = 1,2 .... 4) is a normal by the Stieltjes-Vitali [8, Theorem of the positive (B) -J=iiz+t(n) I -- 6(K) G I B n(z) It follows . IzI < B(K) n = 1,2,3,... 15 and Remark converges uniformly can be computed on K . 3]). FG-Algorithm. 2] (see also D ~ or I 30 An application in Table and G i . 21a) If a triangular row give the desired **
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