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Read or Download Theory and Application of Special Functions. Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin–Madison, March 31–April 2, 1975 PDF

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Extra info for Theory and Application of Special Functions. Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin–Madison, March 31–April 2, 1975

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C . z ~ + 0 1 m 1 '" l-a(t) . (See, in this c o n n e c - tion, Allen, Chui, Madych, Narcowich and Smith [1974]). Similar s t a t e - ments can be obtained for the entries in the lower half of the Pade table by using (4). We remark that in the c a s e of convergence, the continued fraction 1 S (m) l Z S (m) 2 Z in (19), and the analogous continued fraction in (20), serve as "converging factor", being the factor by which the l a s t term c z is to be mul- tiplied in order to obtain the correct limit of the series (1).

In describing these transformations, we limit ourselves to elliptic integrals of the first kind, and must refer to the literature for the others. An early treatment of computational algorithms for elliptic functions and integrals i s King [1924]. We follow more c l o s e l y the work of Carlson [1965], who develops the algorithms in a unified way, at least for integrals of the first two kinds. Hof sommer and van de Riet [1963] have ALGOL programs for integrals of the first and second kind, using Landen transformations, as well as programs for elliptic functions, based on ascending Landen and descending Gauss transformations.

1 oo = -TTI = π (Z) z - a P--1, π P(z) J _ P V L k=l «r ■r z. P (z) - π (t) t z-t P d4>(t) v ' 9 π (t) and ω, k pv ' 25 π the a s s o c i a t e d Christoffel WALTER GAUTSCHI numbers. The polynomials ^,(ζ) are thus the denominators of the con- tinued fraction in (14), the associated orthogonal polynomials CO . the numerators. * (t) Both satisfy the same recurrence formula, V l = (z "ar,yr"bryr-l' where y Q = 1, y ^ = 0 for ^ } , r = 0,1, 2, . . , and y Q = 0, y ^ = -1 for {σ^ . This i s meaningful not only for Stieltjes series, but for any series which has an associated J-fraction, provided orthogonality i s defined algebraically (Wall [1948, p.

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