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Handbook of Continued Fractions for Special Functions by Annie A.M. Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon

By Annie A.M. Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon Waadeland, William B. Jones, F. Backeljauw, C. Bonan-Hamada

Special features are pervasive in all fields of technological know-how and undefined. the main recognized software components are in physics, engineering, chemistry, desktop technology and data. due to their significance, a number of books and internet sites (see for example http: functions.wolfram.com) and a wide selection of papers were dedicated to those services. Of the normal paintings at the topic, specifically the Handbook of Mathematical Functions with formulation, graphs and mathematical tables edited by way of Milton Abramowitz and Irene Stegun, the yank nationwide Institute of criteria claims to have bought over seven-hundred 000 copies!

But up to now no venture has been dedicated to the systematic research of persisted fraction representations for those services. This guide is the results of such an endeavour. We emphasise that basically 10% of the continuing fractions contained during this booklet, can be present in the Abramowitz and Stegun venture or on the Wolfram website!

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A sequence {Rn (z)} of functions meromorphic at the origin and at infinity is said to correspond simultaneously to Λ0 (f (z)) and Λ∞ (f (z)) if and only if both λ (Λ0 (f − Rn )) → ∞, λ (Λ∞ (f − Rn )) → ∞. A continued fraction is said to correspond simultaneously to Λ0 (f (z)) and Λ∞ (f (z)) if its sequence of approximants corresponds simultaneously to Λ0 (f (z)) and Λ∞ (f (z)). Criteria for correspondence. Theorems stated in this chapter help to answer the following questions. For a given continued fraction, does there exist a FPS L(z) to which the continued fraction corresponds?

9 for continued fractions over C also hold for continued fractions over a normed field F. 11 Generalisations of continued fractions Generalisations arise when the partial numerators and denominators of the continued fraction are: vectors in Cn [AK87; BGM96; dBJ87; LF96; LVBB94; Par87; Rob02; Smi02], square matrices with complex elements [BB83; BB80; BVB90; Chu01; Fie84; Gu03; LB96; SVI99], operators in a Hilbert space [BF79; Cuy84; Fai72; Hay74; Sch96], multivariate expressions and/or continued fractions themselves [Cha86; Cuy83; Cuy88; CV88a; CV88b; GS81; HS84; KS87; Kuc78; Kuc87; MO78; O’D74; Sem78; Sie80].

This is in sharp contrast with convergent series where ∞ lim n→∞ ck = 0, k=n+1 and convergent infinite products where ∞ lim n→∞ pk = 1. 1: Consider the convergent 2-periodic continued fraction √ 2−1= ∞ K m=1 (3 + (−1)m )/2 1 = 1 2 1 2 . 1 + 1 + 1 + 1 + ... 9 TAILS OF CONTINUED FRACTIONS It can easily be seen that √ f (2M ) = 2 − 1, f (2M +1) = √ 2, 25 M = 0, 1, 2, . . and hence the sequence {f (M ) } does not converge. 2: We study the continued fraction ∞ K m=1 m(m + 2) 1 = 1·3 2·4 3·5 . 1 + 1 + 1 + ...

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