Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-29T14:08:35.005Z Has data issue: false hasContentIssue false

On the Failure of Heilermann's Theorem

Published online by Cambridge University Press:  20 January 2009

Haripada Datta
Affiliation:
Research Student, Edinburgh University.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The theorem of Heilermann can be stated thus:—

If the series

is converted into a continued fraction of the form

then the elements of the continued fraction are

where

and is obtained from this determinant by deleting the (r + 1)th column and the last row.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1917

References

page 83 note * Journal für Math. 33 (1845), p. 174.Google Scholar

page 87 note * Thus, even if p be ∞, the third relation of (6) will not hold unless A 2 = a 5.

page 89 note * On the Theory of Continued Fractions, Proc. Edin. Math. Soc., 34 Part (2), 19161917.Google Scholar

page 90 note * It is also evident from the relation

F(α, β, γ, x) = (1−x)γ−α−βF(γ−α, γ−β, γ, x) due to Euler.

page 94 note * Sur un certain systéme d'équations linéares. Journ. (de Liouville) de Math. (2), iii., p. 46.Google Scholar

page 95 note * See Section (O), On the Theory of Continued Fractions” (2nd Paper), Proc. Edin. Math. Soc. 35 (Part I.), 19161917, p. 48.Google Scholar

page 96 note * This is evidently the case of due to Gauss.

page 97 note * Evidently (nr) in number, for in others all the elements of the first row are zero.

page 98 note * Memoire sur l'élimination. Annales de l'École Norm. Sup. (2) 7 (1878), p. 151.CrossRefGoogle Scholar

page 98 note † The C's with suffixes higher than n are to be replaced by zeros.

page 99 note * If n is odd, m is odd also ; if n is odd and m even, then at least two of the convergents are equal, but they may not be the n th and the m th convergent.

page 100 note * The conditions are sufficient if none of the a's is zero.

page 102 note * K (a 2a 3 …) denotes the continuant