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Lets say you have a partial fraction of the form:

$$ f(x) = a_0 + \sum_{n=0}^{\infty} \frac{a_n}{\lambda_n + x} $$

Can anyone explain to me, in mildly plain English, how to convert this partial fraction to a continued fraction of the form:

$$ f(x)= b_0+\frac{b_1}{1+}\;\;\frac{b_2x}{1+}\;\;\frac{b_3x}{1+}\;\;... $$

I have tried to find good answers but have only been able to find very difficult academic papers detailing QD algorithms (Rutishauser, 1954), Jacobi matrices (de Boor & Golub, 1978), etc. I have also looked in the textbook Numerical Methods That Work (Acton) but he only has a small section on QD algorithms.

Any help or direction to a good resource is greatly appreciated.

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  • $\begingroup$ please check the solution here math.stackexchange.com/questions/6757/… it may help you $\endgroup$ – Nizar Oct 13 '15 at 19:58
  • $\begingroup$ See Gauss's continued fraction and Euler's continued fraction formula. $\endgroup$ – Lucian Oct 13 '15 at 20:09
  • $\begingroup$ Should it be $b_1 x$ instead of just $b_1$? $\endgroup$ – Hans Lundmark Oct 13 '15 at 22:01
  • $\begingroup$ No, there, the form of the continued fractions which I have looked at are in the form with just $b_1$ at the top, not $b_1x$. $\endgroup$ – Darcy Oct 13 '15 at 22:48
  • $\begingroup$ @Nizar The link you provided gives the solution for finding the partial sum given a continued fraction. I am looking for the reverse process whereby you arrive at a continued fraction given a partial sum. $\endgroup$ – Darcy Oct 13 '15 at 23:13
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This is not really a complete answer, but I'm afraid that the complete story is rather long...

What you're looking for is called a Stieltjes continued fraction, and that's the subject of Stieltjes's famous work Recherches sur les fractions continues. Unfortunately, I don't know any modern exposition of this material in English, but if you can read French, the original text is actually not too difficult. It's on pp. 402–566 of Volume II of his Œuvres Complètes, which can be found online, for example here.

If you expand each term of your sum as a geometric series, then (provided that the order of summation can be changed) you can rewrite it as a Laurent series of the form $$ f(x) = \frac{c_0}{x} - \frac{c_1}{x^2} + \frac{c_2}{x^3} - \dots $$ instead. Stieltjes explains in the first sections, especially in Section 7 (p. 419), how to rewrite a continued fraction as such a Laurent series (at least formally), and then in Section 11 (p. 425) he shows how to go the other way around (which is what you want to do); the outcome is that the coefficients of the continued fraction are given by formulas involving certain determinants where the coefficients $c_k$ appear (formula (7) on p. 427).

(I'm not really familiar with much more of the article than this, but I think that much of the rest is concerned with questions of convergence and such things.)

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