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The following is called a J continued fraction:

$$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$

where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$ for $n\geq 0$. I want to write the J-fraction expansion of the power series $\sum_{n=0}^\infty \alpha_nx^n.$ Is there any algorithm for this? If not, what should be a good start?

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  • $\begingroup$ Sorry, I've edited it. $\endgroup$ Oct 29, 2021 at 7:20
  • $\begingroup$ I got confused between $a_n$ and $\alpha_n$, now I've edited it. $\endgroup$ Oct 29, 2021 at 7:45
  • $\begingroup$ how can we find the first few coefficients of the power series expansion of above J-fraction in terms of $\alpha_0, a_i,b_i$? $\endgroup$ Oct 29, 2021 at 7:47
  • $\begingroup$ I am just using Series in Mathematica. $\endgroup$ Oct 29, 2021 at 8:45
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    $\begingroup$ See the answer - it is much more concise and efficient than what I am doing $\endgroup$ Oct 30, 2021 at 7:11

1 Answer 1

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Sagemath can do that too

  sage: x = PowerSeriesRing(QQ,'x').gen()
  sage: f = sum(x**n/(n+1)**2 for n in range(20)).O(20)
  sage: f.jacobi_continued_fraction()
  ((-1/4, -7/144),
   (-13/28, -647/11025),
   (-8795/18116, -71180289/1172105200),
  etc
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