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Consider the following two continued fractions $A$ and $B$:

$$\frac{\alpha_0}{1+a_0x-\frac{b_1x^2}{1+a_1x-\frac{b_2x^2}{1+a_2x-\cdots}}}$$ $$\frac{\beta_0}{1+c_0x-\frac{d_1x}{1+c_1x-\frac{d_2x}{1+c_2x-\cdots}}}$$

where $a_i,b_i,c_i,d_i$ and $\alpha_0,\beta_0$ are real numbers.
The first fraction $A$ is called J-fraction. Can I rewrite the second fraction $B$ in the form of J-fraction $A$? Any suggestions, please?

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    $\begingroup$ The second fraction is called a T-fraction. $\endgroup$
    – Ira Gessel
    Commented Oct 28, 2021 at 4:22
  • $\begingroup$ J-fraction is associated with a Hankel deteminant. Is T-fraction also associated with a Hankel determinant? $\endgroup$
    – VSP
    Commented Oct 28, 2021 at 9:30
  • $\begingroup$ I don't know much about them, but you can look them up. They're also called Thron-type continued fractions. Two references are projecteuclid.org/journals/… and jstor.org/stable/44237585 . $\endgroup$
    – Ira Gessel
    Commented Oct 28, 2021 at 15:15
  • $\begingroup$ thanks, can u share any link for the relation between the continued fractions and determinants? $\endgroup$
    – VSP
    Commented Oct 28, 2021 at 17:31
  • 2
    $\begingroup$ See arxiv.org/abs/math/9902004 and arxiv.org/abs/math/0503507. $\endgroup$
    – Ira Gessel
    Commented Oct 28, 2021 at 20:41

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