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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 $\alpha_n= \frac{1}{(n+1)^2}$...

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-...