Elementary Analytic Properties of Continued Fractions

It is seen that aj affects the coefficients of and higher powers affects the coefficients of and higher powers and so on. In general, a affects the coefficients of x and higher powers. Hence the series expansion of the nth approximant matches exactly the series expansion of the Stieltjes fraction up to powers of order n, while higher power coefficients in general will be different. We can thus write 

In general the approximants can be obtained using the fundamental recurrence formulas (2.3), which in our specific case are 

The expressions for the coefficients o, y, 8 can be worked out explicitly from the fundamental recurrence formulas (3.5). We have need in particular for a few of them as follows  

If we consider the Stieltjes fraction (3.1), write x= /z, and perform an equivalence transformation by multiplying by z the composing fractions, we obtain the S-fraction of the type 

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