The connection between $P$-fractions and associated fractions

The associated continued fractions of a power series $L$ is a special case of the $P$-fraction of a power series ${L^{\ast }}$. The latter is closely connected with the Padé table of ${L^{\ast }}$. We prove that every $P$-fraction is the limit of the appropriate contraction of associated fractions in the sense that as the coefficients of $L$ approach those of ${L^{\ast }}$ the elements and approximants of the contraction approach the elements and approximants of the $P$-fraction.