  The model of self-avoiding lattice walks and the asymptotic analysis of
power-series have been two of the major research themes of Tony Guttmann. In
this paper we bring the two together and perform a new analysis of the
generating functions for the number of square lattice self-avoiding walks and
some of their metric properties such as the mean-square end-to-end distance.
The critical point $x_c$ for self-avoiding walks is known to a high degree of
accuracy and we utilise this knowledge to undertake a new numerical analysis of
the series using biased differential approximants. The new method is major
advance in asymptotic power-series analysis in that it allows us to bias
differential approximants to have a singularity of order $q$ at $x_c$. When
biasing at $x_c$ with $q\geq 2$ the analysis yields a very accurate estimate
for the critical exponent $\gamma=1.3437500(3)$ thus confirming the conjectured
exact value $\gamma=43/32$ to 8 significant digits and removing a long-standing
minor discrepancy between exact and numerical results. The analysis of the
mean-square end-to-end distance yields $\nu=0.7500002(4)$ thus confirming the
exact value $\nu=3/4$ to 7 significant digits.
