  This article studies the properties of positive definite, radial functions on
free groups following the work of Haagerup and Knudby . We obtain
characterizations of radial functions with respect to the $\ell^{2}$ length on
the free groups with infinite generators and the characterization of the
positive definite, radial functions with respect to the $\ell^{p}$ length on
the free real line with infinite generators for $0 < p \leq 2$. We obtain the
Levi-Khintchine formulas for length-radial conditionally negative functions as
well.
