  We study the general connectivity distribution functions for growing networks
with preferential attachment of fractional power, $\Pi_{i} \propto k^{\alpha}$,
using the Simon's method. We first show that the heart of the previously known
methods of the rate equations for the connectivity distribution functions is
nothing but the Simon's method for word problem. Secondly, we show that the
case of fractional $\alpha$ the $Z$-transformation of the rate equation
provides a fractional differential equation of new type, which coincides with
that for PA with linear power, when $\alpha = 1$. We show that to solve such a
fractional differential equation we need define a transidental function
$\Upsilon (a,s,c;z)$ that we call {\it upsilon function}. Most of all
previously known results are obtained consistently in the frame work of a
unified theory.
