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We study the general connectivity distribution functions for growing networkswith preferential attachment of fractional power, $\Pi_{i} \propto k^{\alpha}$,using the Simon's method. We first show that the heart of the previously knownmethods of the rate equations for the connectivity distribution functions isnothing but the Simon's method for word problem. Secondly, we show that thecase of fractional $\alpha$ the $Z$-transformation of the rate equationprovides a fractional differential equation of new type, which coincides withthat for PA with linear power, when $\alpha = 1$. We show that to solve such afractional differential equation we need define a transidental function$\Upsilon (a,s,c;z)$ that we call {\it upsilon function}. Most of allpreviously known results are obtained consistently in the frame work of aunified theory.Comment: 10 page

General connectivity distribution functions for growing networks with preferential attachment of fractional power