A study on functional independence of the Iwasawa power series

We study a class of functional independences that the Iwasawa power series satisfy for both zero and non-zero characteristics. As results, we prove a generalization of Anglès and Ranieri B. Anglès, G. Ranieri, On the linear independence of p$p$-adic L$L$-functions modulo p$p$, Ann. Inst. Fourier (Grenoble) 60 (5) (2010) 1831–1855] and transcendence of the Iwasawa power series over the rational functions for non-zero characteristics. We also verify that the power series is not a solution of any non-trivial linear differential equation with the coefficients of rational functions over p$p$-adic numbers.