Transcendence and algebraic independence of a family of $p$-adic valuation generating functions

We show that $T_p(z)={\prod }_{j=1}^{\mathrm{\infty }}(1-z^{p^j}{)}^{-1\mathrm{/}{p}^j}$ is transcendental over $\bar{\mathbb{Q}}(z)$, and establish the transcendence of its values at nonzero algebraic points inside the unit disk. Furthermore, we obtain an algebraic independence result for multiplicatively independent algebraic arguments. In summary, this paper extends Mahler’s method beyond the classical automatic setting by studying the function $T_p(z)$, whose coefficients are governed by the unbounded arithmetic function ${\nu }_p(n)$.