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

Kelvin Lam

We show that $T_p(z)={\prod }_{j=1}^{\infty }(1-z^{p^j}{)}^{-1/p^j}$ is transcendental over $\overline{\mathbb{Q}}(z)$, and establish the transcendence of its values at nonzero algeb…