On the norm of Krasner analytic functions with applications to transcendence results

Given a prime number p let Cp be the Tate field i.e. the topological completion of the algebraic closure of the field of p-adic numbers with respect to the p-adic absolute value. We give an estimate of the norm of Krasner analytic functions defined on the complement of a fundamental set X of Cp, which are Cauchy transforms obtained by integrating against strongly Lipschitz distributions on X. The functions from a specific class of the above representations are transcendental over Cp(Z) and, as a consequence, we obtain transcendence results related to the twisted p-adic log gamma function and the trace functions. The twisted p-adic log gamma function and its derivatives are linearly independent over Cp(Z) and, moreover, all their zeros are algebraic.