Regularity properties of k-Brjuno and Wilton functions

Abstract We study functions related to the classical Brjuno function, namely k -Brjuno and Wilton function. Both appear in of boundary regularity properties (quasi) modular forms their integrals. consider various possible versions them, based on $$\alpha $$ α -continued fraction developments. BMO behaviour near rational numbers finite truncations. then complexify functional equations which they fulfill we construct analytic extensions upper half-plane. using an extension continued algorithm complex plane. also prove that harmonic conjugate real function is continuous at all irrational has a decreasing jump $$\pi /q^k$$ π / q k points p / q .