  The dual parametrization and the Mellin-Barnes integral approach represent
two frameworks for handling the double partial wave expansion of generalized
parton distributions (GPDs) in the conformal partial waves and in the
$t$-channel ${\rm SO}(3)$ partial waves. Within the dual parametrization
framework, GPDs are represented as integral convolutions of forward-like
functions whose Mellin moments generate the conformal moments of GPDs. The
Mellin-Barnes integral approach is based on the analytic continuation of the
GPD conformal moments to the complex values of the conformal spin. GPDs are
then represented as the Mellin-Barnes-type integrals in the complex conformal
spin plane. In this paper we explicitly show the equivalence of these two
independently developed GPD representations. Furthermore, we clarify the
notions of the $J=0$ fixed pole and the $D$-form factor. We also provide some
insight into GPD modeling and map the phenomenologically successful
Kumeri\v{c}ki-M\"uller GPD model to the dual parametrization framework by
presenting the set of the corresponding forward-like functions. We also build
up the reparametrization procedure allowing to recast the double distribution
representation of GPDs in the Mellin-Barnes integral framework and present the
explicit formula for mapping double distributions into the space of double
partial wave amplitudes with complex conformal spin.
