Recent results on
Abel-Gončarov polynomial expansions are applied to study the representability of
holomorphic functions as infinite series in a given Pincherle sequence. As a
generalization of the ordinary derivative we consider the so-called Gel’fond-Leont’ev
derivative 𝒟. We take the exponential function with respect to the derivative 𝒟
and use a duality principle in order to investigate the completeness of the
system En(z) = znE(λnz) in the space ℱr of functions holomorphic on the
interior of the disc of radius r ≦∞. Finally we study the uniqueness of the
representability of holomorphic functions as infinite series in the system
En.
