Approximations of Analytic Functions via Generalized Power Product Expansions

Let f(x)=1+∑n=1∞anxn be a formal power series. For a fixed set of nonzero complex numbers {rk}k=1∞ we convert f(x)f(x) into the formal product ∏k=1∞(1+gkxk)rk, namely the Generalized Power Product Expansion  . We provide estimates on the domain of absolute convergence of the infinite product when f(x)=1+∑n=1∞anxn is absolutely convergent. This makes it possible to use the truncated Generalized Power Product Expansions ∏n=1M(1+gkxk)rk as approximations to the analytic function f(x)f(x). The results are made possible by certain intriguing algebraic properties characteristic of the expansions for the case of rk≥1rk≥1. An asymptotic formula for the gkgk associated with the majorizing power series is provided. A combinatorial interpretation of the Generalized Power Product Expansion   with {rk}k=1∞ being integers is also given.