Approximations of analytic functions via generalized power product expansions (Q468456)

Let \(f(x)=1+\sum_{n=1}^\infty a_n x^n\) be an analytic function. The \textit{Generalized Power Product Expansion} (GPPE) of \(f\) is \(f(x)=\prod_{k=1}^\infty (1+g_k x^k)^{r_k}\), where \(\{g_k\}\) are certain coefficients and \(\{r_k\}\) is a sequence of arbitrary nonzero complex numbers. The paper deals with the conversion of the power series of \(f\) into a GPPE. For this, three algebraic representations for the coefficients \(g_k\) in terms of the coefficients \(a_n\) and \(r_n\) are obtained. It is also proved that given the sequences \(\{r_k\}\) and \(\{g_k\}\), then there exists a unique sequence of complex numbers \(\{a_n\}\) such that the corresponding power series equals the GPPE associated with \(\{r_k\}\) and \(\{g_k\}\).NEWLINENEWLINEThe majorizing GPPE is \(M(x)=1-\sum_{n=1}^\infty M_n x^n=\prod_{n=1}^\infty (1-E_nx^n)^{r_n}\), with \(|a_n|\leq M_n\), \(n\in \mathbb{N}\). The following is one of the main results. Let \(r_n\geq 1\), and let \(s=\sup_{n\geq 1}|a_n|^{1/n}\). Then \(f(x)\), \(M(x)\) and their respective GPPEs are all absolutely convergent when \(|x|\leq 1/(2s)\).NEWLINENEWLINEThe rest of the article is devoted to study convergence as well as a combinatorial interpretation for the equation \(f(x)=\prod_{k=1}^\infty (1+g_k x^k)^{r_k}\).