A new application of Lagrange-Bürmann expansions. I. General principle
A method for obtaining a system of approximating functions \(\{s^{(n)}(z)\}\) for a function f(z) which is associated with the series \(\sum_{i=0}f_ iz^ i\) for small z and satisfies the relationship \(f(z)=f(\infty)+O(z^{-1})\) for large z is described. The method proceeds in three stages as follows. (A) With \(e(z)=z^{- 1}\{f(z)-f_ 0\}\) set \(g(z)=\{f(z)-f(\infty)\}/e(z)\) and obtain by algebraic methods the coefficients \(g_ i(1\leq i\leq n)\) in the Lagrange- Bürmann expansion \(g(z)=g(0)+\sum_{i=1}g_ i\{ze'(z)/e(z)\}^ i.\) (B) With \(p(z)=\sum^{n}_{i=1}g_ iz^ i\) obtain, again by algebraic methods, the coefficients \(q_ i\) (1\(\leq i\leq n)\) in the inverse function \(q(z)=\sum_{i=1}q_ iz^ i\) for which \(q\{p(z)\}=z\). (C) With \(q^{(n)}(z)=\sum^{n}_{i=1}q_ iz^ i,\) solve the differential equation \(r^{(n)}(0)=f_ 1\), \(z dr^{(n)}(z)/dz=r^{(n)}(z)q^{(n)}\{((zr^{(n)}(z)-f(\infty)+f_ 0)/r^{(n)}(z))-g(0)\}.\) \(r^{(n)}(z)\) is then an approximation to e(z) and \(s^{(n)}(z)=f_ 0+zr^{(n)}(z)\) is one to f(z). A theory of the convergence of the \(s^{(n)}(z)\) to f(z) is given. In very simple cases, as the author indicates, the \(g_ i\), \(q_ i\) and \(r^{(n)}(z)\) above are available in closed form.
scientific article; zbMATH DE number 3883483