Power-series equivalence of some functional series with applications

In this paper, we show that the Laplace transform of the expansion
h(t) = \sigma_{n = 0}^{\infty} c_{n} g_{n} (t)
for some important sets
g_{n} (t)
is equivalent to a power-series expansion. Techniques based on this result are presented for obtaining the coefficients c. as those of a power series; also, methods are presented for obtaining the functional series inverse. The set of Laguerre functions is discussed in detail and, using the power-series equivalence, the truncation error is obtained. The application of the power-series equivalence to the summing of series is shown and illustrated with the Neumann series. Finally, the extension of the power-series equivalence to the expansion of functions of several variables is given. The areas for which the techniques developed are relevant include the analysis and design of signals and the identification and synthesis of processes and systems.