Applications of continued fractions in one and more variables

Abstract

Elementary properties of continued fractions are derived from sets of three-term recurrence relations and approximation methods are developed from this simple approach.

First, a well-known method for numerical inversion of Laplace transforms is modified in two different ways to obtain exponential approximations. Differential-difference equations arising from certain Markov processes are solved by direct application of continued fractions and practical error estimates are obtained. Approximations of a slightly different form are then derived for certain generalised hypergeometric functions using those hypergeometric functions that satisfy three-term recurrence relations and have simple continued fraction expansions. Error estimates are also given in this case.

The class of corresponding sequence algorithms is then described for the transformation of power series into continued fraction form. These algorithms are shown to have very general application and only break down if the required continued fraction does not exist. A continued fraction in two variables is then shown to exist and its correspondence with suitable double power series made feasible by the generalisation of the corresponding sequence method. A convergence theorem, due to Van Vleck, is adapted for use with this type of continued fraction and a comparison is made with Chisholm rational approximants in two variables. Some of these ideas are further generalised to the multivariate case.

Such corresponding fractions are closely related to other fractions that may be used for point-wise bivariate or multivariate interpolation to function values known on a mesh of points. Interpolation algorithms are described and advantages and limitations discussed.

The work presented forms a basis for a wide range of further research and some possible applications in numerical mathematics are indicated.