Abstract
The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.
Highlights
Questions that arise in economics, physics, biology, etc., lead to mathematical models, which are often formulated in the form of functional equations of various types, in particular, differential, integro-differential and difference equations
One of the fundamental problems in approaches to finding solutions of such equations is the reconstruction of functions of one or several variables, as well as problems that arise in the development and implementation of effective methods and algorithms for representing and approximating the functions of one or several variables
Gauss in 1812 [29] as a composition of their three-term recurrence relations. This contributed to the construction and study of continued fraction expansions of many special functions, including those that are solutions of various differential equations (see more examples in ([8], Part III: Special Functions))
Summary
Questions that arise in economics, physics, biology, etc., lead to mathematical models, which are often formulated in the form of functional equations of various types, in particular, differential, integro-differential and difference equations (see, for example, [1,2,3,4,5,6,7]). Gauss in 1812 [29] as a composition of their three-term recurrence relations This contributed to the construction and study of continued fraction expansions of many special functions, including those that are solutions of various differential equations (see more examples in ([8], Part III: Special Functions)). Note that in [26] the authors found, in particular, the branched continued expansions of three different types of ratios of generalized hypergeometric function: F2. We show an effective approximation of the analytic function, which under certain conditions is the solution of the differential Equation (2), using the constructed expansion
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