Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

Axioms

Roman Dmytryshyn¹  Serhii Sharyn¹  Tamara Antonova² 

[1] Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., 76018 Ivano-Frankivsk, Ukraine;Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, 12 Stepana Bandery Str., 79000 Lviv, Ukraine;

关键词: generalized hypergeometric function;    branched continued fraction;    convergence;    rational approximation;   

DOI  :  10.3390/axioms10040310

来源: DOAJ

【 摘 要 】

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 ${}_3{F}_2$ 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 ${}_3{F}_2.$ To illustrate this, we have given a few numerical experiments at the end.

【 授权许可】

Unknown