Approximation for the Ratios of the Confluent Hypergeometric Function ΦD(N) by the Branched Continued Fractions

The paper deals with the problem of expansion of the ratios of the confluent hypergeometric function of N variables Phi((N))(D)(a,(b) over bar ;c;(z) over bar) into the branched continued fractions (BCF) of the general form with N branches of branching and investigates the convergence of these BCF. The algorithms of construction for BCF expansions of confluent hypergeometric function Phi((N))(D) ratios are based on some given recurrence relations for this function. The case of nonnegative parameters a,b(1),...,b(N-1) and positive c is considered. Some convergence criteria for obtained BCF with elements in R-N and C-N are established. It is proven that these BCF converge to the functions which are an analytic continuation of the above-mentioned ratios of function Phi((N))(D)(a,(b) over bar ;c;(z) over bar) in some domain of C-N.

Automated summary

This paper deals with the expansion problem of confluent hypergeometric function ratios and describes it using branched continued fractions. The convergence of these fractions and their analytical continuation in the complex domain are investigated.