Approximation for the Ratios of the Confluent Hypergeometric Function <inline-formula><math display="inline"><semantics><mrow><msubsup><mo mathvariant="bold">Φ</mo><mi mathvariant="bold-italic">D</mi><mrow><mo mathvariant="bold">(</mo><mi mathvariant="bold-italic">N</mi><mo mathvariant="bold">)</mo></mrow></msubsup></mrow></semantics></math></inline-formula> by the Branched Continued Fractions

Abstract

The paper deals with the problem of expansion of the ratios of the confluent hypergeometric function of N variables ΦD(N)(a,b¯;c;z¯) 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 ΦD(N) ratios are based on some given recurrence relations for this function. The case of nonnegative parameters a,b1,…,bN−1 and positive c is considered. Some convergence criteria for obtained BCF with elements in RN and CN are established. It is proven that these BCF converge to the functions which are an analytic continuation of the above-mentioned ratios of function ΦD(N)(a,b¯;c;z¯) in some domain of CN.

Keywords

• confluent hypergeometric function of several variables
• recurrence relations
• branched continued fraction
• approximant
• uniform convergence