| Summary: | The paper deals with the problem of expansion of the ratios of the confluent hypergeometric function of <i>N</i> variables <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>Φ</mo><mi>D</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mo>;</mo><mi>c</mi><mo>;</mo><mover accent="true"><mi>z</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula> into the branched continued fractions (BCF) of the general form with <i>N</i> branches of branching and investigates the convergence of these BCF. The algorithms of construction for BCF expansions of confluent hypergeometric function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mo>Φ</mo><mi>D</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> ratios are based on some given recurrence relations for this function. The case of nonnegative parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>b</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> and positive <i>c</i> is considered. Some convergence criteria for obtained BCF with elements in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>N</mi></msup></semantics></math></inline-formula> are established. It is proven that these BCF converge to the functions which are an analytic continuation of the above-mentioned ratios of function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>Φ</mo><mi>D</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mo>;</mo><mi>c</mi><mo>;</mo><mover accent="true"><mi>z</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in some domain of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">C</mi><mi>N</mi></msup></semantics></math></inline-formula>.
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