| Summary: | We derive new reduction formulas for the incomplete beta function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">B</mi><mfenced separators="" open="(" close=")"><mo>ν</mo><mo>,</mo><mn>0</mn><mo>,</mo><mi>z</mi></mfenced></mrow></semantics></math></inline-formula> and the Lerch transcendent <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><mi>z</mi><mo>,</mo><mn>1</mn><mo>,</mo><mo>ν</mo></mfenced></mrow></semantics></math></inline-formula> in terms of elementary functions when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>ν</mo></semantics></math></inline-formula> is rational and <i>z</i> is complex. As an application, we calculate some new integrals. Additionally, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.
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