| Summary: | Ebisu and Iwassaki proved that there are three-term relations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> with a group symmetry of order 72. In this paper, we apply some specific three-term relations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> to partially answer the open problem raised by Miller and Paris in 2012. Given a known value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>−</mo><mi>x</mi></mrow></semantics></math></inline-formula> is an integer, then we construct an algorithm to obtain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mn>3</mn></msub><msub><mi>F</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> in an explicit closed form, where <i>n</i> is a positive integer and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></semantics></math></inline-formula> and <i>f</i> are arbitrary complex numbers. We also extend our results to evaluate some specific forms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>F</mi><mi>p</mi></msub><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, for any positive integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>.
|
|---|