| Summary: | This paper concerns the properties of the generalized bi-periodic Fibonacci numbers <inline-formula><math display="inline"><semantics><mrow><mo>{</mo><msub><mi>G</mi><mi>n</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> generated from the recurrence relation: <inline-formula><math display="inline"><semantics><mrow><msub><mi>G</mi><mi>n</mi></msub><mo>=</mo><mi>a</mi><msub><mi>G</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> (<i>n</i> is even) or <inline-formula><math display="inline"><semantics><mrow><msub><mi>G</mi><mi>n</mi></msub><mo>=</mo><mi>b</mi><msub><mi>G</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> (<i>n</i> is odd). We derive general identities for the reciprocal sums of products of two generalized bi-periodic Fibonacci numbers. More precisely, we obtain formulas for the integer parts of the numbers <inline-formula><math display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><mo>∞</mo></msubsup><mfrac><msup><mrow><mo>(</mo><mi>a</mi><mo>/</mo><mi>b</mi><mo>)</mo></mrow><mrow><mi>ξ</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup><mrow><msub><mi>G</mi><mi>k</mi></msub><msub><mi>G</mi><mrow><mi>k</mi><mo>+</mo><mi>m</mi></mrow></msub></mrow></mfrac></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mo>⋯</mo><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><mo>∞</mo></msubsup><mfrac><mn>1</mn><mrow><msub><mi>G</mi><mi>k</mi></msub><msub><mi>G</mi><mrow><mi>k</mi><mo>+</mo><mi>m</mi></mrow></msub></mrow></mfrac></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mspace width="0.277778em"></mspace><mspace width="0.277778em"></mspace><mi>m</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mo>⋯</mo><mo>.</mo></mrow></semantics></math></inline-formula>
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