Let
(1)
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(2)
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(3)
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(OEIS A104457), where is the golden ratio, and
(4)
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(5)
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(OEIS A002390).
Define the Fibonacci hyperbolic sine by
(6)
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(7)
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(8)
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The function satisfies
(9)
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and for ,
(10)
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where is a Fibonacci number. For , 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).
Define the Fibonacci hyperbolic cosine by
(11)
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(12)
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(13)
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This function satisfies
(14)
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and for ,
(15)
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where is a Fibonacci number. For , 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).
Similarly, the Fibonacci hyperbolic tangent is defined by
(16)
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and for ,
(17)
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For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).