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# Fibonacci Hyperbolic Functions

Let

 (1) (2) (3)

(OEIS A104457), where is the golden ratio, and

 (4) (5)

(OEIS A002390).

Define the Fibonacci hyperbolic sine by

 (6) (7) (8)

The function satisfies

 (9)

and for ,

 (10)

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) (12) (13)

This function satisfies

 (14)

and for ,

 (15)

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)

and for ,

 (17)

For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).

Fibonacci Number

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## References

Sloane, N. J. A. Sequences A001519/M1439, A001906/M2741, A002390/M3318, and A104457 in "The On-Line Encyclopedia of Integer Sequences."Stakhov, A. and Tkachenko, I. "Hyperbolic Fibonacci Trigonometry." Dokl. Akad. Nauk Ukrainy, No. 7, 9-14, 1993.Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles." Fib. Quart. 34, 129-138, 1996.

## Referenced on Wolfram|Alpha

Fibonacci Hyperbolic Functions

## Cite this as:

Weisstein, Eric W. "Fibonacci Hyperbolic Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html