Fibonacci Hyperbolic Functions
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A001519/M1439, A001906/M2741, A002390/M3318, and A104457 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
5-12-2020
989
Fibonacci Hyperbolic Functions
Let
(OEIS A104457), where 
is the golden ratio, and
(OEIS A002390).
Define the Fibonacci hyperbolic sine by
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
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).
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.
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