斐波那契双曲函数

斐波那契双曲函数（Fibonoacci hyperbolic functions）是一个与黄金分割有关的特殊函数^[1]。

斐波那契双曲函数

定义如下：^[2]

斐波那契双曲正弦函数

$s F h (x) = \frac{2 * s i n h (2 * x * α)}{\sqrt{5}}$

其中 $α$ 是黄金分割的对数：

$α = l n (ϕ) = l n \frac{1 + \sqrt{5}}{2} = 0.4812118246$

斐波那契双曲余弦函数

$c F h (x) = \frac{2 * s i n h (2 * x * α)}{\sqrt{5}}$

斐波那契双曲正切函数

$t F h (x) = \frac{f s h (x)}{f c h (x)}$

斐氏双曲函数图

关系式

$s F h (- x) = - s F h (x)$
$c F h (- x) = c F h (x - 1)$
$s F h^{2} (x) + c F h^{2} (x) = c F h (2 x)$
$c F h^{2} (x) - s F h^{2} (x) = 1 + s F h (x) c F h (x)$
$s F h (x) + s F h (y) = \sqrt{(} 5) s F h * (\frac{x + y}{2}) c F h (\frac{x - y - 1}{2})$
$c F h (x) + c F h (y) = \sqrt{5} c F h (\frac{x + y}{2}) c F h (\frac{x + y - 1}{2})$

级数展开

$f s h (x) \approx {(4 / 5 \sqrt{5} \ln (1 / 2 + 1 / 2 \sqrt{5}) x + \frac{8}{15} \sqrt{5} {(\ln (1 / 2 + 1 / 2 \sqrt{5}))}^{3} x^{3} + \frac{8}{75} \sqrt{5} {(\ln (1 / 2 + 1 / 2 \sqrt{5}))}^{5} x^{5} + \frac{16}{1575} \sqrt{5} {(\ln (1 / 2 + 1 / 2 \sqrt{5}))}^{7} x^{7} + O (x^{9}))}$

$f c h (x) \approx (1 / 5) * \sqrt{(} 5) * (5 + \sqrt{(} 5)) / (\sqrt{(} 5) + 1) + (2 / 5) * \sqrt{(} 5) * l n (1 / 2 + (1 / 2) * \sqrt{(} 5)) * x + (2 / 5) * \sqrt{(} 5) * (5 + \sqrt{(} 5)) * l n (1 / 2 + (1 / 2) * \sqrt{(} 5))^{2} * x^{2} / (\sqrt{(} 5) + 1) + (4 / 15) * \sqrt{(} 5) * l n (1 / 2 + (1 / 2) * \sqrt{(} 5))^{3} * x^{3} + (2 / 15) * \sqrt{(} 5) * (5 + \sqrt{(} 5)) * l n (1 / 2 + (1 / 2) * \sqrt{(} 5))^{4} * x^{4} / (\sqrt{(} 5) + 1) + O (x^{5})$

$f t h (x) \approx 4 * l n (1 / 2 + (1 / 2) * \sqrt{(} 5)) * (\sqrt{(} 5) + 1) * x / (5 + \sqrt{(} 5)) - 8 * l n (1 / 2 + (1 / 2) * \sqrt{(} 5))^{2} * (\sqrt{(} 5) + 1)^{2} * x^{2} / (5 + \sqrt{(} 5))^{2} + (2 * (- (8 / 3) * l n (1 / 2 + (1 / 2) * \sqrt{(} 5))^{3} + 8 * l n (1 / 2 + (1 / 2) * \sqrt{(} 5))^{3} * (\sqrt{(} 5) + 1)^{2} / (5 + \sqrt{(} 5))^{2})) * (\sqrt{(} 5) + 1) * x^{3} / (5 + \sqrt{(} 5)) + O (x^{4})$

渐近展开

$s F h (x) \approx {1 / 5 \frac{\sqrt{5} {({(\sqrt{5} + 1)}^{x})}^{2}}{{(2^{x})}^{2}} - 1 / 5 \frac{\sqrt{5} {(2^{x})}^{2}}{{({(\sqrt{5} + 1)}^{x})}^{2}}}$

$c F h (x) \approx {2 / 5 \frac{(1 / 4 \sqrt{5} + 1 / 4) \sqrt{5} {({(\sqrt{5} + 1)}^{x})}^{2}}{{(2^{x})}^{2}} + 2 / 5 \frac{\sqrt{5} {(2^{x})}^{2}}{(\sqrt{5} + 1) {({(\sqrt{5} + 1)}^{x})}^{2}}}$

Heun函数表示

$s F h (x) = 4 / 5 \sqrt{5} x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) 𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, 2 \sqrt{\frac{x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1}}) {(\sqrt{5} + 1)}^{- 1} {(e^{2 \frac{x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1}})}^{- 1}$
$c F h (x) = 2 / 5 i \sqrt{5} (2 x + 1) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, \sqrt{2} \sqrt{\frac{(- 2 x - 1) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1} + 1 / 2 i π}) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) {(\sqrt{5} + 1)}^{- 1} {(e^{1 / 2 \frac{- 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x \sqrt{5} + 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x - 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) \sqrt{5} + 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) + i π \sqrt{5} + i π}{\sqrt{5} + 1}})}^{- 1} + 1 / 5 \sqrt{5} π 𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, \sqrt{2} \sqrt{\frac{(- 2 x - 1) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1} + 1 / 2 i π}) {(e^{1 / 2 \frac{- 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x \sqrt{5} + 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x - 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) \sqrt{5} + 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) + i π \sqrt{5} + i π}{\sqrt{5} + 1}})}^{- 1}$
$t F h (x) = 4 x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) 𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, 2 \sqrt{\frac{x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1}}) e^{1 / 2 \frac{- 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x \sqrt{5} + 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x - 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) \sqrt{5} + 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) + i π \sqrt{5} + i π}{\sqrt{5} + 1}} {(\sqrt{5} + 1)}^{- 1} {(e^{2 \frac{x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1}})}^{- 1} {(\frac{2 i (2 x + 1) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1} + π)}^{- 1} {(𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, \sqrt{2} \sqrt{\frac{(- 2 x - 1) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1} + 1 / 2 i π}))}^{- 1}$

斐波那契双曲反函数

反双曲正弦

$a r c s F h (z) = 1 / 2 \frac{𝑎 𝑟 𝑐 𝑠 𝑖 𝑛 ℎ (1 / 2 z \sqrt{5})}{\ln (1 / 2 + 1 / 2 \sqrt{5})}$ 满足 $s F h (a r c s F h (y)) = y$

反双曲余弦

$a r c c F h (z) = - 1 / 2 \frac{\ln (1 / 2 + 1 / 2 \sqrt{5}) - 𝑎 𝑟 𝑐 𝑐 𝑜 𝑠 ℎ (1 / 2 z \sqrt{5})}{\ln (1 / 2 + 1 / 2 \sqrt{5})}$ 满足 $a r c c F h (c F h (z)) = z$

反双曲正切

$a r c t F h (z) = 1 / 4 \ln (- \frac{- z + z \sqrt{5} + 2}{z + z \sqrt{5} - 2}) {(\ln (1 / 2 + 1 / 2 \sqrt{5}))}^{- 1}$ 满足 $a r c t F h (t F h (z)) = z$

Heun函数表示

$a r c s F h (z) = 1 / 2 z \sqrt{5} (\sqrt{5} + 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1 / 2, 0, 0, 1 / 4, 5 \frac{z^{2}}{5 z^{2} + 4}) \frac{1}{\sqrt{5 z^{2} + 4}} {(\sqrt{5} - 1)}^{- 1} {(𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}))}^{- 1}$

$a r c c F h (z) = - 1 / 2 + (1 / 2 \sqrt{- {(- 2 + z \sqrt{5})}^{2}} z \sqrt{5} (\sqrt{5} + 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1 / 2, 0, 0, 1 / 4, 5 / 4 \frac{z^{2}}{5 / 4 z^{2} - 1}) {(- 2 + z \sqrt{5})}^{- 1} \frac{1}{\sqrt{- 5 z^{2} + 4}} {(\sqrt{5} - 1)}^{- 1} - 1 / 4 \frac{\sqrt{- {(- 2 + z \sqrt{5})}^{2}} π (\sqrt{5} + 1)}{(- 2 + z \sqrt{5}) (\sqrt{5} - 1)}) {(𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}))}^{- 1}$

$a r c t F h (x) = z = 4 x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) 𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, 2 \sqrt{\frac{x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1}}) e^{1 / 2 \frac{- 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x \sqrt{5} + 4 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) x - 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) \sqrt{5} + 2 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1}) + i π \sqrt{5} + i π}{\sqrt{5} + 1}} {(\sqrt{5} + 1)}^{- 1} {(e^{2 \frac{x (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1}})}^{- 1} {(\frac{2 i (2 x + 1) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1} + π)}^{- 1} {(𝐻 𝑒 𝑢 𝑛 𝐵 (2, 0, 0, 0, \sqrt{2} \sqrt{\frac{(- 1 - 2 x) (\sqrt{5} - 1) 𝐻 𝑒 𝑢 𝑛 𝐶 (0, 1, 0, 0, 1 / 2, \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{\sqrt{5} + 1} + 1 / 2 i π}))}^{- 1}$

级数展开

$a r c s F h (z) \approx (1 / 4 \frac{\sqrt{5}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z - \frac{5}{96} \frac{\sqrt{5}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{3} + \frac{15}{512} \frac{\sqrt{5}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{5} - \frac{625}{28672} \frac{\sqrt{5}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{7} + \frac{21875}{1179648} \frac{\sqrt{5}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{9} + O (z^{11})) >$

$a r c c F h (z) \approx (- 1 / 2 \frac{\ln (1 / 2 + 1 / 2 \sqrt{5}) + 1 / 2 i 𝑐 𝑠 𝑔 𝑛 (i (1 / 2 z \sqrt{5} - 1)) π}{\ln (1 / 2 + 1 / 2 \sqrt{5})} + 1 / 4 i 𝑐 𝑠 𝑔 𝑛 (i (1 / 2 z \sqrt{5} - 1)) \sqrt{5} {(\ln (\frac{1}{2} + 1 / 2 \sqrt{5}))}^{- 1} z + \frac{5}{96} i \sqrt{5} 𝑐 𝑠 𝑔 𝑛 (i (1 / 2 z \sqrt{5} - 1)) {(\ln (\frac{1}{2} + 1 / 2 \sqrt{5}))}^{- 1} z^{3} + \frac{15}{512} i \sqrt{5} 𝑐 𝑠 𝑔 𝑛 (i (1 / 2 z \sqrt{5} - 1)) {(\ln (\frac{1}{2} + 1 / 2 \sqrt{5}))}^{- 1} z^{5} + \frac{625}{28672} i \sqrt{5} 𝑐 𝑠 𝑔 𝑛 (i (1 / 2 z \sqrt{5} - 1)) {(\ln (\frac{1}{2} + 1 / 2 \sqrt{5}))}^{- 1} z^{7} + \frac{21875}{1179648} i \sqrt{5} 𝑐 𝑠 𝑔 𝑛 (i (1 / 2 z \sqrt{5} - 1)) {(\ln (\frac{1}{2} + 1 / 2 \sqrt{5}))}^{- 1} z^{9} + O (z^{11})) >$

$a r c t F h (z) \approx (1 / 4 \frac{\sqrt{5}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z + 1 / 4 \frac{1 / 2 \sqrt{5} (\sqrt{5} + 1) - 5 / 2}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{2} + 1 / 4 \frac{- 5 / 6 \sqrt{5} - 5 / 2 + 1 / 4 \sqrt{5} {(\sqrt{5} + 1)}^{2}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{3} + 1 / 4 \frac{- \frac{5}{16} {(\sqrt{5} + 1)}^{2} + 1 / 8 \sqrt{5} {(\sqrt{5} + 1)}^{3} - 5 / 8 \sqrt{5} - \frac{25}{8}}{\ln (1 / 2 + 1 / 2 \sqrt{5})} z^{4} + O (z^{5})) >$

渐近展开

$a r c s F h (z) \approx 1 / 2 \frac{1 / 2 \ln (5) + \ln (z)}{\ln (1 / 2 + 1 / 2 \sqrt{5})} + 1 / 10 \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{2}} - \frac{3}{100} \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{4}} + \frac{1}{75} \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{6}} - \frac{7}{1000} \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{8}} + O (z^{- 10})$
$a r c c F h (z) \approx - 1 / 2 \frac{\ln (1 / 2 + 1 / 2 \sqrt{5}) - 1 / 2 \ln (5) - \ln (z)}{\ln (1 / 2 + 1 / 2 \sqrt{5})} - 1 / 10 \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{2}} - \frac{3}{100} \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{4}} - \frac{1}{75} \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{6}} - \frac{7}{1000} \frac{1}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{8}} + O (z^{- 10})$
$a r c t F h (z) \approx 1 / 4 \frac{\ln (\frac{\sqrt{5} - 1}{\sqrt{5} + 1}) + i π}{\ln (1 / 2 + 1 / 2 \sqrt{5})} + 1 / 4 \frac{2 + 2 \frac{\sqrt{5} - 1}{\sqrt{5} + 1}}{(\sqrt{5} - 1) \ln (1 / 2 + 1 / 2 \sqrt{5}) z} + 1 / 4 \frac{8 \frac{\sqrt{5}}{{(\sqrt{5} + 1)}^{2} (\sqrt{5} - 1)} - 2 \frac{\sqrt{5} (2 + 2 \frac{\sqrt{5} - 1}{\sqrt{5} + 1})}{(\sqrt{5} + 1) {(\sqrt{5} - 1)}^{2}}}{\ln (1 / 2 + 1 / 2 \sqrt{5}) z^{2}} + O (z^{- 3})$