Generalized Hyperbolic Functions

In 1757, V. Riccati first recorded the generalizations of the Hyperbolic Functions defined by

$$F_{n,r}^\alpha(x)\equiv \sum_{k=0}^\infty {\alpha^k\over (nk+r)!} x^{nk+r},$$ (1)

for $r=0$, ..., $n-1$, where $\alpha$ is Complex, with the value at $x=0$ defined by

$$F_{n,0}^\alpha(0)=1.$$ (2)

This is called the $\alpha$-hyperbolic function of order $n$ of the $r$th kind. The functions $F_{n,r}^\alpha$ satisfy

$$f^{(k)}(x)=\alpha f(x),$$ (3)

where

$$f^{(k)}(0)=\cases{0 & $k\not=r$, $0\leq k\leq n-1$,\cr 1 & $k=r$.\cr}$$ (4)

In addition,

$${d\over dx} F_{n,r}^\alpha(x)=\cases{ F_{n,r-1}^\alpha(x) &... ... $0<r\leq n-1$\cr \alpha F_{n,n-1}^\alpha(x) & for $r=0$.\cr}$$ (5)

The functions give a generalized Euler Formula

$$e^{{\root n\of\alpha}\,}=\sum_{r=0}^{n-1} ({\root n\of\alpha}\,)^r F_{n,r}^\alpha(x).$$ (6)

Since there are $n$ $n$th roots of $\alpha$, this gives a system of $n$ linear equations. Solving for $F_{n,r}^\alpha$ gives

$$F_{n,r}^\alpha(x)={1\over n}({\root n\of\alpha}\,)^{-r} \sum... ...mathop{\rm exp}\nolimits ({\omega_n}^k {\root n\of\alpha}\,x),$$ (7)

where

$$\omega_n=\mathop{\rm exp}\nolimits \left({2\pi i\over n}\right)$$ (8)

is a Primitive Root of Unity.

The Laplace Transform is

$$\int_0^\infty e^{-st}F_{n,r}^\alpha(at)\,dt={s^{n-r-1} a^r\over s^n+\alpha a_n}.$$ (9)

The generalized hyperbolic function is also related to the Mittag-Leffler Function $E_\gamma(x)$ by

$$F_{n,0}^1(x)=E_n(x^n).$$ (10)

The values $n=1$ and $n=2$ give the exponential and circular/hyperbolic functions (depending on the sign of $\alpha$), respectively.

$$F_{1,0}^\alpha(x) = e^{\alpha x}$$ (11)

$$F_{2,0}^\alpha(x) = \cosh(\sqrt{\alpha}\,x)$$ (12)

$$F_{2,1}^\alpha(x) = {\sinh(\sqrt{\alpha}\,x)\over\sqrt{\alpha}}.$$ (13)

For $\alpha=1$, the first few functions are

$$F_{1,0}^1(x) = e^x$$

$$F_{2,0}^1(x) = \cosh x$$

$$F_{2,1}^1(x) = \sinh x$$

$$F_{3,0}^1(x) = {\textstyle{1\over 3}}[e^x+2e^{-x/2}\cos({\textstyle{1\over 2}}\sqrt{3}\,x)]$$

$$F_{3,1}^1(x) = {\textstyle{1\over 3}}[e^x+2e^{-x/2}\cos({\textstyle{1\over 2}}\sqrt{3}\,x+{\textstyle{1\over 3}}\pi)]$$

$$F_{3,2}^1(x) = {\textstyle{1\over 3}}[e^x+2e^{-x/2}\cos({\textstyle{1\over 2}}\sqrt{3}\,x-{\textstyle{1\over 3}}\pi)]$$

$$F_{4,0}^1(x) = {\textstyle{1\over 2}}(\cosh x+\cos x)$$

$$F_{4,1}^1(x) = {\textstyle{1\over 2}}(\sinh x+\sin x)$$

$$F_{4,2}^1(x) = {\textstyle{1\over 2}}(\cosh x-\cos x)$$

$$F_{4,3}^1(x) = {\textstyle{1\over 2}}(\sinh x-\sin x).$$

See also Hyperbolic Functions, Mittag-Leffler Function

References

Kaufman, H. ``A Biographical Note on the Higher Sine Functions.'' Scripta Math. 28, 29-36, 1967.

Muldoon, M. E. and Ungar, A. A. ``Beyond Sin and Cos.'' Math. Mag. 69, 3-14, 1996.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, 1996.

Ungar, A. ``Generalized Hyperbolic Functions.'' Amer. Math. Monthly 89, 688-691, 1982.

Ungar, A. ``Higher Order Alpha-Hyperbolic Functions.'' Indian J. Pure. Appl. Math. 15, 301-304, 1984.