In 1757, V. Riccati first recorded the generalizations of the Hyperbolic Functions defined by
![\begin{displaymath}
F_{n,r}^\alpha(x)\equiv \sum_{k=0}^\infty {\alpha^k\over (nk+r)!} x^{nk+r},
\end{displaymath}](g_1143.gif) |
(1) |
for
, ...,
, where
is Complex, with the value at
defined by
![\begin{displaymath}
F_{n,0}^\alpha(0)=1.
\end{displaymath}](g_1146.gif) |
(2) |
This is called the
-hyperbolic function of order
of the
th kind. The functions
satisfy
![\begin{displaymath}
f^{(k)}(x)=\alpha f(x),
\end{displaymath}](g_1148.gif) |
(3) |
where
![\begin{displaymath}
f^{(k)}(0)=\cases{0 & $k\not=r$, $0\leq k\leq n-1$,\cr 1 & $k=r$.\cr}
\end{displaymath}](g_1149.gif) |
(4) |
In addition,
![\begin{displaymath}
{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}
\end{displaymath}](g_1150.gif) |
(5) |
The functions give a generalized Euler Formula
![\begin{displaymath}
e^{{\root n\of\alpha}\,}=\sum_{r=0}^{n-1} ({\root n\of\alpha}\,)^r F_{n,r}^\alpha(x).
\end{displaymath}](g_1151.gif) |
(6) |
Since there are
th roots of
, this gives a system of
linear equations. Solving for
gives
![\begin{displaymath}
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),
\end{displaymath}](g_1152.gif) |
(7) |
where
![\begin{displaymath}
\omega_n=\mathop{\rm exp}\nolimits \left({2\pi i\over n}\right)
\end{displaymath}](g_1153.gif) |
(8) |
is a Primitive Root of Unity.
The Laplace Transform is
![\begin{displaymath}
\int_0^\infty e^{-st}F_{n,r}^\alpha(at)\,dt={s^{n-r-1} a^r\over s^n+\alpha a_n}.
\end{displaymath}](g_1154.gif) |
(9) |
The generalized hyperbolic function is also related to the Mittag-Leffler Function
by
![\begin{displaymath}
F_{n,0}^1(x)=E_n(x^n).
\end{displaymath}](g_1156.gif) |
(10) |
The values
and
give the exponential and circular/hyperbolic functions (depending on the sign of
),
respectively.
For
, the first few functions are
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.
© 1996-9 Eric W. Weisstein
1999-05-25