## Generalized Hyperbolic Functions

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

 (1)

for , ..., , where is Complex, with the value at defined by
 (2)

This is called the -hyperbolic function of order of the th kind. The functions satisfy
 (3)

where
 (4)

 (5)

The functions give a generalized Euler Formula
 (6)

Since there are th roots of , this gives a system of linear equations. Solving for gives
 (7)

where
 (8)

is a Primitive Root of Unity.

The Laplace Transform is

 (9)

The generalized hyperbolic function is also related to the Mittag-Leffler Function by
 (10)

The values and give the exponential and circular/hyperbolic functions (depending on the sign of ), respectively.

 (11) (12) (13)

For , the first few functions are

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.