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)

In addition,

(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)
			(11)

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

			(12)
			(13)

In particular

			(14)
			(15)
			(16)

For , the first few functions are

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

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.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

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.