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Generalized Hyperbolic Functions

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In 1757, V. Riccati first recorded the generalizations of the hyperbolic functions defined by

F_(n,r)^alpha(x)=sum_(k=0)^infty(alpha^k)/((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 rth kind. The functions F_(n,r)^alpha satisfy

f^((k))(x)=alphaf(x),
(3)

where

f^((k))(0)={0 k!=r, 0<=k<=n-1,; 1 k=r.
(4)

In addition,

d/(dx)F_(n,r)^alpha(x)={F_(n,r-1)^alpha(x) for 0<r<=n-1; alphaF_(n,n-1)^alpha(x) for r=0.
(5)

The functions give a generalized Euler formula

e^(RadicalBox[alpha, n])=sum_(r=0)^(n-1)(RadicalBox[alpha, n])^rF_(n,r)^alpha(x).
(6)

Since there are n nth roots of alpha, this gives a system of n linear equations. Solving for F_(n,r)^alpha gives

F_(n,r)^alpha(x)=1/n(RadicalBox[alpha, n])^(-r)sum_(k=0)^(n-1)omega_n^(-rk)exp(omega_n^kRadicalBox[alpha, n]x),
(7)

where

omega_n=exp((2pii)/n)
(8)

is a primitive root of unity.

The Laplace transform is

int_0^inftye^(-st)F_(n,r)^alpha(at)dt=(s^(n-r-1)a^r)/(s^n-alphaa^n).
(9)

The generalized hyperbolic function is also related to the Mittag-Leffler function E_n(x) by

F_(n,0)^1(x)=E_n(x^n)
(10)
=sum_(k=0)^(infty)(x^(kn))/((kn)!).
(11)

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

F_(1,r)^alpha(x)=(e^(alphax)x^r)/((xalpha)^r)(Gamma(r)-Gamma(r,alphax))/(Gamma(r))
(12)
F_(2,r)^alpha(x)=(x^r)/(r!)_1F_2(1;1/2(1+r),1+1/2r;1/4alphax^2).
(13)

In particular

F_(1,0)^alpha(x)=e^(alphax)
(14)
F_(2,0)^alpha(x)=cosh(sqrt(alpha)x)
(15)
F_(2,1)^alpha(x)=(sinh(sqrt(alpha)x))/(sqrt(alpha)).
(16)

For alpha=1, the first few functions are

F_(1,0)^1(x)=e^x
(17)
F_(2,0)^1(x)=coshx
(18)
F_(2,1)^1(x)=sinhx
(19)
F_(3,0)^1(x)=1/3[e^x+2e^(-x/2)cos(1/2sqrt(3)x)]
(20)
F_(3,1)^1(x)=1/3[e^x+2e^(-x/2)cos(1/2sqrt(3)x+1/3pi)]
(21)
F_(3,2)^1(x)=1/3[e^x+2e^(-x/2)cos(1/2sqrt(3)x-1/3pi)]
(22)
F_(4,0)^1(x)=1/2(coshx+cosx)
(23)
F_(4,1)^1(x)=1/2(sinhx+sinx)
(24)
F_(4,2)^1(x)=1/2(coshx-cosx)
(25)
F_(4,3)^1(x)=1/2(sinhx-sinx).
(26)

See also

Hyperbolic Functions, Mittag-Leffler Function

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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.

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Generalized Hyperbolic Functions

Cite this as:

Weisstein, Eric W. "Generalized Hyperbolic Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html

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