Generalized Hyperbolic Functions

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

[画像: 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

[画像: f^((k))(0)={0 k!=r, 0<=k<=n-1,; 1 k=r. ]

(4)

In addition,

[画像: 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

[画像: 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

[画像: 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

[画像: omega_n=exp((2pii)/n) ]

(8)

is a primitive root of unity.

The Laplace transform is

[画像: 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)

= [画像: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) = [画像:(e^(alphax)x^r)/((xalpha)^r)(Gamma(r)-Gamma(r,alphax))/(Gamma(r))]

(12)

F_(2,r)^alpha(x) = [画像:(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) = [画像:(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)

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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. https://www2.math.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

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

Generalized Hyperbolic Functions

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