Generalized Hypergeometric Functions

Author

Eric W. Weisstein
August 30, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/GeneralizedHyperbolicFunctions.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

GeneralizedHyperbolicF[a_,{n_,r_},0]:=1
GeneralizedHyperbolicF[a_,{n_,r_},x_]:=Sum[a^k/(n k+r)!x^(n k+r),{k,0,Infinity}]

Definition

GeneralizedHyperbolicF[α,{n,r},x]

Special Cases

Mittag-Leffler

GeneralizedHyperbolicF[1,{n,0},x]

Circular

GeneralizedHyperbolicF[α,{1,r},x]

FunctionExpand[%]//FullSimplify

Hyperbolic

GeneralizedHyperbolicF[α,{2,r},x]//FullSimplify

FunctionExpand[%]//FullSimplify

Examples

FullSimplify[PowerExpand[Table[GeneralizedHyperbolicF[1,{n,r},x],{n,4},{r,0,n-1}]]]//TableForm

Solve[GeneralizedHyperbolicF[a,{n,r},x]==(a^(-r/n)GeneralizedHyperbolicF[a,{n1,r1},a^(1/n)x]), ]

Laplace Transform

Integrate[Exp[-s t]GeneralizedHyperbolicF[alpha,{n,r},a t],{t,0,Infinity}]

Table[Integrate[Exp[-s t]GeneralizedHyperbolicF[alpha,{n,r},a t]/.Thread[{alpha,n,r,a,s}->Table[Random[Complex],{5}]],{t,0,Infinity}],{2}]

Table[{
Integrate[Exp[-s t]GeneralizedHyperbolicF[alpha,{n,r},a t],{t,0,Infinity}],
(((s^(n-r-1))(a^r))/((s^n)-alpha (a^n)))
}/.Thread[{alpha,n,r,a,s}->Table[Random[Integer,{1,5}],{5}]],{5}]//Timing