Properties of generalized trigonometric functions

Decay of (p,q)-Fourier coefficients

“…Moreover, it is now known (see [6]) that this basis property is also possessed by the functions sin p,…”

Two double-angle formulas of generalized trigonometric functions

“…which is said to have been developed by Abu al-Wafa' (940-998), a Persian mathematician and astronomer. In case (p, q) = (2, 4), it is easy to see that sin 2,4 x coincides with the lemniscate function. Since this classical function has the double-angle formula (see, e.g.…”

“…The following double-angle formulas for the two special cases will be established. The double-angle formulas for cos 2,3 x and cos 4/3,2 x are also obtained by differentiating both sides of those for sin 2,3 x and sin 4/3,2 x, respectively.…”

Action-angle variables for the purely nonlinear oscillator

“…For α = 1, the system reduces to the harmonic oscillator and hence, the solutions are then the standard (circular) trigonometric functions. The solutions we obtain are in terms of generalized trigonometric functions being defined [14][15][16][17][18] as parametric equations to the curve,…”