Properties of generalized trigonometric functions (Q657435)

Let \(1< p,q<\infty\) and \(p'= {p\over p-1}\). We define \[ F_{p,q}(x)= \int^x_0 (1-t^q)^{-1/q}dt,\;x\in[0,1]. \] The generalized sine function \(\sin_{p,q}\) is defined by \(\sin_{p,q}= (F_{p,q})^{-1}\) on the interval \([0,\pi_{p,q}/2]\), where \[ \pi_{p,q}= 2 \int^1_0 (1- t^q)^{-1/p}dt. \] Clearly, the usual sine function is given by \(F_{22}= \sin^{-1}\). In this note, the authors establish various properties of the generalized trigonometric functions \(\sin_{p,q}\).