Higher order \(\alpha\)-hyperbolic functions (Q796710)

The functions in question are \[ \sum^{\infty}_{k=0}\alpha^ kz^{nk+r}/(nk+r)!, \] \(r=0,...,n-1\), which are linearly independent solutions of \(y^{(n)}=\alpha y\). References are given to earlier work on these functions in the cases \(\alpha =\pm 1\). For \(n=2\) these give the trigonometric and hyperbolic functions respectively. The author gives, in matrix fom, higher order analogues of the relations \(\sin^ 2x+\cos^ 2x-1\) and the formulas for \(\sin(x\pm y)\), etc.