On a generalization of the Airy, hyperbolic and circular functions

Abstract

The ordinary circular and hyperbolic functions of variable x are generalized using a parameter m, so that: (i) they correspond to zero value, m=0, of the parameter; (ii) for m=1 are obtained the Airy functions. The generalized circular and hyperbolic cosine and sine are integral functions specified by MacLaurin series valid in the finite complex x-plane of the variable for all complex values of the parameter m excluding negative integers. Differentiation formulas are obtained for complex variable x and parameter m, and some inequalities are also obtained for real x and m. The extension to the generalized circular and hyperbolic secant, cosecant, tangent and cotangent is made in the usual way.