p-Trigonometric andp-Hyperbolic Functions in Complex Domain

Abstract

We study extension ofp-trigonometric functionssinpandcospand ofp-hyperbolic functionssinhpandcoshpto complex domain. Our aim is to answer the question under what conditions onpthese functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example,sin(z)=-i·sinh⁡i·z. In particular, we prove in the paper that forp=6,10,14,…thep-trigonometric andp-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series forp-trigonometric andp-hyperbolic functions.