p-Trigonometric and p-Hyperbolic Functions in Complex Domain

Abstract

We study extension of $p$-trigonometric functions ${\mathrm{s}\mathrm{i}\mathrm{n}}_p$ and ${\mathrm{c}\mathrm{o}\mathrm{s}}_p$ and of $p$-hyperbolic functions ${\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}_p$ and ${\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}_p$ to complex domain. Our aim is to answer the question under what conditions on $p$ these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, $\mathrm{s}\mathrm{i}\mathrm{n}(z)=-i·\sinh \left(i·z\right)$. In particular, we prove in the paper that for $p=\mathrm{6,10,14},\dots$ the $p$-trigonometric and $p$-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 for $p$-trigonometric and $p$-hyperbolic functions.