Generalized trigonometric functions in complex domain

Abstract

We study extension of p-trigonometric functions sinp and cosp to complex domain. For p = 4, 6, 8,. . ., the function sinp satisfies the initial value problem which is equivalent to (*) −(u ′) p−2 u ′′ − u p−1 = 0, u(0) = 0, u ′ (0) = 1 in R. In our recent paper, Girg, Kotrla (2014), we showed that sinp(x) is a real analytic function for p = 4, 6, 8,. .. on (−πp/2, πp/2), where πp/2 = 1 0 (1 − s p) −1/p. This allows us to extend sinp to complex domain by its Maclaurin series convergent on the disc {z ∈ C : |z| < πp/2}. The question is whether this extensions sinp(z) satisfies (*) in the sense of differential equations in complex domain. This interesting question was posed by Došlý and we show that the answer is affirmative. We also discuss the difficulties concerning the extension of sinp to complex domain for p = 3, 5, 7,. .. Moreover, we show that the structure of the complex valued initial value problem (*) does not allow entire solutions for any p ∈ N, p > 2. Finally, we provide some graphs of real and imaginary parts of sinp(z) and suggest some new conjectures.