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**Generalized trigonometric functions in complex domain.**
*(English)*
Zbl 1349.33018

Summary: We study extension of \(p\)-trigonometric functions \(\sin_p\) and \(\cos_p\) to complex domain. For \(p=4,6,8,\dots\), the function \(\sin_p\) satisfies the initial value problem which is equivalent to
\[
-(u')^{p-2}u''-u^{p-1} =0, \quad u(0)=0, \quad u'(0)=1, \tag \(*\)
\]
in \(\mathbb{R}\). In our recent paper [P. Girg and L. Kotrla, Electron. J. Differ. Equ. 2014, Conf. 21, 101–127 (2014; Zbl 1291.33021)], we showed that \(\sin_p(x)\) is a real analytic function for \(p=4,6,8,\dots\) on \((-\pi_p/2,\pi_p/2)\), where \(\pi_p/2 =\int_0^1(1-s^p)^{-1/p}\). This allows us to extend \(\sin_p\) to complex domain by its Maclaurin series convergent on the disc \(\{z\in\mathbb{C}\: |z|<\pi_p/2\}\). The question is whether this extensions \(\sin_p(z)\) 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 \(\sin_p\) to complex domain for \(p=3,5,7,\dots\). Moreover, we show that the structure of the complex valued initial value Problem \((*)\) does not allow entire solutions for any \(p\in\mathbb{N}\), \(p>2\). Finally, we provide some graphs of real and imaginary parts of \(\sin_p(z)\) and suggest some new conjectures.

### MSC:

33E30 | Other functions coming from differential, difference and integral equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |

34M99 | Ordinary differential equations in the complex domain |