Variational and Topological Methods: Theory, Applications, 
Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 101-127.

Title: Differentiability properties of p-trigonometric functions

Authors: Petr Girg    (Univ. of West Bohemia, Plzen, Czech Republic)
         Lukas Kotrla (Univ. of West Bohemia, Plzen, Czech Republic)
 
Abstract: 
 p-trigonometric functions are generalizations of the trigonometric functions.
 They appear in context of nonlinear differential equations
 and also in analytical geometry of the p-circle in the plain.
 The most important p-trigonometric function is $\sin_p(x)$.
 For p>1, this function is defined  as the unique solution
 of the initial-value problem
 $$
 (|u'(x)|^{p-2} u'(x))'= (p-1) |u(x)|^{p-2} u(x), \quad u(0)=0, \; u'(0)=1\,,
 $$
 for any $x\in\mathbb{R}$.
 We prove that the $n$-th derivative of $\sin_p(x)$
 can be expressed in the form
 $$
 \sum_{k=0}^{2^{n-2}-1} a_{k,n} \sin_{p}^{q_{k,n}}(x) \cos_p^{1-q_{k,n}}(x)\,,
 $$
 on $(0, \pi_p/ 2)$, where $\pi_p=\int_0^1 (1-s^p)^{-1/p}\rm{d}s$,
 and $\cos_p(x)=\sin_p'(x)$.
 Using this formula, we proved the order of differentiability
 of the function $ \sin_p(x)$. 
 The most surprising (least expected)
 result is that $\sin_p(x)\in C^{\infty}(-{\pi_p/2}, {\pi_p/2})$
 if p is an even integer.
 This result was essentially used in the proof of theorem, which says that the
 Maclaurin series of $\sin_p(x)$
 converges on $(-\pi_p/2, \pi_p/2)$ if
 p is an even integer. This completes previous results that were
 known e.g. by Lindqvist and Peetre where this convergence was conjectured.


Published February 10, 2014.
Math Subject Classifications: 34L10, 33E30, 33F05.
Key Words: p-Laplacian; p-trigonometry; analytic functions; approximation.