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

Differentiability properties of p-trigonometric functions

Petr Girg, Lukas Kotrla

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.

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Petr Girg
Department of Mathematics, University of West Bohemia
Univerzitni 22, 30614
Plzen, Czech Republic
email: pgirg@kma.zcu.cz
Lukas Kotrla
Department of Mathematics, University of West Bohemia
Univerzitni 22, 30614
Plzen, Czech Republic
email: kotrla@students.zcu.cz

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