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.